Existence and Regularity of Branched Minimal Submanifolds

EXISTENCE AND REGULARITY OF BRANCHED MINIMAL SUBMANIFOLDS

A DISSERTATION SUBMITTED TO THE DEPARTMENT OF MATHEMATICS AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

Brian Krummel August 2011

This work is licensed under a Creative Commons Attribution- Noncommercial 3.0 United States License. http://creativecommons.org/licenses/by-nc/3.0/us/

This dissertation is online at: http://purl.stanford.edu/rc085mz1473

ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Leon Simon, Primary Adviser

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Richard Schoen

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Brian White

Approved for the Stanford University Committee on Graduate Studies. Patricia J. Gumport, Vice Provost Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file in University Archives.

iii Abstract

We consider two-valued solutions to elliptic problems, which arise from the study branched minimal submanifolds. Simon and Wickramasekera constructed in [10] examples of two-valued solutions to the Dirichlet problem for the minimal surface ˘2 n−2 equation on the cylinder C = B1 (0) × R with Höldercontinuity estimates on the gradient assuming the boundary data satisfies a symmetry condition. However, their method was specific to the minimal surface equation. We generalize Simon and Wickramasekera’s result to an existence theorems for a more general class elliptic equations and for a class of elliptic systems with small data. In particular, we extend Simon and Wickramasekera’s result to the minimal surface system. Our approach uses techniques for elliptic differential equations such as the Leray-Schauder theory and contraction mapping principle, which have the advantage of applying in more general contexts than codimension 1 minimal surfaces. We also show that for two- valued solutions to elliptic equations with real analytic data, the branch set of their graphs are real analytic (n − 2)-dimensional submanifolds. This is a consequence of using the Schauder estimate for two-valued functions and a technique involving majorants due to Friedman [2] to inductively get estimates on the derivatives of the two-valued solutions.

iv Acknowledgments

I am grateful for my advisor Leon Simon, whose ideas and encouragement were in- valuable to my dissertation work. Leon’s insightful way of looking at mathematics has had a positive inﬂuence on me as a mathematician. I consider myself fortunate to have been his student.

I also thank Rick Schoen, Brian White, Rafe Mazzeo, and Simon Brendle for taking time to meet with me to discuss mathematics throughout my time at Stanford. Their ideas have often been interesting and helpful in my work. In particular, I thank Rick Schoen, Brian White, and Rafe Mazzeo for being on my dissertation committee and Simon Brendle for providing ﬁnancial support for my quarter spent at Cambridge.

I appreciate Neshan Wickramasekera showing his interest in my research and taking time to talk with me about it while I was visiting Cambridge.

Finally I am thankful to my friends and family for their support and interest in my academic endeavors. In particular, I am grateful to my parents for teaching me the value of education from a young age and for their encouragement during my graduate work.

v Contents

Abstract iv

Acknowledgments v

1 Introduction and Statement of Main Results 1 1.1 Notation ...... 1 1.2 Two-valued functions ...... 2 1.3 The function spaces Vk and Vk,µ ...... 10 1.4 Minimal submanifolds ...... 18 1.5 Existence and regularity theorems ...... 21

2 Elliptic theory for two-valued functions 28 2.1 Overview ...... 28 2.2 Maximum principles ...... 29 2.3 Schauder estimates ...... 35 2.4 Global estimates ...... 45 2.5 H¨oldercontinuity estimates ...... 50

3 Existence theorems 60 3.1 Overview ...... 60 3.2 Poisson equation ...... 60 3.3 Elliptic systems ...... 68 3.4 Elliptic equations ...... 73

vi 4 Regularity theorems 88 4.1 C1,µ regularity of C1 solutions ...... 88 4.2 Smoothness of two-valued solutions ...... 112

4.3 Bounding gγ ...... 118 i 4.4 Bounding the H¨oldercoeﬃcient of fγ ...... 123 4.5 Analyticity of two-valued solutions ...... 129

Bibliography 133

vii Chapter 1

Introduction and Statement of Main Results

1.1 Notation

We shall adopt the following notation and conventions throughout this thesis.

n ≥ 3 is a ﬁxed integer.

m m BR (X0) denotes the closed ball of radius R centered at X0 in R .

˘m m BR (X0) denotes the open ball of radius R centered at X0 in R .

n ˘ ˘n BR(X0) = BR(0), BR(X0) = BR(X0).

˘2 n−2 n C = B1 (0) × R denotes an open cylinder in R .

2 n−2 X = (x, y) denotes a point in C, where x ∈ B1 (0) and y ∈ R . We identify x with the point reiθ in C, where r ∈ [0, 1] and θ ∈ R.

1 CHAPTER 1. INTRODUCTION AND STATEMENT OF MAIN RESULTS 2

iθ iθ+iφ Rφ denotes the n × n matrix such that Rφ(re , y) = (re , y). We write

  cos φ − sin φ 0 0 ··· 0   sin φ cos φ 0 0 ··· 0    0 0 1 0 ··· 0 i   Rφ = (R )i,j=1,...,n =   j  0 0 0 1 ··· 0    ......   ......    0 0 0 0 ··· 1

j where Ri denotes the entry in the i-th row and j-th column of Rφ.

Rm×n denotes the space of m × n real matrices.

Hk denotes the k-th dimensional Hausdorﬀ measure.

Ln denotes the Lebesgue measure in Rn.

1.2 Two-valued functions

Let Ω be a domain in Rn. Given an integer q ≥ 2, we say u is a q-valued function m on Ω taking values in R if at each point X ∈ Ω, u(X) = {u1(X), . . . , uq(X)} is an m unordered q-tuple, where uj(X) ∈ R [1]. We can deﬁne a metric G on the space of unordered q-tuples by

q !1/2 X 2 G(u, v) = sup inf |uj − vσ(j)| X∈Ω σ j=1 for unordered q-tuples u = {u1, . . . uq} and v = {v1, . . . vq}, where σ is a permutation of {1, 2, . . . , q}. We say a q-valued function u is continuous at X ∈ Ω if

lim G(u(X), u(Y )) = 0. Y →X CHAPTER 1. INTRODUCTION AND STATEMENT OF MAIN RESULTS 3

We say a q-valued function u from Ω ⊆ Rn to unordered q-tuples in Rm is diﬀerentiable at X ∈ Ω if for some m × n matrices A1,...Aq,

G(u(X + h), {u (X) + A h}) lim j j = 0, h→0 |h| in which case we say Du(X) = {A1,...,Aq} is the derivative of u at X. We deﬁne k-th order derivatives for integers k ≥ 2 inductively by Dku(X) = D(Dk−1u)(X). For integers k ≥ 0, Ck(Ω, Rm) denotes the space of q-valued functions u on Ω taking values in Rm such that Dju exists and is continuous for 0 ≤ j ≤ k. We say u is H¨oldercontinuous with exponent µ ∈ (0, 1] on Ω if

G(u(X), u(Y )) [u]µ,Ω = sup µ < ∞. (1.1) X,Y ∈Ω,X6=Y |X − Y |

For k ≥ 0 and µ ∈ (0, 1], Ck,µ(Ω, Rm) denotes the space of two-valued functions u ∈ Ck(Ω, Rm) such that Dku is H¨oldercontinuous with exponent µ. For Ω ⊆ Rn open, k m k m given k ≥ 0 we let Cc (Ω, R ) denote the space of two-valued functions u ∈ C (Ω, R ) such that u = {0, 0} on Ω \ Ω0 for some Ω0 ⊂⊂ Ω and given k ≥ 0 and µ ∈ (0, 1] we k,µ m k,µ m let Cc (Ω, R ) denote the space of two-valued functions u ∈ C (Ω, R ) such that u = {0, 0} on Ω \ Ω0 for some Ω0 ⊂⊂ Ω. For Ω ⊆ Rn open, k ≥ 0, and µ ∈ (0, 1], we k,µ m k m k,µ 0 m let Cloc (Ω, R ) denote the space of u ∈ C (Ω, R ) such that u ∈ C (Ω , R ) for all Ω0 ⊂⊂ Ω. For Ω ⊆ Rn, we let

kukC0(Ω) = sup G(u(X), 0) X∈Ω for each two-valued function u ∈ C0(Ω, Rm),

X α kukCk(Ω) = kD ukC0(Ω) |α|≤k for each two-valued function u ∈ Ck(Ω, Rm) for k ≥ 1, and

X α X α kukCk,µ(Ω) = kD ukC0(Ω) + [D u]µ,Ω |α|≤k |α|=k CHAPTER 1. INTRODUCTION AND STATEMENT OF MAIN RESULTS 4

for each two-valued function u ∈ Ck,µ(Ω, Rm) for k ≥ 0 and µ ∈ (0, 1). We deﬁne a metric on C0(Ω, Rm) by

ku − vkC0(Ω) = sup G(u(X), v(X)) X∈Ω for two-valued functions u, v ∈ C0(Ω, Rm) and we deﬁne a metric on Ck(Ω, Rm) for k ≥ 1 by X α α ku − vkCk(Ω) = kD u − D vkC0(Ω) |α|≤k for two-valued functions u, v ∈ Ck(Ω, Rm).

Note that there is no well-deﬁned addition or multiplication operators on the space of q-valued functions in C1(Ω). Consider the two-valued functions u and v on C given by u(z) = {± Re z3/2} and v(z) = {± Re(z + 1)3/2} and suppose we had deﬁned u + v as a two-valued function in C1(Ω). On the slit domain {reiθ : θ ∈ (−π, π)}, there are two possibilities for u + v as a C1 two-valued function, either

(u + v)(z) = {Re z3/2 + Re(z + 1)3/2, − Re z3/2 − Re(z + 2)3/2} (1.2) or (u + v)(z) = {Re z3/2 − Re(z + 1)3/2, − Re z3/2 + Re(z + 2)3/2}, (1.3) where we let w3/2 = r3/2e3iθ/2 for w = reiθ where r > 0 and θ ∈ (−π, π). By continuity, both expressions in (1.2) and (1.3) extend to unique continuous two-valued functions on all of C. However, neither of the expressions for u + v in (1.2) and (1.3) are diﬀerentiable along the interval (−1, 0).

We are primarily interested in the case where q = 2. We say a two-valued function u is symmetric if at each point X ∈ Ω, u(X) = {−u1(X), +u1(X)} for some value u1(X). We can write u as

u (X) + u (X) u (X) − u (X) u = u + u , where u (X) = 1 2 and u (X) = ± 1 2 a s a 2 s 2 CHAPTER 1. INTRODUCTION AND STATEMENT OF MAIN RESULTS 5

where u(X) = {u1(X), u2(X)}. Observe that we can regard ua as single-valued and that us is two-valued symmetric. We will call ua the average part of u and us the two-valued symmetric part of u. Given a two-valued function u ∈ C0(Ω, Rm), we deﬁne

Zu = {X ∈ Ω: u1(X) = u2(X)} where we write u(X) = {u1(X), u2(X)} as unordered pairs. Given a two-valued 1 m function u ∈ C (Ω, R ), we deﬁne Bu to be the set of all points X0 ∈ Ω such that there is no δ > 0 such that u(X) = {u1(X), u2(X)} on Bδ(X0) ⊂ Ω for some single- 1 valued functions u1, u2 ∈ C (Ω). We also deﬁne

Ku = {X ∈ Ω: u1(X) = u2(X), Du1(X) = Du2(X)}

where we write u(X) = {u1(X), u2(X)} and Du(X) = {Du1(X), Du2(X)} as unordered pairs.

1 m Lemma 1. For any two-valued function u ∈ C (Ω, R ), Bu ⊆ Ku.

Proof. We will show that if X0 ∈ Ω \Ku then there is a ball Bδ(X0) ⊂ Ω on which 1 u(X) = {u1(X), u2(X)} for C single-valued functions u1 and u2. The case where u1(X0) 6= u2(X0) follows by continuity. Suppose u1(X0) = u2(X0) and Du1(X0) 6=

Du2(X0). Let A 6= 0 be a matrix equal to Du1(X0) − Du2(X0), noting there are two choices for A diﬀering a sign and we choose A to be one of them. For Y ∈ Rn, let

K = {X ∈ n : |(X − Y ) · η| ≤ √1 |X − Y ||η| if Aη = 0}. Y R 2

Choose ε ∈ (0, |A|/4) such that

if ξ ∈ K0 then |Aξ| ≥ 5ε|ξ|. (1.4)

By diﬀerentiability, for every Y ∈ Ω there is a ρ(Y ) > 0 such that Bρ(Y )(Y ) ⊂ Ω and

G(u(X), {uj(Y ) + Duj(Y )(X − Y )}) ≤ ε|X − Y | for X ∈ Bρ(Y )(Y ). (1.5) CHAPTER 1. INTRODUCTION AND STATEMENT OF MAIN RESULTS 6

By continuity, there is a δ ∈ (0, 2ρ(X0)) such that

G(Du(X), Du(X0)) < ε for X ∈ B2δ(X0). (1.6)

By (1.6), on B2δ(X0) we can write Du(X) = {A1(X),A2(X)} where A1,A2 are con- 1 tinuous singled-valued, m×n matrix-valued functions. Let the vj be C single-valued functions on B2δ(X0) such that vj(X0) = u1(X0) and Dvj(X) = Aj(X) on B2δ(X0), j = 1, 2. We claim u(X) = {v1(X), v2(X)} on B2δ(X0).

Suppose Y ∈ B2δ(X0) such that u1(Y ) = u2(Y ) = v1(Y ) = v2(Y ). If X ∈

Bρ(Y )(Y ) ∩ KY , by (1.4), (1.5) and (1.6),

|u1(X) − u2(X)| ≥ |A(X − Y )| − G(Du(X0), Du(Y ))|X − Y |

−G(u(X), u(Y ) + Du(Y )(X − Y )) ≥ 3ε|X − X0|.

1 It follows that on Bρ(Y )(Y ) ∩ KY , we have u(X) = {u1(X), u2(X)} for unique C single-valued functions u1 and u2 and in fact

u(X) = {v1(X), v2(X)} on Bρ(Y )(Y ) ∩ KY . (1.7)

Suppose there is a point X1 ∈ Bδ(X0) such that u(X1) 6= {v1(X1), v2(X1)} and n recall that X1 6∈ KX0 by (1.7). Fix ζ ∈ R orthogonal to the null space of A. Let

X2 = X1 + sζ for the smallest s > 0 such that u(X2) = {v1(X2), v2(X2)}. We claim that for some σ > 0, u(X) = {u1(X), u2(X)} on Bσ(X2) ∩ KX2 for unique single- 1 valued u1, u2 ∈ C (Bσ(X2) ∩ KX2 ). In the case that u1(X2) 6= u2(X2) this follows by continuity, and in the case that u1(X2) = u2(X2) this is immediate from (1.7). However, this contradicts the fact that for small t > 0,

u(X2 − tζ) 6= {v1(X2 − tζ), v2(X2 − tζ)}.

Therefore u(X) = {v1(X), v2(X)} on Bδ(X0).

Let p ≥ 1. We say a measurable two-valued function u on Ω is in Lp(Ω, Rm) for CHAPTER 1. INTRODUCTION AND STATEMENT OF MAIN RESULTS 7

p ≥ 1 if Z 1/p p kukLp(Ω) = G(u(X), 0) dX < ∞. Ω We define Sobolev spaces of two-valued functions as follows. Suppose u ∈ C0(Ω, Rm) is two-valued symmetric function, i.e. u(X) = {u1(X), −u1(X)} for some u1(X) ≥ 0. 1,p m p m×n We say u ∈ W (Ω, R ) if Du1 ∈ Lloc(Ω \Zu, R ) and we define the weak derivative Du by Du = {±Du1} locally on Ω \Zu, where Du1 denotes the weak derivative 0 m of u1, and Du = {0, 0} on Zu. For general two-valued functions u ∈ C (Ω, R ), recall that we can write u = ua + us for ua(X) = (u1(X) + u2(X))/2 and us(X) = 1,p m {±(u1(X) − u2(X))/2}, where u(X) = ua(X) + us(X). We say u ∈ W (Ω, R ) if 1,p m 1,p m Dua ∈ W (Ω, R ) and Dus ∈ W (Ω, R ) and we define the weak derivative of u k,p m by Du = Dua + Dus. For k ≥ 1 and p ≥ 1, we let W (Ω, R ) denote the space of two-valued functions u ∈ Ck−1(Ω, Rm) such that Dk−1u ∈ W 1,p(Ω, Rm). We let 1,p m k−1 m Wloc (Ω, R ) denote the space of two-valued functions u ∈ C (Ω, R ) such that Dk−1u ∈ W 1,p(Ω0, Rm) for all Ω0 ⊂⊂ Ω.

We want to consider diﬀerential systems of the form

i Qκu = DiAκ(X, u, Du) + Bκ(X, u, Du) = 0 on Ω \Bu, κ = 1, . . . , m, (1.8)

1 m i where u ∈ C (Ω, R ) is a two-valued function and Aκ and Bκ are continuous single- n m m×n valued functions on R ×R ×R . By the deﬁnition of Bu, for X0 ∈ Ω\Bu there is a ball Bδ(X0) ⊂ Ω \Bu on which u(X) = {u1(X), u2(X)} for single-valued functions 1 u1, u2 ∈ C (Bδ(X0)). By (1.8) we mean that Qu1 = Qu2 = 0 weakly on Bδ(X0) for all choices of balls Bδ(X0) ⊂ Ω \Bu. The deﬁnition of (1.8) is motivated by the following.

Lemma 2. Let v ∈ C1(Ω, R) be a single-valued function and Ai,B ∈ C1(Ω × R × Rn) be single-valued functions such that

i 2 n DPj A (X,Z,P )ξiξj ≥ λ(X,Z,P )|ξ| for all ξ ∈ R (1.9) CHAPTER 1. INTRODUCTION AND STATEMENT OF MAIN RESULTS 8

0 n n for some positive function λ ∈ C (Ω × R × R ) and L (Sv) = 0. Let

Sv = {X ∈ Ω: v(X) = 0, Dv(X) = 0} denote the singular part of the nodal set of v. Then Z i A (X, v, Dv)Diζ − B(X, v, Dv)ζ = 0 (1.10) Ω\Sv

1 for all ζ ∈ Cc (Ω \Sv) if and only if Z i A (X, v, Dv)Diζ − B(X, v, Dv)ζ = 0 (1.11) Ω

1 for all ζ ∈ Cc (Ω).

Remark 1. Using a similar proof to the proof of Lemma 2 below, we can obtain a similar result for elliptic systems. Given single-valued functions v ∈ C1(Ω, R) and Ai,B ∈ C1(Ω × R × Rn), Z i Aκ(X, v, Dv)Diζ − Bκ(X, v, Dv)ζ = 0 for κ = 1, . . . , m Ω\Sv

1 for all ζ ∈ Cc (Ω \Sv) if and only if Z i Aκ(X, v, Dv)Diζ − Bκ(X, v, Dv)ζ = 0 for κ = 1, . . . , m Ω

1 n i ij for all ζ ∈ C (Ω) provided L (Sv) = 0 and |D λ A (X, u, Du) − δ δ| < ε for some c Pj κ ε = ε(n) > 0 suﬃciently small.

2 2 0 Proof of Lemma 2. First we will show (1.10) implies D v ∈ L (Ω \Sv) for any open 0 2 2 0 0 set Ω ⊂⊂ Ω. By elliptic estimates [3, Theorem 9.11], D v ∈ L (Ω ) for Ω ⊂⊂ Ω \Sv. CHAPTER 1. INTRODUCTION AND STATEMENT OF MAIN RESULTS 9

Replacing ζ with Dkζ in (1.10) and integrating by parts, we obtain Z i i i DPj A Djkv + DZ A Dkv + DXk A Diζ BR(X0) Z + DPj BDjkv + DZ BDkv + DXk B ζ = 0, BR(X0)

i where the derivatives of A and B are evaluated at (X, v, Dv). For δ > 0 let βδ ∈ ∞ C ([0, ∞)) be a function such that βδ(t) = 1 for t > 2δ, βδ(t) = 0 for |t| < δ, and 0 ∞ 0 ≤ βδ(t) ≤ 3/δ for all t ≥ 0. Let BR/2(X0) ⊂ Ω and η ∈ C (Ω) be the cutoﬀ function such that 0 ≤ η ≤ 1, η = 1 on BR/2(X0), η = 0 on Ω \ BR(X0), and |Dη| ≤ 3/R. 2 2 Letting ζ = βδ(|Dv| )Dkv η and summing over k = 1, . . . , n, we obtain Z i i 2 i 2 DPj A DkvDjkv + DZ A |Dv| + DXk A Dkv βδηDiη BR(X0) Z i i i 2 + DPj A DikvDjkv + DZ A DkvDikv + DXk A Dikv βδη BR(X0) Z 1 i 2 2 i 2 2 i 2 0 2 + 2 DPj A Di|Dv| Dj|Dv| + DZ A |Dv| Di|Dv| + DXk A DkvDi|Dv| βδη BR(X0) Z 2 2 + DPj BDkvDjkv + DZ B|Dv| + DXj BDkv βδη = 0 BR(X0)

2 0 where βδ and its derivative are evaluated at |Dv| . Using (1.9), βδ ≥ 0, and Cauchy- Schwartz, we obtain Z Z 2 2 2 2 2 βδ|D v| η ≤ C βδ|Dv| |Dη| BR/2(X0) BR(X0) Z 2 0 2 2 + C |Dv| + 1 βδ + βδ|Dv| η BR(X0)

0 2 for some constant C > 0 depending on n, DA, DB, R. Observe that βδ|Dv| ≤ 6. Using the deﬁnition of η and letting δ ↓ 0, we obtain Z |D2v|2 < ∞. BR/2(X0)\Sv CHAPTER 1. INTRODUCTION AND STATEMENT OF MAIN RESULTS 10

Now for δ > 0 let γδ be a smooth function such that γδ(t) = t for |t| > δ, γδ(t) = 0 0 ∞ for |t| < δ/2, and |γδ(t)| ≤ 3. For any ζ ∈ C (Ω), by integration by parts Z i A (X, v, γδ(Dv))Diζ − B(X, v, γδ(Dv))ζ Ω Z i 0 i = − DPj A (X, v, γδ(Dv)) γδ(Djv)Dijv + DZ A (X, v, γδ(Dv))Div ζ Ω Z i − DXi A (X, v, γδ(Dv)) + B(X, v, γδ(Dv)) ζ, Ω

2 2 0 0 where γδ(Dv) = {γδ(D1v), . . . , γδ(Dnv)}. Since D v ∈ L (Ω \Sv) for all Ω ⊂⊂ Ω, we can let δ ↓ 0 to obtain Z Z i i A (X, v, Dv)Diζ − B(X, v, Dv)ζ = − DPj A (X, v, Dv)Dijv ζ Ω Ω\Sv Z i i − DZ A (X, v, Dv)Div + DXi A (X, v, Dv) + B(X, v, Dv) ζ. (1.12) Ω

By (1.10),

i i i DiA (X, v, Dv) + B(X, v, Dv) = DPj A (X, v, Dv)Dijv + DZ A (X, v, Dv)Div i + DXi A (X, v, Dv) + B(X, v, Dv) = 0 a.e. on Ω \Sv. (1.13)

n By (1.12), (1.13), and the fact that L (Sv) = 0, we obtain (1.11).

1.3 The function spaces Vk and Vk,µ

n iθ 2 2iθ 0 m Let Ω ⊂ R and let Ω0 = {(re , y):(r e , y) ∈ Ω}. We will deﬁne V (Ω, R ) 0 m to be the space of single-valued functions u0 ∈ C (Ω0, R ). Observe that for every 0 m u0 ∈ V (Ω, R ), we can associate a two-valued function

iθ 1/2 iθ/2 1/2 iθ/2 u(re , y) = {u0(r e , y), u0(−r e , y)} (1.14) on Ω. CHAPTER 1. INTRODUCTION AND STATEMENT OF MAIN RESULTS 11

0 m Let u be a two-valued function associated with u0 ∈ V (Ω, R ), where Ω = C ˘2 n−2 or Ω = ∂C for C = B1 (0) × R . Given an odd integer k ≥ 3, we say u is k-fold symmetric if iθ+iπ/k iθ u0(re , y) = u0(re , y)

n−2 for all r ∈ [0, 1], θ ∈ R, and y ∈ R . We say u is periodic with respect to yj with period ρj > 0 if u0 is periodic with respect to each yj with period ρj, i.e.

iθ iθ u0(re , y1 + k1ρ1, . . . , yn−2 + kn−2ρn−2) = u0(re , y1, . . . , yn−2)

n−2 for all r ∈ [0, 1], θ ∈ R, y ∈ R , and kj ∈ Z.

0 m Since each element u0 in V (Ω, R ) is associated with a two-valued function u, we can deﬁne the following spaces and norms. For k ≥ 0, let Vk(Ω, Rm) be the space of 0 m k n−2 m u0 ∈ V (Ω, R ) such that u0 ∈ C (Ω0 \{0} × R , R ) and the two-valued function k m k m u associated with u0 is in C (Ω, R ). We deﬁne the norm on V (Ω, R ) by

ku0kVk(Ω) = kukCk(Ω)

k m where u0 ∈ V (Ω, R ) with corresponding two-valued function u. For k ≥ 1 and k,µ m k m µ ∈ (0, 1], let V (Ω, R ) be the space of u0 ∈ V (Ω, R ) such that the two-valued k,µ m k m function u associated with u0 is in C (Ω, R ). For u0 ∈ V (Ω, R ), deﬁne

iθ1 iθ2 |u0(r1e , y1) − u0(r2e , y2)| hµ,Ω(u0) = sup (1.15) 2 2iθ1 2 2iθ2 µ |(r1e , y1) − (r2e , y2)|

iθ1 iθ2 where the supremum is taken over all distinct points (r1e , y1) and (r2e , y2) in Ω0

iθ1 with |θ1 − θ2| < π/4. The condition |θ1 − θ2| < π/4 is imposed so that (r1e , y1)

iθ2+iπ is not close to (r2e , y2), in which case the ratio in (1.15) may become inﬁnite. k m Observe that for u0, v0 ∈ V (Ω, R ) with hµ,Ω(u0) < ∞ and hµ,Ω(v0) < ∞ and for CHAPTER 1. INTRODUCTION AND STATEMENT OF MAIN RESULTS 12

any single-valued function F ∈ C1(Rm, R),

hµ,Ω(u0 + v0) ≤ hµ,Ω(u0) + hµ,Ω(v0), hµ,Ω(u0v0) ≤ hµ,Ω(u0) hµ,Ω(v0),

hµ,Ω(F (u0)) ≤ sup |DF (u0)| hµ,Ω(u0). Ω

We can deﬁne

ku0kVk,µ(Ω) = ku0kVk(Ω) + hµ,Ω(u0).

k m k m for u0 ∈ V (Ω, R ). Note that for u0 ∈ V (Ω, R ) and the two-valued function u associated with u0, the relationship between the Höldercoefficient [u]µ,Ω defined in

(1.1) and hµ,Ω(u0) is given in Lemma 3 below. As a consequence, hµ,Ω(u0) < ∞ if k,µ m and only if [u]µ,Ω < ∞, so ku0kVk,µ(Ω) is a norm on V (Ω, R ).

n k m Lemma 3. Let Ω ⊆ R be a convex domain. For any u0 ∈ V (Ω, R ), if u is the two-valued function associated with u0, then

1 √ √ [u]µ,Ω ≤ hµ,Ω(u0) ≤ 4m [u]µ,Ω. 8

iθ1 Proof. First we will show hµ,Ω(u0) ≤ 2[u]µ,Ω. Let X1 = (r1e , y1) and X2 =

iθ2 (r2e , y2) be distinct points in Ω and assume |θ1 − θ2| ≤ π. If |θ1 − θ2| < π/2, then

If π/2 ≤ |θ1 − θ2| < π, take X3 = (X1 + X2)/2 and observe

G(u(X1), u(X2)) ≤ G(u(X1), u(X3)) + G(u(X3), u(X2)) √ µ µ ≤ 2 hµ,Ω(u0)(|X1 − X3| + |X3 − X2| ) √ µ ≤ 2 2 hµ,Ω(u0) |X1 − X2| . CHAPTER 1. INTRODUCTION AND STATEMENT OF MAIN RESULTS 13

The case where |θ1 − θ2| = π follows by continuity.

Next we will show [u]µ,Ω ≤ 2hµ,Ω(u0) if m = 1. The case m > 1 follows. Let

iθ1 iθ2 (r1e , y1) and (r2e , y2) be distinct points in Ω0 with |θ1 − θ2| < π/4. Let X1 =

2 2iθ1 2 2iθ2 (r1e , y1) and X2 = (r2e , y2). Suppose

iθ1 iθ2 2 iθ1 iθ2 2 |u0(r1e , y1) − u0(−r2e , y2)| + |u0(−r1e , y1) − u0(r2e , y2)|

iθ1 iθ2 2 iθ1 iθ2 2 < |u0(r1e , y1) − u0(r2e , y2)| + |u0(−r1e , y1) − u0(−r2e , y2)| .

Then

iθ1 iθ1 iθ2 iθ2 u0(r1e , y1) 6= u0(−r1e , y1) and u0(r2e , y2) 6= u0(−r2e , y2) and without loss of generality

iθ1 iθ1 iθ2 iθ2 u0(r1e , y1) < u0(−r1e , y1) and u0(r2e , y2) > u0(−r2e , y2).

Let Γ ⊂ Ω be the line segment between X1 and X2 and Γ0 ⊂ Ω0 be the arc

iθ 2 2iθ Γ0 = {(re , y):(r e , y) ∈ Γ}.

iθ3 iθ3 iθ3 By continuity, for some point X3 = (r3e , y3) ∈ Γ0, u0(r3e , y3) = u0(−r3e , y3). Thus

= G(u(X1), u(X3)) + G(u(X3), u(X2)) µ µ ≤ [u]µ,Ω (|X1 − X3| + |X3 − X2| ) µ ≤ 2[u]µ,Ω|X1 − X2| . CHAPTER 1. INTRODUCTION AND STATEMENT OF MAIN RESULTS 14

k m k m Note that for k ≥ 0, we will let Vc (Ω, R ) denote the space of u0 ∈ V (Ω, R ) such k,µ m that u0 has compact support in Ω0. Similarly for k ≥ 0 and µ ∈ (0, 1], Vc (Ω, R ) k,µ m denotes the space of u0 ∈ V (Ω, R ) such that u0 has compact support in Ω0. For n k m k m Ω ⊆ R open and k ≥ 0, Vloc(Ω, R ) denote the Fr´echet space V (Ω, R ) with semi- 0 n k,µ m norms k kVk(Ω0) for Ω ⊂⊂ Ω. For Ω ⊆ R open, k ≥ 0, and µ ∈ (0, 1], Vloc (Ω, R ) k m k,µ 0 m the Fr´echet space of functions u0 ∈ V (Ω, R ) such that u0 ∈ V (Ω , R ) for all 0 k,µ m 0 Ω ⊂⊂ Ω and equip Vloc (Ω, R ) with the seminorms k kVk,µ(Ω0) for Ω ⊂⊂ Ω.

n 0 m n−2 Given an open set Ω ⊆ R , u0 ∈ V (Ω, R ), h ∈ R and η ∈ R , we can define iθ 0 m the translation u0(re , y + ηh) in V ({X : dist(X, ∂Ω) > h}, R ) and associate with iθ iθ u0(re , y +ηh) the two-valued function denoted by u(re , y +ηh). We can also define the difference quotient iθ iθ u0(re , y + hη) − u0(re , y) h in V0({X : dist(X, ∂Ω) > h}, Rm) and associate with this the difference quotient the two-valued function denoted by δh,ηu.

n 1/2 iθ/2 1/2 iθ/2 Let Ω ⊂ R be open and let Ω0 = {(r e , y):(r e , y) ∈ Ω}. We want to consider linear and nonlinear diﬀerential systems of two-valued functions u asso- k m ciated with u0 ∈ V (Ω, R ) for some appropriate k ≥ 1. We will do this using the 1/2 change of variable x1 + ix2 = (ξ1 + iξ2) , under which u(x1, x2, y) transforms to u0(ξ1, ξ2, y). Let M(ξ, y) denote the Jacobian matrix of the map from (x1, x2, y) to CHAPTER 1. INTRODUCTION AND STATEMENT OF MAIN RESULTS 15

(ξ1, ξ2, y) coordinates. We can express M(ξ, y) as

i M(ξ, y) = (Mj (ξ, y))i,j=1,...,n  1 −1 −1 −1  2 Re(ξ1 + iξ2) 2 Im(ξ1 + iξ2) 0 0 ··· 0  1 −1 1 −1   Im(ξ1 + iξ2) Re(ξ1 + iξ2) 0 0 ··· 0  2 2     0 0 1 0 ··· 0 =   ,  0 0 0 1 ··· 0    ......   ......    0 0 0 0 ··· 1

i where Mj (ξ, y) denotes the entry in the i-th row and j-th column of M(ξ, y). Us- 1/2 ing the change of variables x1 + ix2 = (ξ1 + iξ2) , the derivative operator D : Ck(Ω, Rm) → Ck−1(Ω, Rm×n) on two-valued function, where k ≥ 1, induces a linear k m k−1 m×n 0 operator mapping u0 ∈ V (Ω, R ) to Du0 M ∈ V (Ω, R ). Given u0 ∈ V (Ω) and the two-valued function u associated with u0, we can interpret the integral Z u(x, y) dx dy Ω as Z 4 u0(ξ, y) 4|ξ| dξ dy, Ω0 1/2 ij i using the change of variable x1 + ix2 = (ξ1 + iξ2) . Now suppose u, a , b , cκ and 2 ij i 0 f are the two-valued functions associated with u0 ∈ V (Ω) and a0 , b0, c0, f0 ∈ V (Ω) 1/2 respectively. Using the change of variable x1 + ix2 = (ξ1 + iξ2) , we shall interpret

ij i n−2 a Dxixj u + b Dxi u + cu = (≥, ≤) f on Ω \{0} × R to mean that

ij k l ij k l i k a0 Mi MjDξkξl u0 + a0 Mi Dξk Mj Dξl u0 + b0Mi Dξk u0 + c0u0 n−2 = (≥, ≤) f0 on Ω0 \{0} × R . CHAPTER 1. INTRODUCTION AND STATEMENT OF MAIN RESULTS 16

0 1,2 Suppose u ∈ C (Ω) ∩ W (Ω) is the two-valued function associated with u0 ∈ V0(Ω), and aij, bi, cj, d, f i, and g are the two-valued functions associated with ij i j i 0 a0 , b0, c0, d0, f0, g0 ∈ V (Ω) respectively. Then, using the change of variable x1 +ix2 = 1/2 (ξ1 + iξ2) , we shall interpret

ij i j i n−2 Dxi (a Dxj u + b u) + c Dxj u + du = (≥, ≤) Dxi f + g on Ω \{0} × R to mean that

2 k l ij 2 k 2 l j 2 Dξk (4|ξ| Mi Mja0 Dξl u0 + 4|ξ| Mi b0u0) + 4|ξ| Mjc0Dlu0 + 4|ξ| d0u0 2 k i 2 n−2 = (≥, ≤) Dξk (4|ξ| Mi f0) + 4|ξ| g0 weakly on Ω0 \{0} × R . (1.16)

1 n−2 Let ζ0 ∈ Vc (Ω \{0} × R ) and ζ be the two-valued function associated with ζ0. If we multiply (1.16) by ζ0 and integrated by parts, we obtain Z 2 k l ij k i l j 4|ξ| Mi Mja0 Dξl u0 + Mi b0u0 Dξk ζ0 − Mjc0Dlu0 + d0u0 ζ0 Ω0 Z 2 k i = (≥, ≤) 4|ξ| Mi f0Dξk ζ0 − g0ζ0 Ω0 which we can regard as Z ij i j a Dxj u + b u Dxi ζ − c Dju + du ζ Ω Z i = (≥, ≤) f Dxi ζ − gζ (1.17) Ω

1/2 via the change of variable x1 + ix2 = (ξ1 + iξ2) . Note that (1.17) holds if instead 0 1,2 ζ0 ∈ Vc (Ω) and the two-valued function ζ associated with ζ0 is in W (Ω). To see this, 1 for δ > 0, let χδ ∈ C (R) be the single-valued function such that 0 ≤ χδ ≤ 1, χδ(r) = 1 if r ≥ δ, χδ(r) = 0 if r ≤ δ/2, and |Dχδ| ≤ 3/δ. We can regard χδ as a function on n (x, y) ∈ R that depends only on r = |x|. Replace ζ with ζχδ in (1.17), interpreting 1/2 iθ/2 1/2 iθ/2 1 ζχδ as the two-valued function associated with ζ0(r e , y)χδ(r e , y) in V (Ω). CHAPTER 1. INTRODUCTION AND STATEMENT OF MAIN RESULTS 17

This yields Z ij i j a Dxj u + b u Dxi ζχδ − c Dju + du ζχδ Ω Z Z i i ij i = (≥, ≤) f Dxi ζ − gζ χδ + f − a Dxj u − b u ζDxi χδ. (1.18) Ω Ω

1,2 Using the deﬁnition of χδ and fact that u, ζ ∈ W (Ω), we can let δ ↓ 0 in (1.18) 1 m to obtain (1.17). Suppose u0 ∈ V (Ω, R ), u is the two-valued function associated i 0 m m×n with u0, and Aκ,Bκ ∈ C (Ω × R × R ). Using the change of variable x1 + ix2 = 1/2 (ξ1 + iξ2) , then we shall interpret the nonlinear system

i n−2 Qu = Dxi Aκ(X, u, Du) + Bκ(X, u, Du) = 0 on Ω \{0} × R , for κ = 1, . . . , m (1.19) to mean that

2 k i 2 Dξk (4|ξ| Mi Aκ(ξ, u0, Du0M)) + 4|ξ| Bκ(ξ, u0, Du0M) = 0 (1.20)

n−2 weakly on Ω0 \{0} × R for κ = 1, . . . , m. Note that if u satisfies (1.19) in the sense that (1.20) is true, then u satisfies (1.19) as a two-valued function in the sense discussed in Section 1.2. Conversely, if u ∈ C1(Ω) is a two-valued function satisfying i (1.19) in the sense discussed in Section 1.2 and Aκ and Bκ are real analytic, then u 1 is the two-valued function associated with some u0 ∈ V (Ω) and u0 satisfies (1.20). n−2 To see this, recall that for every X0 ∈ Ω \{0} × R , there is an ball BR(X0) ⊂ n−2 Ω \{0} × R on which u = {u1, u2} and Qu1 = Qu2 = 0 for single-valued functions 1 u1, u2 ∈ C (BR(X0)). By elliptic regularity, u1 and u2 are real analytic and thus u = {u1, u2} on BR(X0) for unique real analytic single-valued functions u1 and u2. It follows from unique continuation that u is the two-valued function associated with 1 some u0 ∈ V (Ω) satisfying (1.20). In fact, u is the two-valued function associated 1 iθ with precisely two functions in V (Ω), the other being u0(−re , y). CHAPTER 1. INTRODUCTION AND STATEMENT OF MAIN RESULTS 18

1.4 Minimal submanifolds

We will be interested in elliptic problems of the form

i Di(Aκ(X, u, Du)) + Bκ(X, u, Du) = 0 weakly in C\Bu, κ = 1, . . . , m, u = ϕ on ∂C, (1.21)

i 0 m m×n where m ≥ 1 is an integer, Aκ,Bκ ∈ C (C × R × R ) are single-valued functions, and u ∈ C0(C, Rm)∩C1(C, Rm) and ϕ ∈ C0(∂C, Rm) are two-valued functions. As will be discussed in detail in the next section, this is primarily motivated by the special case where (1.21) is the minimal surface system.

Minimal submanifolds arise in studying the ﬁrst variation of area. Let Σ be a N 1 N N smooth n-dimensional submanifold in R . Let X ∈ Cc (R , R ) such that ∂Σ ∩ N N spt X = ∅ and let F : R × (−1, 1) → R be a smooth function such that {Ft =

F (., t)}t∈(−1,1) is a family of diﬀeomorphisms with Ft(p) = p for all p outside a compact N ∂F set K, F0(p) = p for all p ∈ R , and X(p) = ∂t (p, 0). By the area formula, Z Area(Ft(Σ)) = − JΣFt(p)dp, (1.22) Σ

N where JΣFt denotes the Jacobian of Ft|Σ :Σ → R . The ﬁrst variation of the area of Σ is

d Area(Ft(Σ)) dt t=0 which can be shown using (1.22) to be given by

Z d Area(Ft(Σ)) = − divΣ X(p) dp dt t=0 Σ where n X divΣ X(p) = Dτj X(p) · τj j=1 for any orthonormal basis {τ1, . . . , τn} for the tangent space of Σ at p. We say Σ is a CHAPTER 1. INTRODUCTION AND STATEMENT OF MAIN RESULTS 19

minimal submanifold if the ﬁrst variation of the area of Σ is zero, i.e. Z divΣ X(p) dp = 0 Σ

1 N for all X ∈ Cc (R ) such that ∂Σ ∩ spt X = ∅. We also have the second variation of area for a minimal submanifold Σ, which is deﬁned to be

2 d 2 Area(Ft(Σ)) dt t=0 and which can be shown using (1.22) to be given by

2 Z n n ! d X ⊥ 2 X 2 Area(Ft(Σ)) = |(Dτi X(p)) | − |X(p) · Ap(τi, τj)| dp dt2 t=0 Σ i=1 i,j=1

1 N for X ∈ Cc (R ) such that ∂Σ∩spt X = ∅ and X|Σ is normal to Σ, where {τ1, . . . , τn} ⊥ for the tangent space of Σ at p,(Dτi X(p)) denotes the orthogonal projection of

Dτi X(p) onto the normal space of Σ at p, and A denotes the second fundamental form of Σ. We say a minimal submanifold Σ is stable if the second variation of area is nonnegative; that is,

Z n n ! X ⊥ 2 X 2 |(Dτi X) | − |X · A(τi, τj)| dp ≥ 0 (1.23) Σ i=1 i,j=1

1 N for X ∈ Cc (R ) such that ∂Σ ∩ spt X = ∅ and X|Σ is normal to Σ. Observe that Σ is stationary and stable if Σ is area minimizing, that is Area(Σ) ≤ Area(Σ0) for any 0 0 0 smooth n-dimensional submanifold Σ such that ∂Σ = ∂Σ and Σ ∩BR(0) = Σ∩BR(0) for some R > 0.

1 N−n Suppose Σ = Σu is the graph of a single-valued function u ∈ C (Ω, R ), where n Ω is a domain in R . Then we can write the area functional of Σu as

Z p Area(Σu) = det G(Du) dX, Ω CHAPTER 1. INTRODUCTION AND STATEMENT OF MAIN RESULTS 20

where G(P ) = (Gij(P ))i,j=1,...,n is the n × n matrix given by

κ κ Gij(P ) = δij + Pi Pj

κ for every (N − n) × n matrix P = (Pi )i=1,...,n, κ=1,...,N−n. The graph Σu is minimal if the ﬁrst variation of area is zero; that is,

Z d p ij κ κ Area(Σu+tζ ) = det G(Du) G (Du)Dju Diζ = 0 dt t=0 Σu for all smooth functions ζ :Ω → RN−n with compact support, where (Gij(P )) is the inverse matrix of G(P ). Thus we obtain the minimal surface system

p ij κ Di det G(Du) G (Du)Dju = 0 on Ω, κ = 1,...,N − n.

In the special case that N = n + 1 we get the minimal surface equation

! Diu Di = 0 on Ω. p1 + |Du|2

It is well know that in the case that N = n + 1, graph u is stable [13, Proposition 6.2.2]. In the case that N > n+1, the graph of u is not generally stable [4]. However, if N > n + 1 and |Du| ≤ /2 for > 0 suﬃciently small,

2 Z d p κ λ Area(Σu+tζ ) = D κ λ det G(P ) Diζ Djζ dX 2 Pi Pj dt Ω P =Du+tDζ Z ≥ (1 − C) |Dζ|2 dX ≥ 0 Ω for t ∈ [0, 1] and for any smooth function ζ :Ω → RN−n with compact support such that |Dζ| ≤ 2, where C = C(n, m) > 0. It follows that Area(Σu) ≤ Area(Σv) for v ∈ C1(Ω, RN−n) with |Dv| ≤ and v = u on ∂Ω by taking ζ = v − u. Thus given 1 a normal vector ﬁeld X ∈ Cc (Σu) such that ∂Σu ∩ spt X = ∅, for small t the set 1 N−n {P + tX(P ): P ∈ Σu} is the graph of some function v ∈ C (Ω, R ) with |Dv| ≤ CHAPTER 1. INTRODUCTION AND STATEMENT OF MAIN RESULTS 21

and v = u on ∂Ω, so

Area(Σu) ≤ Area({P + tX : P ∈ Σu})

for all t suﬃciently small. Consequently Σu is a stable minimal submanifold.

Suppose that instead Σ is the graph of a two-valued function u ∈ C1(Ω, RN−n).

Then on any ball BR(x0) ⊂ Ω \Bu, u = {u1, u2} for smooth single-valued function N−n u1, u2 : BR(x0) → R and

p ij κ Di det G(Dul) G (Dul)Djul = 0 on BR(x0), κ = 1,...,N − n, l = 1, 2; in other words,

p ij κ Di det G(Du) G (Du)Dju = 0 on Ω \Bu, κ = 1,...,N − n.

In particular, if N = n + 1,

! Diu Di = 0 on Ω \Bu. p1 + |Du|2

1.5 Existence and regularity theorems

Two-valued functions naturally arise in the study of branched minimal submanifolds. For example, the holomorphic variety

2 2 3 Σ = {(z, w) ∈ C : w = z }, which is an area-minimizing submanifold in R4 with a branch point at the origin, is the graph of a C1,1/2 two-valued function, {±z3/2}. On a neighborhood of the origin, Σ cannot be expressed as the graph of a smooth single-valued function but can be expressed as the graph of the two-valued function w = ±z3/2. To deal with the branching behavior of minimal submanifolds, Almgren [1] developed a theory of CHAPTER 1. INTRODUCTION AND STATEMENT OF MAIN RESULTS 22

multi-valued functions in his proof that the singular set of an n-dimensional area- minimizing rectiﬁable current has Hausdorﬀ dimension at most n − 2.

Wickramasekera [12] proved the following result regarding the extent to which codimension one minimal submanifolds are graphs of two-valued functions.

Theorem 1. For each δ ∈ (0, 1), there exists a number ε ∈ (0, 1) depending only on n and δ, such that the following is true. If M is an orientable immersed stable n+1 n−2 Hn(M) minimal hypersurface of B (0), with H (sing M) < ∞, 0 ∈ M, n ≤ 3 − δ 2 ωn2 R n+1 2 and |x | ≤ ε, then M1 ∩B1(0)× = graph u where M1 is the connected M∩B1(0)×R R component of M ∩ B1/2(0) × R containing the origin and u is either a single-valued 1,µ or two-valued C function on B1/2(0), where µ ∈ (0, 1) depends only on n and δ.

It is natural to ask whether there are examples of minimal submanifolds which are the graphs of two-valued functions. Simon and Wickramasekera constructed in [10] the following examples of two-valued solutions u to the Dirichlet problem for the minimal surface equation.

2 n−2 Theorem 2. Let C = B1 (0)×R and k ≥ 3 be an odd integer. Let ϕ be a two-valued function on ∂C satisfying the k-fold symmetry condition that ϕ(eiθ+i2π/k, y) = ϕ(eiθ, y) for all θ ∈ [0, 2π). For some µ = µ(n, k, sup∂C |ϕ|) ∈ (0, 1), there are two-valued 1,µ 0 n−2 solutions u ∈ Cloc (C) ∩ C (C) with Bu ⊆ {0} × R to the Dirichlet problem for the minimal surface equation

! Diu Di = 0 in C\Bu, p1 + |Du|2 u = ϕ on ∂C.

We will consider the question of whether there are similar solutions to more general elliptic problems. Simon and Wickramasekera method was specific to the minimal surface equation and did not readily generalize to other elliptic differential equations or to elliptic systems. We will construct similar examples using techniques for elliptic differential equations, solving the Dirichlet problem for various elliptic systems for CHAPTER 1. INTRODUCTION AND STATEMENT OF MAIN RESULTS 23

two-valued functions of the form (1.14) provided the boundary data is k-fold symmetric. This approach has the advantage that it applies in a more general context than codimension 1 minimal surfaces. In particular, this work extends the results in [10] by giving examples of minimal submanifolds with codimension greater than one. The proof of these results will be given in Chapter 3.

i 2 m×n Theorem 3. Let m be an integer, µ ∈ (0, 1/2), and ν > 0. Let Fκ ∈ C (R ) and 1 m m×n m×n Gκ ∈ C (R × R ) be single-valued functions, where R is the space of m × n i i matrices, such that Fκ(0) = 0, DFκ(0) = 0, Gκ(0) = 0, DGκ(0) = 0, and

i j j Fκ(P R2π/k) = Ri Fκ (P ),Gκ(Z,P R2π/k) = Gκ(Z,P ) (1.24)

m×n i 1,µ m for all P ∈ R , where R2π/k = (Rj) is as in Section 1.1. Let ϕ ∈ C (∂C, R ) be a two-valued function of the form (1.14) with k-fold symmetry. For some ε = 1,µ m ε(m, n, µ, ν) > 0, if kϕkC1,µ(∂C) ≤ ε, then exists a two-valued solution u ∈ C (C, R ) n−2 with Bu ⊆ {0} × R to

κ i n−2 ∆u = Di(Fκ(Du)) + Gκ(u, Du) weakly in C\{0} × R , κ = 1, . . . , m, u = ϕ on ∂C.

In particular, as in (1.14), u is of the form

iθ 1/2 iθ/2 1/2 iθ/2 u(re , y) = {u0(r e , y), u0(−r e , y)} (1.25)

0 m 1 n−2 m for some u0 ∈ C (C, R ) ∩ C (C\{0} × R , R ), u has k-fold symmetry, and kukC1,µ(C) ≤ ε.

Note that this theorem applies to the minimal surface system. For suﬃciently small = (n, m) > 0, the solutions to the minimal surface system constructed using Theorem 3 are stable in the sense that

Z n n ! X ⊥ 2 X 2 |(Dτi X) | − |X · A(τi, τj)| ≥ 0 Σu i=1 i,j=1 CHAPTER 1. INTRODUCTION AND STATEMENT OF MAIN RESULTS 24

0 n+m 1,2 n+m for all normal vector ﬁelds X ∈ Cc (Σu, R ) ∩ W (Σu, R ), where Σu = graph u and we regard Σu as an immersed submanifold. Theorem 3 also applies to all systems R 2 which arise as Euler-Lagrange equations for functionals of the form C(|Du| +f(Du)) where f ∈ C3(Rm×n, R) is a single-valued function such that Df(0) = 0, D2f(0) = 0, m×n and f(P R2π/k) = f(P ) for all P ∈ R .

Theorem 4. Let Ai ∈ C2(Rn), B ∈ C1(R × Rn) be single-valued functions such that

i j j A (P R2π/k) = Ri A (P ),B(Z,P R2π/k) = B(Z,P ), (1.26)

i where R2π/k = (Rj) is as in Section 1.1. Suppose

2 i 2 n 0 < λ(P )|ξ| ≤ DPj A (P )ξiξj ≤ Λ(P )|ξ| for all ξ ∈ R for some continuous positive functions λ and Λ, the structure conditions

B(Z,P ) sgn Z/λ(P ) ≤ β1|P | + β2, (1.27) 2 |Λ(P )| + |B(Z,P )| ≤ β3λ(P )|P | if |P | ≥ 1, (1.28)

for some constants β1, β2, β3 > 0, and B(Z,P ) is non-increasing in Z for ﬁxed P ∈ Rn. Let ϕ ∈ C2(∂C) be a two-valued function of the form (1.14) with k-fold symmetry 1,µ and kϕ0kV2(∂C) < ∞. Then there exists a two-valued function u such that u ∈ C (C) n−2 for all µ ∈ (0, 1/2), Bu ⊆ {0} × R , and

i n−2 Di(A (Du)) + B(u, Du) = 0 in C\{0} × R , u = ϕ on ∂C. (1.29)

In particular, u is of the form

iθ 1/2 iθ/2 1/2 iθ/2 u(re , y) = {u0(r e , y), u0(−r e , y)} (1.30)

0 1 n−2 for some u0 ∈ C (C) ∩ C (C\{0} × R ) and u has k-fold symmetry.

Corollary 1. Let Ai ∈ C2(Rn), B ∈ C1(R × Rn) be single-valued functions satisfying CHAPTER 1. INTRODUCTION AND STATEMENT OF MAIN RESULTS 25

2 1/2 ij ij 2 (1.26). Let v(P ) = (1 + |P | ) and g (P ) = δ − PiPj/(1 + |P | ) and suppose the structure conditions (1.27),

i PiA (P ) ≥ v(P ) − γ1, (1.31)

|A(P )| ≤ γ2, |B(Z,P )| ≤ γ3/v(P ) (1.32) i ij n v(P )DPj A (P )ξjξj ≥ g (P )ξiξj for ξ ∈ R , (1.33) i ij 1/2 ij 1/2 n v(P )|DPj A (P )ξiηj| ≤ γ4(g (P )ξiξj) (g (P )ηiηj) for ξ, η ∈ R , (1.34)

n hold for all Z ∈ R and P ∈ R for some constants γ1 ∈ [0, 1) and γ2, γ3, γ4 > 0. Also suppose B(Z,P ) is non-increasing in Z for ﬁxed P ∈ Rn. Let ϕ ∈ C0(∂C) be a two-valued function of the form (1.14) with k-fold symmetry such that sup∂C |ϕ| < ∞.

Then for some µ ∈ (0, 1/2) depending on n, β1, β2, γ1, γ2, γ3, γ4, and sup∂C |ϕ|, 1,µ n−2 there exists a two-valued solution u ∈ C (C) with Bu ⊆ {0} × R to (1.29). In particular, u is of the form (1.30) and u has k-fold symmetry. We also consider the regularity of the branch set of graphs of two-valued solutions to elliptic problems, with a particular interest in the minimal surface equation. Alm- gren showed in [1] that the branch set of an area-minimizing n-dimensional integral current has Hausdorﬀ dimension at most n − 2. Simon and Wickramasekera more recently showed in [11] that the branch set of a n-dimensional stationary varifold represented as the graph of a two-valued C1,µ function has Hausdorﬀ dimension at most n − 2. However, there have been no results establishing the regularity of the branch set of a minimal submanifold, i.e. whether the branch set is locally a smooth or real analytic (n − 2)-dimensional submanifold. We will show that for C1,µ solutions u to n−2 elliptic equations with Bu ⊆ {0}×R , the branch set of graph of u is a real analytic submanifold of dimension (n − 2). In particular, this establishes that the branch sets of the minimal hypersurfaces constructed by Simon and Wickramasekera in [10] are real analytic. The proof of this theorem will be discussed in Chapter 4. 1 ˘ n−2 Theorem 5. Let u ∈ Cloc(B1(0)) be a two-valued function with Bu ⊆ {0} × R and

Bu 6= ∅. Suppose u is a solution to the non-linear elliptic diﬀerential equation

i ˘ Qu = Di(A (X, u, Du)) + B(X, u, Du) = 0 in B1(0) \B CHAPTER 1. INTRODUCTION AND STATEMENT OF MAIN RESULTS 26

i ˘ where A (X,Z,P ) and B(X,Z,P ) are single-valued, real analytic functions on B1(0)× R × Rn such that

i 2 ˘ n n (DjA )(X,Z,P )ξiξj ≥ λ(X,Z,P )|ξ| for (X,Z,P ) ∈ B1(0) × R × R , ξ ∈ R

0 ˘ n for some positive single-valued function λ ∈ C (B1(0) × R × R ). Then u(x, y) is real ˘ analytic in y; that is, for BR(x0, y0) ⊂ B1(0),

γ p −p |Dy u(x, y)| ≤ p!C R for (x, y) ∈ BR/2(x0, y0), p = |γ| ≥ 3, for some C > 0 depending on n, µ, Ai, and sup u. In particular, the branch BR(x0,y0) set {(0, y, u(0, y)) : y ∈ Bu} of the graph of u is a real analytic submanifold.

Remark 2. Under the hypotheses of this theorem, the assumption that Bu ⊆ {0} × n−2 R and Bu 6= ∅ automatically implies that u is of the form

iθ 1/2 iθ/2 1/2 iθ/2 u(re , y) = {u0(r e , y), u0(−r e , y)}

1 1/2 iθ/2 iθ for some u0 ∈ V (B1(0)) that is locally real analytic in {(r e , y):(re , y) ∈ n−2 n−2 B1(0) \{0} × R } and that Bu = {0} × B1 (0). This is a consequence of unique n−2 continuation using the fact that for every X ∈ C \ {0} × R , on any ball BR(X0) ⊂ n−2 B1(0) \{0} × R , u = {u1, u2} and Qu1 = Qu2 = 0 for unique real analytic single- 1 valued functions u1, u2 ∈ C (BR(X0)) (see the discussion at the end of Section 1.3). We also obtain a similar result for elliptic systems.

1,µ ˘ m Theorem 6. Let u ∈ Cloc (B1(0), R ) be a two-valued function of the form (1.14). Suppose u is a solution to the non-linear elliptic diﬀerential equation

i ˘ Di(Aκ(X, u, Du)) + Bκ(X, u, Du) = 0 in B1(0) \B, κ = 1, . . . , l

i ˘ where A (X,Z,P ) and B(X,Z,P ) are single-valued, real analytic functions on B1(0)× Rm × Rm×n such that

i ij ˘ m l×n |(D λ A )(X,Z,P ) − δ δκλ| < ε for (X,Z,P ) ∈ B1(0) × × , Pj κ R R CHAPTER 1. INTRODUCTION AND STATEMENT OF MAIN RESULTS 27

where ε > 0 is as in Theorem 11 (see Section 2.3). Then u(x, y) is real analytic in y.

We will show that Theorem 6 holds under the weaker assumption that u ∈ 1 ˘ m C (B1(0), R ) in the case that u is a solution to

i Di(Aκ(Du)) = 0 on B1(0) \Bu for κ = 1, . . . , m, (1.35) where Ai(P ) are single-valued, real analytic functions on Rm×n. Our approach is to show that any C1 solution to (1.35) is in fact in C1,µ for some µ ∈ (0, 1/2). The proof of this result will be given in Section 1 of Chapter 4.

(Note that showing two-valued C1 solutions to the minimal surface equation are in C1,µ for some µ ∈ (0, 1/2) is of importance since by [11] it would imply the branch set of a n-dimensional stationary varifold represented as the graph of a two-valued function in C1 would have Hausdorﬀ dimension at most n − 2.)

i Theorem 7. Let m ≥ 1 be an integer and µ ∈ (0, 1/2). Let Aκ be a real analytic m×n i ij function on with D λ A (0) = δ δκλ. There are constants γ > 0 and C > R Pj κ i 1 m 0 depending on n, m, µ, and Aκ such that if u ∈ C (B1(0), R ) is a two-valued solution to (1.35), such that Bu = graph g ∩ B1(0) for some single-valued function g ∈ C1(Rn−2, R2) with |Dg| ≤ γ, and

sup |Du| ≤ γ, B1(0)

1,µ m then u ∈ C (B1/4(0), R ) with

1,µ 2 kukC (B1/4(0)) ≤ CkDukL (B1(0)).

1,1/2 Remark 3. In the case of the minimal surface system, u ∈ C (B1/4(0)) by [11]. Chapter 2

Elliptic theory for two-valued functions

2.1 Overview

The proof of the main results Theorems 3, 4, and 5 use standard theorems for elliptic differential equation such as the maximum principle and the Schauder estimates. This chapter is concerned with generalizing those theorems from single-valued functions to two-valued functions of the form (1.14). The linear differential equations in this chapter will be interpreted as discussed in Section 1.3. In Section 2.2 we will prove some maximum principles for homogeneous elliptic equations. In Section 2.3 we will prove the Schauder estimates. Section 2.4 is concerned with obtaining estimates on supC u for solutions u to inhomogeneous elliptic equations. This necessary both to obtain global Schauder estimates on u that do not depend on supC u and for relaxing regularity assumptions on the boundary data. In Section 2.5 we will obtain a Hölder continuity estimate using the De Giorgi, Nash, and Moser theory.

28 CHAPTER 2. ELLIPTIC THEORY FOR TWO-VALUED FUNCTIONS 29

2.2 Maximum principles

In order to consider maximum principles for two-valued functions, we need to deﬁne the maximum value of a two-valued function. Given a two-valued function u ∈ C0(Ω, R), we deﬁne

sup u = sup max{u1(X), u2(X)}. Ω X∈Ω where u(X) = {u1(X), u2(X)} at X ∈ Ω. Note that we similarly deﬁne

inf u = inf min{u1(X), u2(X)}. Ω X∈Ω

We say u attains its maximum value at X0 ∈ Ω if

sup u = max{u1(X0), u2(X0)}. Ω

We say u has a local maximum at X0 if u attains its maximum value on an open 1 neighborhood of X0 in Ω. Note that in the case that u ∈ C (Ω) and X0 ∈ Ω \Bu, u attains a local maximum at X0 means that on some ball BR(X0) ⊂ Ω\Bu we can write 1 u = {u1, u2} for two single-valued functions u1, u2 ∈ C (BR(X0)) and one of u1 and u2 attains its maximum value at X0. The ﬁrst maximum principle we will consider is a generalization of the strong maximum principle [3, Theorem 3.5] to two-valued functions.

Theorem 8. Let u, aij, bi, c, and ϕ be the two-valued functions associated with 0 1 2 n−2 ij i 0 0 u0 ∈ V (C) ∩ V (C) ∩ V (C\{0} × R ), a0 , b0, c0 ∈ V (C), and ϕ0 ∈ V (∂C) respectively. Suppose

ij i n−2 Lu = a Diju + b Diu + cu ≥ 0 in C\{0} × R , u = ϕ on ∂C, which we interpret as discussed in Section 1.3, and

ij 2 n a0 (X)ξiξj ≥ λ|ξ| for X ∈ C, ξ ∈ R CHAPTER 2. ELLIPTIC THEORY FOR TWO-VALUED FUNCTIONS 30

for some constant λ > 0 and c0 ≤ 0. Then u cannot attain a nonnegative maximum value in C unless u is constant.

n−2 Proof. Suppose u attains a nonnegative maximum at X0 ∈ C \ {0} × R . Take a n−2 ball BR(X0) ⊂ C \ {0} × R and on BR(X0), write u = {u1, u2} for single-valued 2 u1, u2 ∈ C (BR(X0)) such that Lu1 ≥ 0 and Lu2 ≥ 0. Without loss of generality u1 attains the nonnegative maximum value of u at X0. By the strong maximum principle for single-valued functions [3, Theorem 3.5], u1 is constant on BR(X0). It follows that the nonnegative maximum value of u is attained at X0 if and only if u is constant. If u is not constant, then u can only attain a nonnegative maximum on {0}×Rn−2 ∪∂C.

1 ij i 0 Suppose u0 ∈ V (C) and a0 , b0 ∈ V (C), and u attains a nonnegative maximum n−2 maximum at X0 ∈ {0} × R . Consider a ball BR(X1) ⊂ C that is tangent to n−2 {0} × R at X0. On BR(X1), u = {u1, u2} for single-valued functions u1, u2 ∈ 1 2 ˘ 1 C (BR(X0)) ∩ C (BR(X0)) such that Lu1 ≥ 0 and Lu2 ≥ 0. Since u ∈ C (C) and u attains its nonnegative maximum value at X0, u(X0) ≥ 0 and Du(X0) = 0. But this contradicts the Hopf boundary point lemma [3, Lemma 3.4], which when applied to u1 on BR(X1) implies Du(X0) 6= 0 unless u is constant. Therefore if u is not constant, u can only attain its maximum on ∂C.

Corollary 2. Let u, aij, bi, c, and ϕ be the two-valued functions associated with 0 1 2 n−2 ij i 0 0 u0 ∈ V (C) ∩ V (C) ∩ V (C\{0} × R ), a0 , b0, c0 ∈ V (C), and ϕ0 ∈ V (∂C) ij i respectively. Suppose u0, a0 , b0, c0, and ϕ0 are periodic with respect to yj with period

ρj. Suppose

ij i n−2 Lu = a Diju + b Diu + cu ≥ 0 in C\{0} × R , u = ϕ on ∂C, which we interpret as discussed in Section 1.3, and

ij 2 n a0 (X)ξiξj ≥ λ|ξ| for X ∈ C, ξ ∈ R CHAPTER 2. ELLIPTIC THEORY FOR TWO-VALUED FUNCTIONS 31

for some constant λ > 0 and c0 ≤ 0. Then

sup u ≤ sup ϕ+, C ∂C

+ 0 where ϕ is the two-valued function associated with max{ϕ0, 0} ∈ V (C).

Remark 4. If Lu = 0, we can replace u with −u to conclude that

sup |u| ≤ sup |ϕ|, C ∂C

0 where we interpret |u| and |ϕ| as the two-valued function associated with |u0| ∈ V (C) 0 and |ϕ0| ∈ V (∂C).

Proof of Corollary 2. Follows immediately from Theorem 8 and the fact that since u0 is periodic with respect to yj, u must attain its maximum on C.

The next maximum principle generalizes the weak maximum principle for single- valued functions [3, Theorem 8.1]. The proof requires proving a Sobolev inequality for two-valued functions.

Lemma 4. Let 1 ≤ p < n. Let ρj > 0 for j = 1, . . . , n − 2 and R = [0, ρ1] × · · · × 0 1,p [0, ρn−2]. Suppose u ∈ C (C) ∩ W (R) is a two-valued function such that u has the form (1.14), u is periodic with respect to yj with period ρj, and u = 0 near ∂C. Then

kukLnp/(n−p)(R) ≤ CkDukLp(R) (2.1) for some C = C(n, p) > 0.

Remark 5. A similar proof shows that if u ∈ C0(C) ∩ W 1,p(C) is any two-valued function of the form (1.14) that has compact support in C, then

kukLnp/(n−p)(C) ≤ CkDukLp(C) for some C = C(n, p) > 0. CHAPTER 2. ELLIPTIC THEORY FOR TWO-VALUED FUNCTIONS 32

Proof of Lemma 4. For δ > 0, let γδ be an odd smooth function such that γδ(t) = t 0 for |t| > δ, γδ(t) = 0 for |t| < δ/2, and |γδ(t)| ≤ 3 for all t ∈ R. Approximate u by 1,p (1) (2) vδ = ua +γδ(us) so that as δ ↓ 0, vδ → u in W (R). Write as vδ = {vδ , vδ } on C for (1) (2) 0 1,p single-valued functions vδ = ua +γδ(|us|) and vδ = ua −γδ(|us|) in C (C)∩W (R). By the Sobolev inequality for single-valued functions,

(l) (l) kvδ kLnp/(n−p)(R) ≤ CkDvδ kLp(R) for l = 1, 2.

Hence, letting q = np/(n − p),

Z 1/q (1) q (2) q kvδkLq(R) = |vδ | + |vδ | R Z q/p Z q/p!1/q 1/q (1) p (2) p ≤ C |Dvδ | + |Dvδ | R R Z Z 1/p 1/q (1) p (2) p ≤ C |Dvδ | + |Dvδ | R R 1/q = C kDvδkLp(R),

Using the identity as + bs ≤ (a + b)s for a, b > 0 and s = q/p > 1. Now (2.1) follows by letting δ ↓ 0.

Theorem 9. Let u, aij, bi, cj, d, and ϕ be the two-valued functions associated with 0 1 ij i j 0 0 u0 ∈ V (C) ∩ V (C), a0 , b0, c0, d0 ∈ V (C), and ϕ0 ∈ V (∂C) respectively. Suppose u0, ij i j a0 , b0, c0, d0, and ϕ0 are periodic with respect to yj with period ρj. Suppose

ij i i n−2 Di(a Dju + b u) + c Diu + du ≥ 0 in C\{0} × R , u = ϕ on ∂C, which we interpret as discussed in Section 1.3, and

ij 2 n a0 (X)ξiξj ≥ λ|ξ| for X ∈ C, η ∈ R (2.2) CHAPTER 2. ELLIPTIC THEORY FOR TWO-VALUED FUNCTIONS 33

for some constant λ > 0. Suppose Z j −b Dxj ζ + dζ ≤ 0 (2.3) C

1 for all two-valued function ζ associated with some ζ0 ∈ Vc (C) such that ζ0 ≥ 0, where we interpret (2.3) as Z 2 j k 4|ξ| −b0Mj Dξk ζ0 + d0ζ0 ≤ 0 (2.4) C 1/2 using the change of variable ξ1 + iξ2 = (x1 + ix2) like in Section 1.3. Then

sup u ≤ sup ϕ+ C ∂C

+ 0 where ϕ is the two-valued function associated with max{ϕ0, 0} ∈ V (∂C).

Remark 6. If Lu = 0, we can replace u with −u to conclude that

sup |u| ≤ sup |ϕ|, C ∂C

0 where we interpret |u| and |ϕ| as the two-valued function associated with |u0| ∈ V (C) 0 and |ϕ0| ∈ V (∂C) respectively.

Proof of 9. The proof is similar to the proof of Theorem 8.1 in [3]. Suppose sup∂C ϕ < supC u. Let + ζ0 = max{u0, 0} where sup ϕ < l < sup u, ∂C C and let ζ be the two-valued function associated with ζ0. Using the weak equation Z ij j i a DjuDiζ + b uDjζ − c Diuζ − duζ ≤ 0, R

2 where R = B1 (0) × [0, ρ1] × · · · × [0, ρn−2] and we interpret the integral as discussed in Section 1.3, and arguing as in [3, Theorem 8.1], we obtain

Z Z ij i i a DiζDjζ ≤ (b + c )ζDiζ. R R CHAPTER 2. ELLIPTIC THEORY FOR TWO-VALUED FUNCTIONS 34

Deﬁne the measurable function ψ0 on C by ψ0 = 1 wherever Dζ0 6= 0 and ψ0 = 0 elsewhere. Associate with ψ0 the two-valued function

iθ 1/2 iθ/2 1/2 iθ/2 ψ(re , y) = {ψ0(r e , y), ψ0(−r e , y)}.

By (2.2) and the fact that

ψ|b + c|2ζ2 |(bi + ci)ζD ζ| = ψ|(bi + ci)ζD ζ| ≤ + λ|Dv|2/2, i i 2λ

1/2 interpreted in the appropriate way using the change of variable ξ1 +iξ2 = (x1 +ix2) , we obtain Z Z 2 1 i i 2 2 |Dζ| ≤ 2 kb0 + c0kV0(C) ψ ζ , (2.5) R λ R i i If b0 + c ≡ 0, then (2.5) implies Dζ ≡ 0 and so ζ ≡ 0, contradicting v 6≡ 0. If bi + ci 6≡ 0, then applying the Sobolev inequality and H¨olderinequality to (2.5) implies

C i i kζk 2n/(n−2) ≤ CkDζk 2 ≤ kb + c k 0 kψk n kζk 2n/(n−2) L (R) L (R) λ 0 0 V (C) L (R) L (R) for C = C(n, ρ1, . . . , ρn−2) > 0, so

λ i i −1 kψk n ≥ kb + c k . L (R) C 0 0 V0(C)

Since Z 1/n 2 kψkLn(R) = ψ0(ξ, y) 4|ξ| dξdy , R it follows that

λn Ln({X ∈ R : Dζ (X) 6= 0}) = Ln({X ∈ R : ψ (X) = 1} ≥ kbi + ci k−n . 0 0 4Cn 0 0 V0(C)

Letting l increase to supC u implies that u0 attains its maximum on a set S ⊂ R for n which L (S) > 0 and Du0 6= 0 on S, which is a contradiction. Therefore supC u ≤ + sup∂C ϕ . CHAPTER 2. ELLIPTIC THEORY FOR TWO-VALUED FUNCTIONS 35

2.3 Schauder estimates

In this section we will prove various Schauder estimates for two-valued functions. One approach to proving Schauder estimates for single-valued functions is to use integral kernels [3, Chapter 4]. For two-valued functions we cannot use integral kernels, so instead we use a scaling argument from [11, Lemma 3.2] to prove the Schauder estimates. First we have a few lemmas.

n Lemma 5. Let BR(X) be any ball in R , k ∈ R, θ ∈ (0, 1), γ > 0, ν > 0, and let

S be any nonnegative subadditive function on the class of convex subsets of BR(X). There is an ε = ε(n, k, θ) such that if

k k ρ S(Bρ/2(Y )) ≤ ερ S(Bρ/2(Y )) + γ (2.6)

whenever Bρ(Y ) ⊆ BR(X) and ρ ≤ νR, then

k R S(BR/2(X)) ≤ Cγ for some constant C = C(n, k, θ, ν).

Proof. See the proof of Lemma 2 in Section 2.8 in [9]. Note that [9] omits the assumption that ρ ≤ νR for (2.6) to hold. To get the full result, ﬁrst show

k ρ S(Bρ/2(X)) ≤ Cγ (2.7)

for Bρ(Y ) ⊆ BR(X) with ρ ≤ νR using the proof in [9] with slight modiﬁcation.

Then cover BR(X) by a ﬁnite collection {BθνR(Yj)}j=1,...,N of open balls such that

BνR(Yj) ⊂ BR(X) and N ≤ C for some constant C = C(n, k, θ, µ) > 0. By (2.7),

N k X k R S(BθR(X)) ≤ R S(BθνR(Yj)) ≤ Cγ j=1 for C = C(n, k, θ, ν) > 0. CHAPTER 2. ELLIPTIC THEORY FOR TWO-VALUED FUNCTIONS 36

Lemma 6. Suppose u ∈ C1,µ(Rn) is a two-valued function with µ ∈ (0, 1/2) such n that ∆u = 0 on R \Ku and [Du]µ,Rn < ∞. Then u is aﬃne, i.e. u(X) = {a0 + Pn Pn i=1 aixi, b0 + i=1 bixi} for some constants ai, bi ∈ R. The proof of Lemma 6 involves frequency functions and is given [11, Sections 2 and 4].

n i Lemma 7. Consider the ball BR(X0) ⊆ R . Let µ ∈ (0, 1/2) and u, f , and g be 1,µ 0,µ the two-valued functions associated with u0 ∈ V (BR(X0)), f ∈ V (BR(X0)), and 0 ij g ∈ V (BR(X0)) respectively. Let a be constants such that

ij 2 n ij a ζiζj ≥ λ|ζ| for ζ ∈ R , |a | ≤ Λ, for some constants λ, Λ > 0. Suppose

ij i ˘ n−2 a Diju = Dif + g in BR(X0) \{0} × R , which we interpret as discussed in Section 1.3. Then

0 1+µ 2 kuk 1,µ ≤ C kuk 0 + R [f] + R kgk 0 C (BR/2(X0)) C (BR(X0)) µ,BR(X0) C (BR(X0)) for some constant C = C(n, µ, λ, Λ) > 0. Proof. By rescaling, assume R = 1. By standard interpolation inequalities and Lemma 5, it suﬃces to show that for every δ > 0,

0 0 [Du]µ,B1/2(X0) ≤ δ[Du]µ,B1(X0) + C kukC (B1(X0)) + kDukC (B1(X0)) 0 +[f]µ,B1(X0) + kgkC (B1(X0)) for some constant C = C(n, µ, λ, Λ, δ) > 0. Suppose not, then for some δ > 0 ij and every positive integer k, there is a ball B1(Xk), constants ak , and two-valued i 1,µ 0,µ functions uk, fk, and gk associated with u0,k ∈ V (BR(X0)), f0,k ∈ V (BR(X0)), 0 and g0,k ∈ V (BR(X0)) respectively, such that

ij 2 n ij ak ζiζj ≥ λ|ζ| for ζ ∈ R , |a | ≤ Λ CHAPTER 2. ELLIPTIC THEORY FOR TWO-VALUED FUNCTIONS 37

and ij i ˘ n−2 ak Dijuk = Difk + gk in B1(Xk) \{0} × R and

0 0 [Duk]µ,B1/2(Xk) > δ[Duk]µ,B1(Xk) + k kukkC (B1(Xk)) + kDukkC (B1(Xk)) 0 +[fk]µ,B1(Xk) + kgkkC (B1(Xk)) . (2.8)

0 n−2 Select Yk,Yk ∈ B1/2(Xk) \{0} × R and

G(Du (Y ), Du (Y 0)) 1 k k k k ≥ [Du ] . (2.9) 0 µ k µ,B1/2(Xk) |Yk − Yk| 2

0 Let ρk = |Yk − Yk|. By (2.8) and (2.9),

1 2 2 0 [Duk]µ,B1/2(Xk) ≤ µ kDukC (B1(Xk)) < µ [Duk]µ,B1/2(Xk), 2 ρk kρk

µ ρk ≤ 4/k for all positive integers k and thus ρk → 0 as k → ∞.

0 n−2 Suppose dist({Yk,Yk}, {0}×R )/ρk ≤ c for some constant c > 0. Then for some n−2 Zk ∈ {0}×R , |Yk −Zk| ≤ 2cρk. By translating, we may suppose Zk = 0. Let Rk =

1/2ρk −2c > 0 for k suﬃciently large so that BRk (0) ⊆ B1/2ρk (Yk/ρk) ⊆ B1/ρk (Xk/ρk) 0 0 0 and Rk → ∞. Rescale letting ξk = Yk/ρk and ξk = Yk/ρk so that |ξk − ξk| = 1 and

uˆ (X) = ρ−1−µ[Du ]−1 (u (ρ X) − u (0) − Du (0) · ρ X), 0,k k k µ,B1(0) 0,k k 0,k 0,k k fˆ (X) = ρ−µ[Du ]−1 (f (ρ X) − f (0)), 0,k k k µ,B1(0) 0,k k 0,k gˆ (X) = ρ1−µ[Du ]−1 g (ρ X), 0,k k k µ,B1(0) 0,k k

ˆ ˆ andu ˆk, fk, andg ˆk be the two-valued functions associated withu ˆ0,k, f0,k, andg ˆ0,k respectively so that

ij ˆi ˘ n−2 ak Dijuˆk = Difk +g ˆk in BRk (0) \{0} × R . CHAPTER 2. ELLIPTIC THEORY FOR TWO-VALUED FUNCTIONS 38

and

0 δ [Duˆk]µ,B (0) ≤ 1, G(Duˆk(ξk),Duˆk(ξ )) ≥ Rk k 2

0 0 0 Since {ξk} and {ξk} are bounded, after passing to a subsequence, ξk → ξ and ξk → ξ 0 n ij ij for some points ξ, ξ ∈ R . Since |ak | ≤ Λ, after passing to a subsequence, {ak } converges to some constanta ˆij. By (2.8),

ˆi ˆ µ−1 f (0) = 0, [fk]µ,B (0) + ρ kgˆkkC0(B (0)) ≤ 1/k, k Rk k Rk

ˆi n so {f0,k} and {gˆ0,k} both converge to zero uniformly on compact subsets of R . Since 1 n [Duˆk]µ,B (0) ≤ 1, after passing to a subsequence, {u0,k} converges in V ( ) to Rk loc R 1,µ n some functionu ˆ0 ∈ V (R ). The two-valued functionu ˆ associated withu ˆ0 satisﬁes ij n n−2 aˆ Dijuˆ = 0 on R \{0} × R ,[Duˆ]µ,Rn ≤ 1, and

δ G(Duˆ(ξ),Duˆ(ξ0)) ≥ , 2 which contradicts Lemma 6.

0 n−2 Suppose instead that dist({Yk,Yk}, {0} × R )/ρk is unbounded. Without loss of 0 n−2 generality, we may suppose that dist({Yk,Yk}, {0} × R )/ρk → ∞. Then for some 0 n−2 Rk → ∞, Rk < dist({Yk,Yk}, {0} × R )/ρk and Rk < 1/2ρk. Restrict uk, fk, and gk to their single-valued branches on BρkRk (Yk) such that

ij i ˘ n−2 ak Dijuk = Difk + gk in B1(Xk) \{0} × R and |Du (Y ) − Du (Y 0)| 1 k k k k ≥ [Du ] . 0 µ k µ,B1/2(Xk) |Yk − Yk| 4

As a slight abuse of notion we will denote the single-valued branches of uk, fk, and 0 gk as uk, fk, and gk respectively. Rescale letting ξk = (Yk − Yk)/ρk so that |ξk| = 1 CHAPTER 2. ELLIPTIC THEORY FOR TWO-VALUED FUNCTIONS 39

and letting

uˆ (X) = ρ−1−µ[Du ]−1 (u (Y + ρ X) − u (Y 0) − Du (Y ) · ρ X), k k k µ,B1(0) k k k k k k k k fˆ (X) = ρ−µ[Du ]−1 (f (Y + ρ X) − f (Y 0)), k k k µ,B1(0) k k k k k gˆ (X) = ρ1−µ[Du ]−1 g (Y + ρ X) k k k µ,B1(0) k k k

on BRk (0). Using a similar argument as above, after passing to a subsequence, {ξk} ij ij ˆi converges to some ξ, {ak } converges to somea ˆ , {fk} and {gˆk} both converge to zero n 1 n uniformly on compact subsets of R , andu ˆk converges in Cloc(R ) to some function ij n uˆ. Moreover,u ˆ satisﬁesa ˆ Dijuˆ = 0 on R ,[Duˆk]µ,Rn−2 ≤ 1, and

δ G(Duˆ(ξ),Duˆ(0)) ≥ , 2 which contradicts the Liouville theorem for single-valued harmonic functions.

n ij i i j Theorem 10. Consider the ball BR(X0) ⊆ R . Let µ ∈ (0, 1/2) and u, a , b , f , c , 1,µ ij i i d, and g be the two-valued function associated with u0 ∈ V (BR(X0)), a0 , b0, f0 ∈ 0,µ j 0 V (BR(X0)), and c0, d0, g ∈ V (BR(X0)) respectively. Suppose

ij i j i ˘ n−2 Di(a Dju + b u) + c Dju + du = Dif + g in BR(X0) \{0} × R , which we interpret as discussed in Section 1.3, and

ij iθ 2 2 2iθ n a0 (re , y)ζiζj ≥ λ|ζ| whenever (r e , y) ∈ BR(X0), ζ ∈ R , ij 0 ka k 0,µ ≤ Λ, C (BR(X0)) i 0 j 2 Rkb k 0,µ + Rkc k 0 + R kdk 0 ≤ ν, (2.10) C (BR(X0)) C (BR(X0)) C (BR(X0)) for some constants λ, Λ, ν > 0. Then

0 1+µ 2 kuk 1,µ ≤ C kuk 0 + R [f] + R kgk 0 (2.11) C (BR/2(X0)) C (BR(X0)) µ,BR(X0) C (BR(X0)) for some constant C = C(n, µ, λ, Λ, ν) > 0. CHAPTER 2. ELLIPTIC THEORY FOR TWO-VALUED FUNCTIONS 40

n−2 Proof. Let Br(X) ⊆ BR(X0). Suppose {0} × R ∩ Br(X) 6= ∅ and let Z ∈ {0} × n−2 R ∩ Br(X). Observe that

ij ij ij i i j ˘ n−2 a (Z)Diju = Di((a (Z)−a )Dju−b u+f )−c Dju−du+g in Br(X)\{0}×R , so by Lemma 7,

1+µ 1+µ ij ij 0 r [Du]µ,Br/2(X) ≤ C kukC (Br(X)) + r [(a (Z) − a )Dju]µ,Br(X) 1+µ i 2 j 2 0 0 +r [b u]µ,Br(X) + r kc DjukC (Br(X)) + r kdukC (Br(X)) 1+µ 2 0 +r [f]µ,Br(X) + r kgkC (Br(X)) for some constant C = C(n, µ, λ, Λ) > 0. By (2.10),

µ 1+2µ 1+µ 2 Λr r [Du] ≤ C kuk 0 + [Du] µ,Br/2(X) C (BR(X0)) Rµ µ,Br(X) 0 1+µ 2 +(Λ + ν)kuk 1 + R [f] + R kgk 0 . (2.12) C (BR(X0)) µ,BR(X0) C (BR(X0))

n−2 Note that if {0} × R ∩ Br(X) = ∅, then (2.12) holds by the Schauder estimates for single-valued functions. By applying Lemma 5 with r ≤ ηR for η = η(n, µ, λ, Λ, ν) > 0 suﬃciently small and applying interpolation, we obtain (2.11).

n ij i i Theorem 11. Consider the ball BR(X0) ⊆ R . Let µ ∈ (0, 1/2) and u, aκ,λ, bκ,λ, fκ, j 1,µ m cκ,λ, dκ,λ, and gκ be the two-valued functions associated with u0 ∈ V (BR(X0), R ), ij i i 0,µ j 0 a0,κ,λ, b0,κ,λ, f0,κ ∈ V (BR(X0)), and c0,κ,λ, d0,κ,λ, g0,κ ∈ V (BR(X0)) respectively. Suppose

ij λ i λ j λ λ i ˘ Di(aκλDju + bκλu ) + cκλDju + dκλu = Difκ + gκ in BR(X0) \B, κ = 1, . . . , m, which we interpret as

2 k l ij λ 2 k i λ 2 l j λ 2 λ Dξk (4|ξ| Mi Mja0,κλDξl u0 + 4|ξ| Mi b0,κλu0 ) + 4|ξ| Mjc0,κλDlu0 + 4|ξ| d0,κλu0 2 k i 2 n−2 = Dξk (4|ξ| Mi f0,κ) + 4|ξ| g0,κ weakly on Ω0 \{0} × R . CHAPTER 2. ELLIPTIC THEORY FOR TWO-VALUED FUNCTIONS 41

1/2 using the change of coordinates ξ1 + iξ2 = (x1 + ix2) as in Section 1.3. Suppose

µ ij i 0 j 2 R [a ] + Rkb k 0,µ + Rkc k 0 + R kd k 0 ≤ ν, κλ µ,BR(X0) κλ C (BR(X0)) κλ C (BR(X0)) κλ C (BR(X0)) (2.13) for some constant ν > 0. For some ε = ε(n, m, µ, ν) > 0, if

ij ij 0 kaκλ − δ δκλkC (BR(X0)) ≤ ε, (2.14)

ij where δ and δκλ denote Kronecker deltas, then

0 1+µ 2 kuk 1,µ ≤ C kuk 0 + R [f] + R kgk 0 (2.15) C (BR/2(X0)) C (BR(X0)) µ,BR(X0) C (BR(X0)) for some constant C = C(n, m, µ, ν) > 0.

Proof. Let Br(X) ⊆ BR(X0). Observe that

κ i ij ij λ i λ j λ λ ˘ ∆u = Di(fκ − (aκλ − δ δκλ)Dju − bκλu ) + gκ − cκλDju − dκλu in BR(X0) \B, for κ = 1, . . . , m, so by Lemma 7,

1+µ κ κ 1+µ ij ij λ 0 r [Du ]µ,Br/2(X) ≤ C ku kC (Br(X)) + r [(δ δκλ − aκλ)Dju ]µ,Br(X) 1+µ i λ 2 j λ 0 +r [bκλu ]µ,Br(X) + r kcκλDju kC (Br(X)) 2 λ 1+µ 2 0 0 +r kdκλu kC (Br(X)) + r [fκ]µ,Br(X) + r kgκkC (Br(X)) .

By (2.13) and (2.14),

1+µ κ 1+µ 0 r [Du ] ≤ C kuk 0 + εr [Du] + νkuk 1 µ,Br/2(X) C (BR(X0)) µ,Br(X) C (BR(X0)) 1+µ 2 0 +R [f]µ,BR(X0) + R kgkC (BR(X0)) .

Summing over κ,

1+µ 1+µ 0 r [Du] ≤ C kuk 0 + εr [Du] + νkuk 1 µ,Br/2(X) C (BR(X0)) µ,Br(X) C (BR(X0)) 1+µ 2 0 +R [f]µ,BR(X0) + R kgkC (BR(X0)) . CHAPTER 2. ELLIPTIC THEORY FOR TWO-VALUED FUNCTIONS 42

By applying interpolation and Lemma 5 with ε suﬃciently small, we obtain (2.15).

The following Schauder estimate is need for the proof of Theorem 4.

n ij Theorem 12. Consider the ball BR(X0) ⊆ R . Let 0 < µ < τ < 1/2 and u, a , and f 1,τ ij 0,µ be the two-valued functions associated with u0 ∈ V (BR(X0)), a0 , f0 ∈ V (BR(X0)) respectively. Suppose

ij ˘ n−2 a Diju = f in BR(X0) \{0} × R , which we interpret as discussed in Section 1.3, and

ij iθ 2 2 2iθ n ij 0,µ a0 (re , y)ζiζj ≥ λ|ζ| whenever (r e , y) ∈ BR(X0), ζ ∈ R , ka kC (BR(X0)) ≤ Λ, for some constants λ, Λ > 0. Then

0 kuk 1,τ ≤ C kuk 0 + kfk 0,µ C (BR/2(X0)) C (BR(X0)) C (BR(X0)) for some constant C = C(n, µ, τ, λ, Λ) > 0.

Proof. The proof is similar to the proof of Lemma 7. By rescaling, assume R = 1. By the interpolation inequalities and Lemma 5, it suﬃces to show that for every δ > 0,

0 [Du]τ,B1/2(X0) ≤ δ[Du]τ,B1(X0) + C kukC (B1(X0)) 0 0,µ +kDukC (B1(X0)) + kfkC (B1(X0)) for some constant C = C(n, µ, λ, Λ, δ) > 0. Suppose not, then for some δ > 0 and ij every positive integer k, there is a ball B1(Xk) and two-valued functions uk, ak , and 1,τ ij 0,µ fk associated with u0,k ∈ V (BR(X0)), a0,k, f0,k ∈ V (BR(X0)) respectively, such that ij ˘ n−2 ak Dijuk = fk in B1(Xk) \{0} × R

ij 0,µ where ak , fk ∈ C (B1(X0)) are two-valued functions of the form (1.14) such that

ij 2 n ij 0,µ ak (X)ζiζj ≥ λ|ζ| for ζ ∈ R ,X ∈ B1(Xk), ka kC (B1(Xk)) ≤ Λ, CHAPTER 2. ELLIPTIC THEORY FOR TWO-VALUED FUNCTIONS 43

and

0 [Duk]τ,B1/2(Xk) > δ[Duk]τ,B1(Xk) + k kukkC (B1(Xk)) 0 0,µ +kDukkC (B1(Xk)) + kfkkC (B1(Xk)) . (2.16)

0 n−2 Select Yk,Yk ∈ B1(Xk) \{0} × R and

G(Du (Y ), Du (Y 0)) 1 k k k k ≥ [Du ] (2.17) 0 τ k τ,B1/2(Xk) |Yk − Yk| 2

0 and let ρk = |Yk − Yk|. As in the proof of Lemma 7, it follows from (2.16) and (2.17) that ρk → 0 as k → ∞.

0 n−2 Suppose dist({Yk,Yk}, {0}×R )/ρk ≤ c for some constant c > 0. Then for some n−2 Zk ∈ {0} × R , |Yk − Zk| ≤ 2cρk and, by translating, we may suppose Zk = 0. Let 0 0 Rk = 1/2ρk − 2c > 0 for k suﬃciently large. Rescale letting ξk = Yk/ρk, ξk = Yk/ρk,

uˆ (X) = ρ−1−τ [Du ]−1 (u(ρ X) − u(0) − Du(0) · ρ X), k k k τ,B1(0) k k ij ij aˆk (X) = ak (ρkX), fˆ (X) = ρ1−τ [Du ]−1 f(ρ X), k k k τ,B1(0) k so that

0 δ [Duˆk]τ,B (0) ≤ 1, |Duˆk(ξk) − Duˆk(ξ )| ≥ Rk k 2 and ij ˆ ˘ n−2 aˆk Dijuˆk = fk in BRk (0) \{0} × R .

0 0 0 Since {ξk} and {ξk} are bounded, after passing to a subsequence, ξk → ξ and ξk → ξ for some points ξ, ξ0 ∈ Rn. Since

ij −µ ij kaˆ kC0(B (0)) + ρ [ˆa ]µ,B (0) ≤ Λ, (2.18) k Rk k k Rk CHAPTER 2. ELLIPTIC THEORY FOR TWO-VALUED FUNCTIONS 44

ij ij after passing to a subsequence, {aˆk } converges to some constant functiona ˆ uniformly on compact subsets of Rn. By (2.16),

ˆ −µ ˆ 1−τ kfkkC0(B (0)) + ρ [fk]µ,B (0) ≤ ρ /k, (2.19) Rk Rk k

ˆ so {fk} converges to zero uniformly. Since [Duˆk]τ,B (0) ≤ 1, after passing to a Rk 1 n 1,τ n subsequence, {u0,k} converges in Vloc(R ) to some functionu ˆ0 ∈ C (R ) such that the two-valued functionu ˆ associated withu ˆ0 satisﬁes [Duˆ]τ,Rn ≤ 1. Moreover, by the interior Schauder estimates for single-valued functions using (2.18) and (2.19), n n−2 for every closed ball Bσ(X) ⊂ R \{0} × R ,

2 0 2,µ 0 ˆ 0,µ kuˆkkC (Bσ/2(X)) ≤ C kuˆkkC (Bσ(X)) + σ kfkkC (Bσ(X)) 1+τ 2 1−τ 2+µ 1−τ+µ ≤ C (σ + |X|) + σ ρk /k + σ ρk /k ≤ C (σ + |X|)1+τ + σ2 + σ2+µ

2 n for sufficiently large k, so after passing to a subsequence,u ˆ0,k → uˆ0 in Vloc(R \{0} × n−2 ij n n−2 R ). Henceu ˆ satisfies the differential equationa ˆ Dijuˆ = 0 on R \{0} × R .

However, [Duˆ]τ,Rn ≤ 1 and

δ |Duˆ(ξ) − Duˆ(ξ0)| ≥ , 2 contradicting Lemma 6.

0 n−2 Suppose instead that dist({Yk,Yk}, {0} × R )/ρk is unbounded. Without loss of 0 n−2 generality, we may suppose that dist({Yk,Yk}, {0} × R )/ρk tends to inﬁnity. Then 0 n−2 for some Rk → ∞, Rk < dist({Yk,Yk}, {0} × R )/ρk and Rk < 1/2ρk. Restrict uk, ij ak , and fk to their single-valued branches on BρkRk (Yk) such that

ij ˘ n−2 ak Dijuk = fk in B1(Xk) \{0} × R and |Du (Y ) − Du (Y 0)| 1 k k k k ≥ [Du ] . 0 µ k τ,B1/2(Xk) |Yk − Yk| 4 CHAPTER 2. ELLIPTIC THEORY FOR TWO-VALUED FUNCTIONS 45

ij As a slight abuse of notion we will denote the single-valued branches of uk, ak , and ij 0 fk as uk, ak , and fk respectively. Rescale letting ξk = (Yk − Yk)/ρk and

uˆ (X) = ρ−1−τ [Du ]−1 (u (Y 0 + ρ X) − u (Y 0) − Du (Y 0) · ρ X), k k k τ,B1(0) k k k k k k k k ij ij 0 aˆk (X) = ak (Yk + ρkX), fˆ (X) = ρ−1−τ [Du ]−1 f ij(Y 0 + ρ X) k k k τ,B1(0) k k k

on BRk (0). Similar to above, after passing to a subsequence, {ξk} converges to some ij 0 n ij ˆ ξ, {aˆk } converges in Cloc(R ) to some constant functiona ˆ and {fk} converges to zero uniformly. Also, by the interior Schauder estimates for single-valued functions, 2,µ n {uˆk} is uniformly bounded in C on compact subsets of R , so after passing to a 2 n subsequence, {uˆk} converges in Cloc(R ) to some functionu ˆ. Moreover,u ˆ satisﬁes ij n aˆ Dijuˆ = 0 on R ,[Duˆk]τ,Rn−2 ≤ 1, and

δ |Duˆ(ξ) − Duˆ(0)| ≥ , 2 which contradicts the Liouville theorem for single-valued harmonic functions.

2.4 Global estimates

Our goal is to bound supC u for solutions u to inhomogeneous equations in terms of sup∂C u. We follow the approach of [3, Section 8.5].

Theorem 13. Let u, aij, bi, cj, d, f i, g, and ϕ be the two-valued functions asso- ij i j i 0 0 ciated with u0, a0 , b0, c0, d0, f0, g0 ∈ V (C) and ϕ0 ∈ V (∂C) respectively. Suppose 1,2 ij i j i u ∈ W (C) and u0, a0 , b0, c0, d0, f0, g0, and ϕ0 are periodic with respect to yj with period ρj. Suppose

ij i j i n−2 Lu = Di a u + b u + c Dju + du = Dif + g in C\{0} × R , u = ϕ on ∂C, CHAPTER 2. ELLIPTIC THEORY FOR TWO-VALUED FUNCTIONS 46

which we interpret as discussed in Section 1.3. Suppose

ij 2 n a0 (X)ξiξj ≥ λ|ξ| for X ∈ C, ξ ∈ R , (2.20) for some constant λ > 0,

i i kb kC0,µ(C) + kc kC0,µ(C) ≤ ν, (2.21) for some constant ν ≥ 0, and Z j −b Djζ + dζ ≤ 0 (2.22) C

1 n−2 for all two-valued functions ζ associated with ζ0 ∈ Cc (C\{0}×R ) such that ζ0 ≥ 0, which is interpreted as in (2.3) in the statement of Theorem 9. Then

sup |u| ≤ sup |ϕ| + C kfkC0(C) + kgkC0(C) C ∂C for some constant C = C(n, λ, ν, ρ1, . . . , ρn−2) > 0.

Proof. The proof is nearly identical to Theorem 8.16 in [3] and uses the weak equation

Z ij i i j a Diu + b u − f Diζ − c Dju + du − g ζ = 0 C

1,2 0 where ζ ∈ W (C) is a two-valued function associated with some ζ0 ∈ V (C) and where we interpret the integral as discussed in Section 1.3.

Observe that in the estimates above depend on the periods ρ1, . . . , ρn−2. For the proof of Theorem 3, we would like to get a bound on supC u in terms of sup∂C u that is independent of ρ1, . . . , ρn−2. The following theorem gives us this in the special case of the Poisson equation.

i Theorem 14. Let u, f , g, and ϕ be the two-valued functions associated with u0 ∈ 1,µ i 0,µ 0 0 i V (C), f0 ∈ V (C), g0 ∈ V (C) and ϕ0 ∈ V (∂C) respectively. Suppose u0, f0, g0, CHAPTER 2. ELLIPTIC THEORY FOR TWO-VALUED FUNCTIONS 47

and ϕ0 are periodic with respect to yj with period ρj. Suppose

i n−2 ∆u = Dif + g in C\{0} × R ,

u0 = ϕ0 on ∂C, which we interpret as discussed in Section 1.3. Then

sup |u| ≤ C sup |ϕ| + [f]µ,C + kgkC0(C) . C ∂C for some constant C = C(n, µ) > 0 independent of ρ1, . . . , ρn−2.

1,µ i 0,µ Proof. Suppose instead for every integer k ≥ 1 there are u0,k ∈ V (C), f0,k ∈ V (C), 0 0 i g0,k ∈ V (C) and ϕ0,k ∈ V (∂C) such that u0,k, f0,k, g0,k, and ϕ0,k are periodic with i respect to yj with period ρj, fk, gk, and ϕk are the two-valued functions associated i with u0,k, f0,k, g0,k, and ϕ0,k,

i n−2 ∆uk = Difk + gk in C\{0} × R ,

u0,k = ϕ0,k on ∂C, and sup |uk| > k sup |ϕk| + [fk]µ,C + kgkkC0(C) . C ∂C

Without loss of generality, suppose supC |uk| = 1 and, by translation, that |u0,k(xk, 0)| 2 = 1 for some xk ∈ B1 (0). Moreover, suppose fk(0, 0) = 0 for all k. By Bolzano-

Weierstrass, after passing to a subsequence, {xk} converges to somex ˆ as k → ∞. Since

sup |ϕk| + [fk]µ,(C + kgkkC0(C) ≤ 1/k, ∂C letting k → ∞, f0,k → 0 and g0,k → 0 uniformly on compact subsets of C and ϕ0,k → 0 uniformly on compact subsets of ∂C. By the interior Schauder estimates Theorem 10 CHAPTER 2. ELLIPTIC THEORY FOR TWO-VALUED FUNCTIONS 48

from Section 2.3, for BR(x0, y0) ⊂ C,

0 ku k 1,µ k C (BR/2(x0,y0)) 1+µ 2 0 0 ≤ C kukkC (BR(x0,y0)) + R [fk]µ,BR(x0,y0) + R kgkkC (BR(x0,y0)) ≤ C(1 + 1/k) < 2C

1 for some constant C = (n, µ) > 0 independent of k, so {u0,k} converges in Vloc(C) to someu ˆ0. Letu ˆ denote the two-valued function associated withu ˆ0. Let (x0, y0) ∈ ∂C iθ and ﬁx the single-valued branches of uk, fk, gk, and ϕ0 given by uk(re , y) = 1/2 iθ/2 iθ 1/2 iθ/2 iθ 1/2 iθ/2 u0,k(r e , y), fk(re , y) = f0,k(r e , y), gk(re , y) = g0,k(r e , y), and

iθ iθ/2 iθ0 ϕk(e = u0,k(e , y) for |θ − θ0| < π/2, where x0 = e . As a slight abuse of notation, we will denote these single-valued branches of uk, fk, gk, and ϕk by uk, fk, gk, and ϕk respectively. By [3, Theorem 8.27], for R ∈ (0, 1/2], ! α 0 oscC∩BR(x0,y0) uk ≤ CR sup |uk| + [fk]µ,B1/4(x0,y0) + kgkkC (B1/4(x0,y0)) C∩BR(x0,y0)

+C osc∂C∩B√ (x ,y ) ϕk R/2 0 0 ≤ 2CRα + 2C/k for some constants α ∈ (0, 1) and C > 0 dependent on and independent of k. So for

(x, y) ∈ C ∩ BR(x0, y0), since |uk(x, y) − ϕk(x0, y0)| ≤ oscC∩BR(x0,y0) uk,

α |uk(x, y)| ≤ 2C(R + 1/k).

Letting k → ∞ implies |uˆ(x, y)| ≤ 2CRα, sou ˆ extends to a continuous on C with uˆ = {0, 0} on ∂C. Therefore the two-valued functionu ˆ satisﬁes

n−2 ∆ˆu = 0 in C\{0} × R , uˆ = {0, 0} on ∂C.

Since supC |uˆ| ≤ 1 and |uˆ0(ˆx, 0)| = 1, |uˆ| has attains its maximum value of 1 at (ˆx, 0). CHAPTER 2. ELLIPTIC THEORY FOR TWO-VALUED FUNCTIONS 49

However, sup |uˆ| = 0 < 1 = sup |uˆ|. ∂C C Thusu ˆ is a harmonic function with an interior maximum atx ˆ, contradicting maximum principle Theorem 2.

Observe that using the local Schauder estimates for two-valued function, the local boundary Schauder estimates for single-valued functions [3, Section 8.11], and Theorems 13 and 14, we obtain the following Schauder estimates.

Corollary 3. Let µ ∈ (0, 1/2) and u, aij, bi, f i, cj, d, g, and ϕ be the two-valued 1,µ ij i i 0,µ j 0 functions associated with u0 ∈ V (C), a0 , b0, f0 ∈ V (C), c0, d0, g0 ∈ V (C) and 0 ij i j i ϕ0 ∈ V (∂C) respectively. Suppose u0, a0 , b0, c0, d0, f0, g0, and ϕ0 are periodic with respect to yj with period ρj. Suppose

ij i j i n−2 Di(a Dju + b u) + c Dju + du = Dif + g in C\{0} × R u = ϕ on ∂C which we interpret as discussed in Section 1.3. Further suppose that

ij 2 n a0 (X)ξiξj ≥ λ|ξ| for X ∈ C, ξ ∈ R , ij 0 i j ka kC0,µ(C) ≤ Λ, kb kC0(C) + kc kC0(C) + kdkC0(C) ≤ ν, for some constants λ, Λ, ν > 0 and Z j −b Djζ + dζ ≤ 0 C

1 n−2 for all two-valued functions ζ associated with ζ0 ∈ Cc (C\{0}×R ) such that ζ0 ≥ 0, which is interpreted as in (2.3) in the statement of Theorem 9. Then

kukC1,µ(C) ≤ C [f]µ,C + kgkC0(C) + kϕkC1,µ(C) .

for some constant C = C(n, µ, λ, Λ, ρ1, ··· , ρn−2) > 0. Moreover, in the special case i that ∆u = Dif + g, the constant C is independent of ρ1, . . . , ρn−2. CHAPTER 2. ELLIPTIC THEORY FOR TWO-VALUED FUNCTIONS 50

2.5 H¨oldercontinuity estimates

In this section we establish H¨oldercontinuity estimates for two-valued solutions to linear elliptic equations using the standard approach involving the De Giorgi-Nash- Moser theory. First we must prove a Poincar´einequality for two-valued functions.

An important feature of this Poincar´einequality is that it assumes the domain BR(0) is centered at a point on {0} × Rn−2.

0 1,2 Lemma 8. Suppose u ∈ C (BR(0)) ∩ W (BR(0)) is a two-valued function of the form (1.14). Then

2 2 ku − `kL (BR(0)) ≤ CRkDukL (BR(0)), where Z ` = u BR(0) for some C = C(n) > 0.

Proof. By scaling, we may suppose R = 1. If we write u = ua + us where ua(X) =

(u1(X) + u2(X))/2 and us = {±(u1(X) − u2(X))/2}, then

2 2 2 ku − `k 2 = ku − `k 2 + ku k 2 L (B1(0)) a L (B1(0)) s L (B1(0)) 2 2 2 kDuk 2 = kDu k 2 + kDu k 2 . L (B1(0)) a L (B1(0)) s L (B1(0))

2 2 Since kua − `kL (B1(0)) ≤ CkDuakL (B1(0)) for some C = C(n) > 0 by the Poincar´e inequality for single-valued functions, it suﬃces to suppose u is two-valued symmetric.

Suppose that for every integer k ≥ 1 there is a two-valued symmetric uk ∈ 0 1,2 C (B1(0)) ∩ W (B1(0)) of the form

iθ 1/2 iθ/2 1/2 iθ/2 uk(re , y) = {u0,k(r e , y), u0,k(−r e , y)} (2.23)

1/2 iθ/2 iθ for some continuous singe-valued function u0,k on Ω = {(r e , y):(re , y) ∈

B1(0)} such that

2 2 kukkL (B1(0)) > kkDukkL (B1(0)). CHAPTER 2. ELLIPTIC THEORY FOR TWO-VALUED FUNCTIONS 51

2 By scaling we may suppose kukkL (B1(0)) = 1 so that

2 2 kukkL (B1(0)) = 1, kDukkL (B1(0)) < 1/k. (2.24)

We want to show that, after passing to a subsequence, uk converges to a constant two-valued symmetric function u of the form (1.14). The only such function u is the

2 zero function, contradicting kukkL (B1(0)) = 1 in (2.24). We do this by considering iθ open subsets Ωj = {(re , y) ∈ B1(0) : |θ − jπ/2| < π/2} for j = 0,..., 7 and consider (j) (j) iθ 1/2 iθ/2 the single-valued branch uk of uk on Ωj given by uk (re , y) = u0,k(r e , y) for |θ − jπ/2| < π/2. Since by (2.24),

(j) (j) 2 2 kuk kL (B1(0)) ≤ 1, kDuk kL (B1(0)) < 1/k,

(j) by Rellich’s theorem for single-valued functions, after passing to a subsequence, {uk } (j) 2 converges to some function u strongly in L (Ωj) strongly, pointwise a.e. on Ωj, and 1,2 weakly in W (Ωj) with (j) 2 kDu kL (B1(0)) = 0. (2.25)

(j) (j) (j+1) By (2.25), each u are constant functions. Since uk = uk on Ωj ∩ Ωj+1 for (0) (7) (j) (j+1) j = 0,..., 6 and uk = uk on Ω0 ∩ Ω7, letting k → ∞ implies u = u on (0) (7) (j) Ωj ∩ Ωj+1 for j = 0,..., 6 and u = u on Ω0 ∩ Ω7. Hence u ≡ c for j = 0,..., 7 (0) (4) for some constant c ∈ R. Since each uk is two valued symmetric, uk = −uk , so (0) (4) (j) 2 c = u = −u = −c, thus c = 0. Since {uk } converges to 0 strongly in L (Ωj) for 2 all j = 0,..., 7, {uk} converges to 0 strongly in L (B1(0)), which implies

lim kukkL2(B (0)) = 0, k→∞ 1

2 contradicting kukkL (B1(0)) = 1 for all k in (2.24).

n−2 ij i j i Theorem 15. Let R > 0 and y0 ∈ R . Let u, a , b , c , d, f , and g be the two- ij i j i 0 valued functions associated with u0, a0 , b0, c0, d0, f0, g0 ∈ V (B5R(0, y0)) respectively. CHAPTER 2. ELLIPTIC THEORY FOR TWO-VALUED FUNCTIONS 52

1,2 Suppose u0 ≥ 0 and u ∈ W (B5R(0, y0)). Suppose

ij i j i ˘ n−2 Di(a Dju + b u) + c Dju + du ≤ Dif + g in B5R(0, y0) \{0} × R , which we interpret as discussed in Section 1.3, and

ij iθ 2 2 2iθ n a0 (re , y)ζiζj ≥ λ|ζ| whenever (r e , y) ∈ B5R(0, y0), ξ ∈ R , (2.26) ij i j 2 0 0 0 0 ka kC (B5R(0,y0)) ≤ Λ, kb kC (B5R(0,y0)) + Rkc kC (B5R(0,y0)) + R kdkC (B5R(0,y0)) ≤ ν, for some constants λ, Λ, ν > 0. Then

Z −n 2 0 0 R u ≤ C inf u + RkfkC (B5R(0,y0)) + R kgkC (B5R(0,y0)) (2.27) B (0,y0) B2R(0,y0) R for some constant C = C(n, Λ/λ, Rν/λ) > 0.

Proof. Translate and rescale so that y0 = 0 and R = 1. By replacing u with u + ε for

ε > 0 and at the end of the proof letting ε ↓ 0, we may suppose u ≥ ε on B5(0). The proof begins nearly identically to the proof of Theorem 8.18 in [3]. We use the weak equation Z Z ij i j i a DjuDiζ + b uDiζ − c Djuζ − duζ ≥ f Diζ − gζ , (2.28) B5(0) B5(0)

1 ˘ where ζ is the two-valued function associated with ζ0 ∈ Vc (B5(0)) and we interpret β 2 (2.28) as discussed in Section 1.3. Let ζ =u ¯ η χδ for β < 0, where

0 0 u¯ = u + kfkC (B5(0)) + kgkC (B5(0)),

1 ˘ and η ∈ Cc (B5(0)) is a single-valued function. Like in the proof of Theorem 8.18 in [3]. in the case that β 6= −1, we let w =u ¯(β+1)/2 and obtain

Z Z |Dw|2η2 ≤ C(1 + β−2) w2 η2 + |Dη|2 . (2.29) B5(0) B5(0) CHAPTER 2. ELLIPTIC THEORY FOR TWO-VALUED FUNCTIONS 53

If instead β = −1, we let w = log(u) and obtain

Z Z |Dw|2η2 ≤ C η2 + |Dη|2 . (2.30) B5(0) B5(0)

Using (2.30) and Moser iteration as discussed in the proof of Theorem 8.18 in [3], we obtain

Z −1/p −p 1 p u¯ ≤ C inf u,¯ ku¯kL (B2(0)) ≤ Cku¯kL (B3(0)), (2.31) B (0) B3(0) 1 for p ∈ (0, 1) and some constant C = C(Λ/λ, ν/λ, n, p) > 0. We want to show that

Z Z u¯p u¯−p ≤ C. (2.32) B3(0) B3(0) for some C = C(Λ/λ, ν/λ, n, p) > 0. Then we can conclude (2.27) holds by combining (2.31) and (2.32). The proof of (2.32) in [3] involves integral kernels, so instead we using the following approach.

Let w = log(¯u). Using (2.30) with η as the cutoff function such that 0 ≤ η ≤ 1, n η = 1 on B4(0), η = 0 on R \ B5(0), and |Dη| ≤ 2, then Z |Dw|2 ≤ C, (2.33) B4(0) where C = C(Λ/λ, ν/λ, n) > 0. Let φ be the cutoff function such that 0 ≤ φ ≤ 1, n 1 φ = 1 on B3(0), φ = 0 on R \ B4(0), and |Dφ| ≤ 2. Letχ ˆδ ∈ C (B4(0)) be the logarithmic cutoff function given by   0 if r ≤ δ2  iθ 2 2 χˆδ(re , y) = − log(r/δ )/ log(δ) if δ < r < δ   1 if r ≥ δ.

q−1 α(κq−1) Let η = |w − `| φ χˆδ in (2.30), where ` ∈ R, q ≥ 2, and κ = n/(n − 2), CHAPTER 2. ELLIPTIC THEORY FOR TWO-VALUED FUNCTIONS 54

α > 1/(κq − 1) will be chosen later, to get

Z 2q−2 2 2α(κq−1) 2 |w − `| |Dw| φ χˆδ B4(0) Z 2q−2 2α(κq−1) 2 2 2q−4 2α(κq−1) 2 2 ≤ C |w − `| φ χˆδ + q |w − `| φ |Dw| χˆδ B4(0) Z 2 2q−2 2α(κq−1)−2 2 2 2q−2 2α(κq−1) 2 + C q |w − `| φ |Dφ| χˆδ + |w − `| φ |Dχˆδ| . B4(0)

Since Z Z 2q−2 2α(κq−1) 2 2q−2 2α(κq−1) 2 |w − `| φ |Dχˆδ| ≤ sup |w − `| φ |Dχˆδ| B4(0) B4(0) B4(0) C ≤ sup |w − `|2q−2φ2α(κq−1) · , B4(0) − log(δ) letting δ ↓ 0 yields Z |w − `|2q−2|Dw|2φ2α(κq−1) B4(0) Z ≤ C |w − `|2q−2φ2α(κq−1) + q2|w − `|2q−4φ2α(κq−1)|Dw|2 B4(0) Z + C q2|w − `|2q−2φ2α(κq−1)−2|Dφ|2. (2.34) B4(0) where C = C(Λ/λ, ν/λ, α) > 0. By Young’s Inequality, the deﬁnition of φ, and (2.33),

Z 1 Z q2|w − `|2q−4φ2α(κq−1)|Dw|2 ≤ Cq2q + |w − `|2q−2φ2α(κq−1), B4(0) 2 B4(0) so by Young’s Inequality and the deﬁnition of φ, (2.34) yields

Z Z |w − `|2q−2|Dw|2φ2α(κq−1) ≤ Cq2q + C |w − `|2qφ2α(κq−1)−2. B4(0) B4(0) CHAPTER 2. ELLIPTIC THEORY FOR TWO-VALUED FUNCTIONS 55

By the Sobolev Inequality,

1/κ Z 2κ Z |w − `|qφα(κq−1) ≤ Cq2q + C |w − `|2qφ2α(κq−1)−2. B4(0) B4(0)

Letting α = 1/(κ − 1),

Z 1/κ Z (|w − `|φακ)2κq φ−2ακ ≤ Cq2q + C (|w − `|φακ)2q φ−2ακ. B4(0) B4(0)

Thus for i ≥ 1, letting q = κi and using (a + b)1/q ≤ a1/q + b1/q for a, b ≥ 0,

Ψ(i + 1) ≤ Cκ2i + CΨ(i). (2.35) where 1/κi Z i Ψ(i) = (|w − `|φακ)2κ φ−2ακ . B4(0)

i0−1 and C = C(Λ/λ, ν/λ, n) > 0. Iterating over i we get for i ≥ i0, where κ ≥ 2,

i−1 X C Ψ(i) ≤ C κ2j + CΨ(i ) ≤ κ2i + CΨ(i ). (2.36) 0 κ2 − 1 0 j=i0

By (2.35) either Ψ(i0) ≤ C or Ψ(i0) ≤ CΨ(i0 − 1). If Ψ(i0) ≤ CΨ(i0 − 1), by using the H¨older’sinequality with respect to the measure φ−2ακdX implies that for every ε ∈ (0, 1),

Z 1/κi0−1 ακ 2κi0−1 −2ακ Ψ(i0) ≤ C (|w − `|φ ) φ B4(0) (1−ε)/κi0 1/s Z i Z ≤ C (|w − `|φακ)2κ 0 φ−2ακ (|w − `|φακ)2sε φ−2ακ B4(0) B4(0) where s = κi0 /(κ − 1 + ε) and thus

Z 1/sε 1/ε ακ 2sε −2ακ Ψ(i0) ≤ C (|w − `|φ ) φ . B4(0) CHAPTER 2. ELLIPTIC THEORY FOR TWO-VALUED FUNCTIONS 56

Since we could have either Ψ(i0) ≤ C or Ψ(i0) ≤ CΨ(i0 − 1),

Z 1/sε 1/ε ακ 2sε −2ακ Ψ(i0) ≤ C + C (|w − `|φ ) φ . B4(0)

Choosing ε so that εs = 1, Z 2 Ψ(i0) ≤ C + C |w − `| . (2.37) B4(0)

By Poincar´einequality Lemma 8, for some ` ∈ R, Z Z |w − `|2 ≤ C |Dw|2. (2.38) B4(0) B4(0) so Ψ(i0) ≤ C by (2.33). Hence by (2.36), (2.37), and (2.38) for integers i ≥ 1,

Ψ(i) ≤ Cκ2i.

By the deﬁnitions of Ψ(i) and φ,

Z 1/κi |w − `|2κi ≤ Cκ2i. (2.39) B3(0)

Thus for every integer j ≥ 1, choosing i such that κi−1 < j/2 ≤ κi in (2.39) and using H¨older’sinequality, Z 1/j |w − `|j ≤ Cj. B3(0) z/e P∞ j j Using e ≤ 1 + j=1 z /j with z = |wˆ − `|/2, Z ep|w−`| ≤ C, where p = 1/2e. B3(0)

So Z Z ep(w−`) ≤ C and e−p(w−`) ≤ C. B3(0) B3(0) CHAPTER 2. ELLIPTIC THEORY FOR TWO-VALUED FUNCTIONS 57

Thus Z Z ep(w−`) e−p(w−`) ≤ C. B3(0) B3(0) Using w = log(¯u), we obtain (2.32); that is,

Z Z u¯p u¯−p ≤ C. B3(0) B3(0) for some C = C(Λ/λ, ν/λ, n, p) > 0.

Theorem 16. Let u, aij, bi, cj, d, f i, and g be the two-valued functions associated ij i j i 0 1,2 with u0, a0 , b0, c0, d0, f0, g0 ∈ V (BR0 (0)) respectively. Suppose u ∈ W (BR0 (0)). Suppose

ij i j i ˘ n−2 Di(a Dju + b u) + c Dju + du ≤ Dif + g in BR0 (0) \{0} × R , (2.40) which we interpret as discussed in Section 1.3, and

ij iθ 2 2 2iθ n a0 (re , y)ζiζj ≥ λ|ζ| whenever (r e , y) ∈ BR0 (0), ξ ∈ R , (2.41) ij i j 2 ka k 0 ≤ Λ, kb k 0 + Rkc k 0 + R kdk 0 ≤ ν, C (BR0 (0)) C (BR0 (0)) C (BR0 (0)) C (BR0 (0)) for some constants λ, Λ, ν > 0. Then for some constants µ ∈ (0, 1/2) and C > 0 depending on n, λ, Λ, and R0ν, ! µ 2 R [u]µ,B (0) ≤ C sup |u| + R0kfkC0(B (0)) + R kgkC0(B (0)) . (2.42) 0 R0/2 R0 0 R0 BR0 (0)

Proof. Rescale so that R0 = 1. Arguing as in the proof of Theorem 8.22 in [3], replacing the weak Harnack inequality [3, Theorem 8.18] with Theorem 15, we obtain

! ! µ µ oscBR(0,y0) u ≤ CR sup u + k ≤ CR sup u + k , (2.43) B1/2(0,y0) B1(0) where −1 −1 0 0 k = λ kfkC (B1(0)) + λ kgkC (B1(0)), CHAPTER 2. ELLIPTIC THEORY FOR TWO-VALUED FUNCTIONS 58

n−2 for all y0 ∈ B1/2 (0), R ∈ (0, 1/2] and for some C > 0 and µ ∈ (0, 1/2) depending on n,Λ/λ, and R0ν/λ.

Let X1 and X2 be distinct points in B1/2(0). Without loss of generality, suppose n−2 0 ≤ |x1| ≤ |x2|. If |X1 − X2| < |x2|/2, then B2|X1−X2|(X2) is disjoint from {0} × R , so by the H¨oldercontinuity estimates for single-valued functions [3, Theorem 8.22] applied on B2|X1−X2|(X2),

! τ G(u(X1), u(X2)) ≤ C|X1 − X2| sup |u| + k , (2.44) B1(0) where G denotes the metric on the space of unordered pairs from Section 1.2, for some constants C = C(n, Λ/λ, R0ν/λ) > 0 and τ = τ(n, Λ/λ, R0ν/λ) ∈ (0, 1/2). Suppose

µ ≤ τ. If |x2|/2 ≤ |X1 − X2| ≤ |x2|, by (2.43), ! G(u(X ), u(X )) ≤ 2 osc u ≤ C|x |µ sup |u| + k 1 2 B2r2 (0,y2) 2 B1(0) ! µ ≤ C|X1 − X2| sup |u| + k . B1(0)

Finally we must consider |X1 − X2| ≥ |x2|. First note that if x1 = x2 = 0, by (2.43), ! µ G(u(X1), u(X2)) ≤ 2 oscB (0,y¯) u ≤ C|X1 − X2| sup |u| + k 3|X1−X2|/4 B1(0) wherey ¯ = (y1 + y2)/2. Hence for |X1 − X2| ≥ |x2|, we have

G(u(X1), u(X2)) = G(u(X1), u(0, y1)) + G(u(0, y1), u(0, y2)) + G(u(0, y2), u(X2)) ! µ µ µ ≤ C (|x1| + |y1 − y2| + |x2| ) sup |u| + k B1(0) ! µ ≤ 3C|X1 − X2| sup |u| + k . B1(0) CHAPTER 2. ELLIPTIC THEORY FOR TWO-VALUED FUNCTIONS 59 Chapter 3

Existence theorems

3.1 Overview

In this chapter we will prove the existence results, Theorems 3 and 4. The development of an existence theory for two-valued functions is similar to the development of the existence theory for single-valued functions. In Section 3.2 we will solve the Dirichlet problem for the Poisson equation using Fourier analysis. Then using this result and the elliptic theory from the previous section, we will prove Theorem 3 in Section 3.3 and Theorem 4 in Section 3.4, which establish the existence of solutions to Dirichlet problems for certain classes of quasilinear elliptic systems and equations.

3.2 Poisson equation

Recall from Section 1.1 that for φ ∈ [0, 2π), Rφ is the n × n matrix such that iθ i(θ+φ) n−2 Rφ(re , y) = (re , y) for r ≥ 0, θ ∈ [0, 2π), y ∈ R . Let k ≥ 3 be an odd i integer and let φ = 2π/k. Write R2π/k = (Rj) where the i-th component of R2π/kX i j is RjX . We want to prove the following theorem.

i i Theorem 17. Let f , g, and ϕ be the two-valued functions associated with f0 ∈ 0,µ 0 0,µ i V (C), g0 ∈ V (C), and ϕ0 ∈ V (∂C) respectively. Suppose f , g, and ϕ are

60 CHAPTER 3. EXISTENCE THEOREMS 61

periodic with respect to yj with period ρj and satisfy

j j l f (R2π/kX) = Rl f (X), g(R2π/kX) = g(X), ϕ(RX) = ϕ(X) (3.1)

i for all X ∈ C, where R2π/k = (Rj) is as in Section 1.1. Then there is a unique 0 1,µ u0 ∈ V (C) ∩ Vloc (C) such that that if u is the two-valued function u associated with u0 then

j n−2 ∆(x,y)u = Djf + g in C\{0} × R ,

u0 = ϕ0 on ∂C, (3.2) which we interpret as discussed in Section 1.3. Moreover, u is periodic with respect to yj with period ρj and u has k-fold symmetry, i.e. u(R2π/kX) = u(X) for all X ∈ C.

Observe that if a solution u to (3.2) exists, then uniqueness immediately follows by maximum principle Theorem 9 from Section 2.2. Also, we can reduce to the case where u, f j, g, and ϕ are all two-valued symmetric as follows. Recall from Section 1.2 that we can write u = ua + us where ua(X) = (u1(X) + u2(X))/2 is the average part of u and us(X) = {±(u1(X)−u2(X))/2} is the two-valued symmetrical part of u. By expressing u, f j, g, and ϕ in terms of their single-valued and two-valued symmetric parts and using linearity, it suﬃces to consider the cases where u, f j, g, and ϕ are either all single-valued or all two-valued symmetric. The existence of single-valued solutions to the Poisson equation is well known [3, Theorem 8.3], so it suﬃces to only consider the case where u, f, g, and ϕ are two-valued symmetric.

0 We shall use the following notation. Suppose v0 ∈ V (Ω), where Ω = C or Ω = ∂C, and let v be the two-valued function associated with v0. Let Π = ρ1Z × · × ρn−2Z.

We will write the Fourier series of v0 with respect to the y variable as Z X il·y −il·y v0(x, y) = v˜0,l(x)e where v0,l(x) = e v0(x, y)dy. n−2 l∈Π R CHAPTER 3. EXISTENCE THEOREMS 62

(m) 0 For each positive integer m, we will let v0 ∈ V (Ω) denote the partial sum

(m) X il·y v0 = v˜0,l(x)e . l∈Π,|l|≤m

(m) (m) and we will let v denote the two-valued function associated with v0 . For now j suppose that f = 0 and that g0 and ϕ0 are smooth single-valued functions. Conse- quently, −2 −2 |g0,l(ξ)| ≤ C(1 + |l1|) ··· (1 + |ln−2|) kg0kC2(C)

(m) (m) for some constant C = C(n) > 0 and thus g0 → g0 uniformly, so g → g uniformly. Similarly ϕ(m) → ϕ uniformly. We want to solve

(m) (m) n−2 ∆u = g in C\{0} × R , (m) (m) u0 = ϕ0 on ∂C, (3.3) from which we will obtain the solution u to (3.2) as a limit of the u(m). Recall from 1/2 Section 1.3 that we interpret (3.3), using the change of variables ξ1+iξ2 = (x1+ix2) , to mean

(m) 2 (m) 2 (m) n−2 ∆ξu0 + 4|ξ| ∆yu0 = 4|ξ| g0 in C\{0} × R , (m) (m) u0 = ϕ0 on ∂C. (3.4)

(3.4) is equivalent to

2 2 2 ˘2 ∆ξu0,l − 4|ξ| |l| u0,l = 4|ξ| g0,l in B1 (0), 2 u0,l = ϕ0,l on ∂B1 (0), (3.5) for all l ∈ Π. By standard elliptic theory [3, Theorem 6.14], there exists a unique ∞ 2 solution u0,l ∈ C (B1 (0)) to (3.5). By maximum principle Theorem 8 from Section CHAPTER 3. EXISTENCE THEOREMS 63

2.2, ul is two-valued symmetric with k-fold symmetry since if we replace ul with

2k−1 1 X (−1)ju (reiθ+jπ/k, y) 2k 0,l j=0 in (3.5), by (3.1) u0,l still solves (3.5). Since u0,l is smooth near 0 and ul is two-fold α symmetric with k-fold symmetry, D u0,l(0) = 0 for |α| = 0, 1, 2. By the Schauder estimates, for all µ ∈ (0, 1/2),

2 2 2 2+2µ |u (ξ)| + |ξ||Du (ξ)| + |ξ| |D u (ξ)| ≤ [D u ] 2 |ξ| 0,l 0,l 0,l 0,l 2µ,B1 (0) 2+2µ ≤ C kg k 0,2µ 2 + kϕ k 2,2µ 2 |ξ| , 0,l C (B1 (0)) 0,l C (∂B1 (0) for some constant C = C(n, µ, |l|) > 0, hence

2 2 1+µ |u (x)| + |x||Du (x)| + |x| |D u (x)| ≤ C kg k 0,2µ 2 + kϕ k 2,2µ 2 |x| . l l l 0,l C (B1 (0)) 0,l C (∂B1 (0)

1,µ for some constant C = C(n, µ, |l|) > 0. Thus ul ∈ C (B1(0)) for all l ∈ Π and µ ∈ (0, 1/2) with

ku k 1,µ 2 ≤ C kg k 0,2µ 2 + kϕ k 2,2µ 2 . l C (B1 (0)) 0,l C (B1 (0)) 0,l C (∂B1 (0) for some constant C = C(n, µ, |l|) > 0. Therefore u(m) ∈ C1,µ(C) for all m and all µ ∈ (0, 1/2). By the Schauder estimate Corollary 3 from Section 2.4,

(m) (m) (m) ku kC1,µ(C) ≤ C kg kC0(C) + kϕ kC1,µ(∂C)

(m) for some constant C = C(n, µ) > 0, so after passing to a subsequence of {u0 }, also (m) (m) 1 1,µ denoted by {u0 }, u0 → u0 in V (C) for some u0 ∈ V (C). Moreover, if u is the two-valued function associated with u0, then u and u0 satisfy (3.2).

If g0 and ϕ0 are merely continuous, then using mollification we can find smooth functions g0,m and ϕ0,m that are periodic with respect to yj with period ρj using mollification such that g0,m → g0 and ϕ0,m → ϕ0 uniformly as m → ∞. Let gm and CHAPTER 3. EXISTENCE THEOREMS 64

ϕm denote the two-valued functions associated with g0,m and ϕ0,m respectively. By using rotationally symmetric molliﬁers to construct g0,m and ϕ0,m, we may suppose gm and ϕm are two-valued symmetric with k-fold symmetry. We can solve (3.2) if we replace g by gm and ϕ by ϕm to obtain a two-valued solution um that is associated 0 1,µ with some u0,m ∈ V (C) ∩ V (C). Then by Theorem 13 from Section 2.4,

ku0,m − u0,m0 kV0(C) ≤ C kϕ0,m − ϕ0,m0 kV0(∂C) + kg0,m − g0,m0 kV0(C)

0 0 for integers m, m ≥ 1, so {u0,m} is Cauchy in the subspace of V (C). Thus {u0,m} 0 converges to some u0 ∈ V (C) such that u0 = ϕ0 on ∂C. Let u be the two-valued function associated with u0. By the local Schauder estimates in Theorem 10 from 1,µ Section 2.3, we can show {u0,m} is bounded in V on compact subsets of C and thus 1 1,µ we can pass to a subsequence of {u0,m} that converges in Vloc(C) to u0 ∈ Vloc (C) such that ∆u = g on C\{0} × Rn−2.

If f j 6= 0, we ﬁrst approximate f j by smooth functions.

0,µ Lemma 9. Let f0 ∈ V (C) and f be the two-valued function associated with f0.

Then for every > 0 there is a smooth function f0, on C such that f0, = 0 on n−2 an open neighborhood of {0} × R and, letting f denote the two-valued function associated with f0,,

kf0 − f0,εkV0(C) < ε, [fε]µ,C ≤ C[f]µ,C (3.6) for some constant C = C(n, µ) > 0. Moreover, if f is k-fold symmetric and is periodic with respect to yj with period ρj, then f also has k-fold symmetry and is periodic with respect to yj with period ρj.

Proof. By an extension theorem for single-valued functions in C1,µ [3, Lemma 6.37], 1,µ n we can extend f to a function in C (R ) such that [f]µ,Rn ≤ C[f]µ,C. Let χδ ∈ ∞ C ([0, 1]) be a smooth function such that 0 ≤ χδ ≤ 1, χδ(r) = 0 if r ≤ δ/2, χδ(r) = 1 if r ≥ δ, and |Dχδ| ≤ 3/δ. We can regard χδ as a function of (x, y) ∈ C that depends CHAPTER 3. EXISTENCE THEOREMS 65

−µ j 0,µ only on |x|. Observe that [χδ]µ,C ≤ 3δ . Consider χδf0 ∈ V (C) and the two- j j valued function χδf associated with χδf0 . For X = (x, y) ∈ C, χδ(X)f0(X) = f0(X) if |x| ≥ δ and

µ |f0(X) − χδ(X)f0(X)| ≤ |1 − χδ(X)||f0(X)| ≤ |f0(x, y) − f0(0, y)| ≤ [f]µ,Cδ

if |x| ≤ δ, so χδf converges to f uniformly. Let X1 = (x1, y1),X2 = (x2, y2) ∈ C. If

|x1|, |x2| ≥ δ,

µ G(χδ(X1)f(X1), χδ(X2)f(X2)) = G(f(X1), f(X2)) ≤ [f]µ,C|X1 − X2| , where G denotes the metric on the space of unordered pairs from Section 1.2, and if

|x1| ≤ δ,

G(χδ(X1)f(X1), χδ(X2)f(X2))

≤ |χδ(X1) − χδ(X2)|G(f(X1), 0) + |χδ(X2)|G(f(X1), f(X2)) −µ µ µ µ ≤ 3δ |X1 − X2| · [f]µ,Cδ + 1 · [f]µ,C|X1 − X2| µ ≤ 4[f]µ,C|X1 − X2| .

Therefore [χδf]µ,Rn ≤ 4C[f]µ,C.

Let ε > 0. For some δ > 0, kf0 − χδf0kV0(C) < ε/2. Let {ψσ}σ>0 be a family n −n of smooth rotationally symmetric mollifiers on R such that ψσ(X) = σ ψ1(σX), 0,µ and spt ψ1 ⊂ B1(0). We define f0, ∈ V (C) as follows. Let σ ∈ (0, δ/4), r0 ∈ n−2 [0, 1], θ0 ∈ [0, 2π), and y0 ∈ R . If r0 ≥ δ/4, we can take the single-valued iθ 1/2 iθ/2 branch (χδf)(re , y) = (χδf0)(r e , y) of χδf where |θ − 2θ0| < π/2 and define

iθ0 f0,ε(r0e , y0) to be the value of convolution of ψσ and the single-valued branch of χδf

2 2iθ0 iθ0 at (r0e , y0). If r0 ≤ δ/4, let f0,ε(r0e , y0) = 0. Choosing σ ∈ (0, δ/2) suﬃciently small, kχδf0 − fεkC0(C) < ε/2. It follows that (3.6) is true.

j j Now replace f with zero and g with Djfε + g in (3.2) and ﬁnd a solution uε to CHAPTER 3. EXISTENCE THEOREMS 66

(3.2). By Theorem 13 from Section 2.4,

ku0,ε − u0,ε0 kV0(C) ≤ Ckf0,ε − f0,ε0 kV0(C) (3.7)

0 for ε, ε > 0. By the local Schauder estimates Theorem 10 for Section 2.3, {u0,ε} are 1,µ bounded in C on compact subsets of C. Thus for some sequence εm ↓ 0, {u0,εm } 0 1 0 1,µ converges in both the V (C) and Vloc(C) topologies to some u0 ∈ V (C) ∩ Vloc (C) that the two-valued function u associated with u0 solves (3.2) for the original f, g and ϕ.

Finally we want to remove the condition that f j, g, and ϕ are periodic.

i i Corollary 4. Let f , g, and ϕ be the two-valued functions associated with f0 ∈ 0,µ 0 0 V (C), g0 ∈ V (C), and ϕ0 ∈ V (∂C) respectively. Suppose g and ϕ are bounded and

j j l f (R2π/kX) = Rl f (X), g(R2π/kX) = g(X), ϕ(RX) = ϕ(X)

i for all X ∈ C, where R2π/k = (Rj) is as in Section 1.1. Then there is a u0 ∈ 0 1,µ V (C) ∩ Vloc (C) such that the two-valued function u associated with u0 satisﬁes

j n−2 ∆(x,y)u = Djf + g in C\{0} × R , u = ϕ on ∂C.

Moreover, u has k-fold symmetry and

sup |u| ≤ C kϕkC0(∂C) + [f]µ,C + kgkC0(C) . (3.8) C for some constant C = C(n, µ) > 0.

j j 0,µ 0 Proof. Approximate f0 , g0, and ϕ0 by f0,m ∈ V (C), g0,m ∈ V (C), and ϕ0,m ∈ 0 V (∂C) respectively such that the following holds. Let fm, gm, and ϕm denote the two- j valued functions associated with f0,m, g0,m, and ϕ0,m, respectively. We shall choose j f0,m, g0,m, and ϕ0,m to be k-fold symmetric and period with respect to yj with period j j 0 ρm, where ρm → ∞. As m → ∞, f0,m → f0 and g0,m → g0 in Vloc(C) and ϕ0,m → ϕ0 CHAPTER 3. EXISTENCE THEOREMS 67

0 in Vloc(∂C) with ϕm = ϕ on each compact subset K ⊂ ∂C for m suﬃciently large j j depending on K. Finally we will require that kfmkC0,µ(C) ≤ Ckf kC0,µ(C) for some constant C = C(n, µ) > 0, kgmkC0(∂C) ≤ kgkC0(∂C), and kϕmkC0(∂C) ≤ kϕkC0(∂C). By 0 1,µ Theorem 17 from Section 3.2, there exists a unique u0,m ∈ V (C) ∩ Vloc (C) such that, letting um denote the two-valued function associated with u0,m, u0,m and um satisfy j j (3.2) if we replace f by fm, g by gm, and ϕ by ϕm. By Theorem 14 from Section 2.4, sup |um| ≤ C kϕkC0(∂C) + [f]µ,C + kgkC0(C) (3.9) C for some constant C > 0 depending on n and m and independent of the period n−2 of ρmZ . By the local Schauder estimates Theorem 10 from Section 2.2, {u0,m} is uniformly bounded in V1,µ on compact subsets of C. Thus, after passing to 1 1,µ a subsequence, {u0,m} converges in Vloc(C) to some u0 ∈ Vloc (C) such that the i two-valued function u associated with u0 is a weak solution to ∆u = Dif + g in n−2 C\{0}×R . By letting m → ∞ in (3.9), (3.8) holds. Fix (x0, y0) ∈ ∂C and ﬁx single- iθ 1/2 iθ/2 iθ 1/2 iθ/2 valued branches of um(re , y) = u0,m(r e , y), fm(re , y) = f0,m(r e , y), iθ 1/2 iθ/2 iθ iθ/2 gm(re , y) = g0,m(r e , y), and ϕm(e , y) = ϕ0,m(e , y) with |θ − θ0| < π/2,

iθ0 where x0 = e . As a slight abuse of notation, we such use um, fm, gm, and ϕm to denote these single-valued branches of um, fm, gm, and ϕm respectively. By [3, Theorem 8.27], for 0 < R ≤ 1/2 and m suﬃciently large,

α osc u ≤ CR ku k 0 + kf k 0 C∩BR(x0,y0) m m C (C∩BR(x0,y0)) m C (B1/4(x0,y0)) 0 +kgmkC (B1/4(x0,y0)) + osc∂C∩BR(x0,y0) ϕm

α 0 0 ≤ CR kϕkC (∂C) + kfkC0,µ(C) + kgkC (∂C) + osc∂C∩BR(x0,y0) ϕ

for some C > 0 and α ∈ (0, 1) depending on n and µ. For (x, y) ∈ C ∩ BR(x0, y0),

|um(x, y) − ϕ(x0, y0)| ≤ oscC∩BR(x0,y0) um for m suﬃciently large. Hence u extends to a continuous function on C such that u = ϕ on ∂C. CHAPTER 3. EXISTENCE THEOREMS 68

3.3 Elliptic systems

In this section we will prove Theorem 3, which involves solving the Dirichlet problem

κ i n−2 ∆u = Di(Fκ(Du)) + Gκ(u, Du) weakly in C\{0} × R , κ = 1, . . . , m,

u0 = ϕ0 on ∂C, (3.10) where u and ϕ are two-valued functions with k-fold symmetry that are associated 1,µ m 1,µ m with u0 ∈ V (C, R ) and ϕ0 ∈ V (∂C, R ) respectively and we interpret (3.10) as discussed in Section 1.3. First we will consider the case where u0 and ϕ0 are 1,µ m periodic with respect to yj with period ρj > 0. Let W (C, R ) denote the space 1,µ m of u0 ∈ V (C, R ) that are periodic with respect to yj with period ρj such that the two-valued function u associated with u0 has k-fold symmetry. Deﬁne the map 1,µ m 1,µ m T : W (C, R ) → W (C, R ) by letting u0 = T v0 if, letting u and v denote the the two-valued functions associated with u0 and v0,

j n−2 ∆u = DjFκ (Dv) + Gκ(v, Dv) in C\{0} × R , κ = 1, . . . , m,

u0 = ϕ0 on ∂C in the sense discussed in Section 1.3. Recall from Section 2.2 that by maximum principle Theorem 9, u0 is unique and thus T is well-deﬁned. Suppose u0,1 = T v0,1 1,µ m and u0,2 = T v0,2 for v0,1, v0,2 ∈ (∂C, R ) such that kv0,1kV1,µ(C), kv0,2kV1,µ(C) ≤ ε.

Let u1, u2, v1, and v2 denote the two-valued functions associated with u0,1, u0,2, v0,1, and v0,2 respectively. Then by the Schauder estimate Corollary 3,

ku0,1kV1,µ(C) ≤ C [F (Dv1)]µ,C + kG(v1, Dv1)kC0(C) + kϕkC1,µ(∂C) , (3.11) ku0,1 − u0,2kV1,µ(C) ≤ C [F (Dv1) − F (Dv2)]µ,C +kG(v1, Dv1) − G(v2, Dv2)kC0(C) , (3.12)

where C > 0 depends on n, m, and µ and is independent of ρ1, . . . , ρn−2. Note that this independence of ρ1, . . . , ρn−2 necessary for later removing the condition that ϕ0 CHAPTER 3. EXISTENCE THEOREMS 69

be periodic. Deﬁne σ : (0, ∞) → (0, ∞) deﬁned by

σ(t) = sup |DG(P )| |Z|+|P |≤ε

2 m×n and note that limt↓0 σ(t) = 0. Using the fact that F ∈ C (R ) with DF (0) = 0 and G ∈ C1(Rm × Rm×n) with G(0) = 0 and using Theorem 3 and the properties of hµ,C from Section 1.3, we obtain

2 [F (Dv1)]µ,C ≤ C sup |DF (P )| [Dv1]µ,C ≤ Cε , kG(v1, Dv1)kC0(C) ≤ Cσ(ε)ε, |P |≤ε

2 for some constant C > 0 depending on n, m, and sup|Z|+|P |≤1 |D F (P )|, so (3.11) yields 2 ku1kC1,µ(C) ≤ C(σ(ε) + )ε ≤ ε

2 if C(σ(ε) + ) ≤ 1, where C > 0 depends on n, m, and sup|Z|+|P |≤1 |D F (P )|. Using the fact that G ∈ C1(Rm × Rm×n) with G(0) = 0,

kG(v1, Dv1) − G(v2, Dv2)kC0(C) ≤ C sup |DG(P )| kv1 − v2kC1(C) |Z|+|P |≤ε

≤ Cσ(ε)kv0,1 − v0,2kV1(C) (3.13)

for some constant C = C(n, m) > 0, where we interpret v1−v2 as being the two-valued iθ function associated with v0,1 − v0,2. Let X1,X2 ∈ {(re , y) ∈ C : |θ − θ0| < π/2} iθ for some θ0 ∈ R and restrict v1 and v2 to their single-valued branches v1(re , y) = 1/2 iθ/2 iθ 1/2 iθ/2 v0,1(r e , y) and v2(re , y) = v0,2(r e , y) where |θ − θ0| < π/2. As a slight abused of notation, we shall denote these single-valued branches of v1 and v2 by v1 CHAPTER 3. EXISTENCE THEOREMS 70

2 m×n and v2 respectively. Using the fact that F ∈ C (R ) with DF (0) = 0,

|F (Dv2(X2)) − F (Dv1(X2)) − F (Dv2(X1)) + F (Dv1(X1))| Z 1 Z 1 2 ∂ ≤ C F ((1 − s)(1 − t)Dv1(X1) + (1 − s)tDv1(X2) 0 0 ∂s∂t

+s(1 − t)Dv2(X1) + stDv2(X2))| ds dt

2 ≤ C sup |D F (P )| [Dv1]µ,C + [Dv2]µ,C kDv2 − Dv1kC0(C) |p|<ε ! µ + sup |DF (P )| [Dv2 − Dv1]µ,C |X1 − X2| |p|<ε

µ ≤ Cεkv0,2 − v0,1kV1,µ(C) |X1 − X2| ,

2 where C > 0 depends on n, m, and sup|Z|+|P |≤1 |D F (P )|. Hence, using Theorem 3 and the deﬁnition of hµ,C from Section 1.3,

[F (Dv1) − F (Dv2)]µ,C ≤ Cεkv0,1 − v0,2kV1,µ(C), (3.14) for C = C(m, n, ν) > 0. Combining (3.12), (3.13), and (3.14)

1 ku − u k 1,µ ≤ C(σ(ε) + ε)kv − v k 1,µ < kv − v k 1,µ 0,1 0,2 V (C) 0,0 0,1 V (C) 2 0,1 0,2 V (C) if C(σ(ε) + ε) ≤ 1/2, where C = C(m, n, µ, ν) > 0, so T is a contraction mapping. Therefore by the contraction mapping principle, for some ε = ε(m, n, µ, ν) > 0, there 1,µ m exists a ﬁxed point u0 ∈ W (C, R ) of T with ku0kV1,µ(C) ≤ ε. In other words, the two-valued function u associated with u0 solves (3.10), where (3.10) is interpreted as discussed in Section 1.3. Consequently u satisﬁes

κ i n−2 ∆u = Di(Fκ(Du)) + Gκ(u, Du) weakly in C\{0} × R , κ = 1, . . . , m, u = ϕ on ∂C, in the sense discussed in Section 1.2 for two-valued functions.

Now we want to remove the condition that ϕ0 is periodic. Approximate ϕ0 by CHAPTER 3. EXISTENCE THEOREMS 71

1,µ m ϕ0,l ∈ V (∂C, R ) that are periodic with respect to yj with period ρl, where ρl → ∞, the two-valued function ϕl associated with ϕ0,l has k-fold symmetry, ϕ0,l → ϕ0 in 1,µ m Vloc (∂C, R ), and kϕ0,lkV1,µ(∂C) ≤ Ckϕ0kV1,µ(∂C) for some constant C = C(n, m, µ) > 1,µ m 0. Provided kϕ0kV1,µ(∂C) < ε/C, there exists a unique u0,l ∈ V (C, R ) such that the two-valued function ul associated with u0,l is a solution to (3.10) if we replace

ϕ by ϕl. Since the u0,l constructed above satisfy ku0,lkV1,µ(C) ≤ ε, after passing to 1 m 1,µ m a subsequence, {u0,l} converges in Vloc(C, R ) to some u0 ∈ Vloc (C, R ) such that the two-valued function u associated with u0 satisﬁes (3.10) with the original ϕ and ku0kV1,µ(C) ≤ ε.

Finally we will establish the stability of the solutions to the minimal surface system constructed in Theorem 3. Let Σu = graph u. We want to show that

Z n n ! X ⊥ 2 X 2 |(Dτi X) | − |X · A(τi, τj)| ≥ 0, (3.15) Σu i=1 i,j=1

0 n+m 1,2 n+m for all normal vector ﬁelds X ∈ Cc (Σu, R ) ∩ W (Σu, R ), where we regard Σu as an immersed submanifold. Observe that in the single-valued case, u would be C2 1 n+m and thus we would suppose X ∈ Cc (Σu, R ). However, u in the two-valued case 1,µ is merely C , so we must allow X to not be continuously diﬀerentiable along Σu ∩ n−2 m 1 n−2 m n+m {0}×R ×R . First we claim (3.15) holds if X ∈ Cc (Σu \{0}×R ×R , R ). This follows from the fact that

2 Z d p κ λ Area(Σu+tζ ) = D κ λ det G(P ) Diζ Djζ dX 2 Pi Pj dt C P =Du(X)+tDζ(X) Z ≥ (1 − C) |Dζ|2 dX ≥ 0 C

1 n−2 m for t ∈ [0, 1] and for any two-valued function ζ ∈ Cc (C\{0} × R , R ) of the form (1.14) with |Dζ| ≤ 4, where C = C(n, m) > 0. Arguing as we did in Section 1.3 when showing the graphs of single-valued graphs with small gradient are stable, it follows 1 n−2 m n+m that (3.15) holds for normal vector ﬁelds X ∈ Cc (Σu \{0} × R × R , R ). By 1,2 n−2 m n+m 1 approximating X ∈ W0 (Σu \{0}×R ×R , R ) by vector ﬁelds in Cc (Σu \{0}× CHAPTER 3. EXISTENCE THEOREMS 72

n−2 m n+m 1,2 n−2 R ×R , R ), we obtain (3.15) for normal vector fields X ∈ W0 (Σu\{0}×R × m n+m 0 n+m 1,2 n+m R , R ). In (3.15), replace X byχ ˆδX where X ∈ Cc (Σu, R ) ∩ W (Σu, R ) 1 m is any normal vector field andχ ˆδ ∈ C (C × R ) is the logarithmic cutoff function given by   0 if r ≤ δ2  iθ 2 2 χˆδ(re , y, Z) = − log(r/δ )/ log(δ) if δ < r < δ   1 if r ≥ δ.

By (3.15),

Z n n ! X ⊥ 2 X 2 2 |χˆδ(Dτi X) + Dτi χˆδX| − |X · A(τi, τj)| χˆδ ≥ 0, (3.16) Σu i=1 i,j=1 so

Since Z n X 2 lim |Dτ χˆδ| = 0, δ↓0 i Σu i=1 we have

Z n Z n X 2 2 X ⊥ 2 2 |X · A(τi, τj)| χˆδ ≤ 2 |(Dτi X) | + 2 sup |X| Σu i,j=1 Σu i=1 for δ suﬃciently small and thus we can let δ ↓ 0 in (3.16) to obtain (3.15) for all 0 n+m 1,2 n+m normal vector ﬁelds X ∈ Cc (Σu, R ) ∩ W (Σu, R ), regardless of whether X n−2 m vanishes near Σu ∩ {0} × R × R . CHAPTER 3. EXISTENCE THEOREMS 73

3.4 Elliptic equations

In this section we will prove Theorem 4, which involves solving the Dirichlet problem

i n−2 Qu = Di(A (Du)) + B(u, Du) = 0 in C\{0} × R ,

u0 = ϕ0 on ∂C, (3.17) where u and ϕ are two-valued functions with k-fold symmetry that are associated 0 1,µ 2 with u0 ∈ V (C) ∩ Vloc (C) and ϕ0 ∈ V (∂C) respectively and we interpret (3.17) as discussed in Section 1.3. It will be convenient to also write (3.17) as

i n−2 Qu = (DPj A )(Du)Diju + B(u, Du) = 0 in C\{0} × R , u = ϕ on ∂C, which we interpret as discussed in Section 1.3. The proof will involve using the Leray- 2,µ Schauder theory. For now suppose that ϕ0 ∈ V (C) and ϕ0 is periodic with respect 1,µ to yj with period ρj. Recall from Section 3.3 that for µ ∈ (0, 1/2), W (C) is the 1,µ space of u0 ∈ V (C) that are periodic with respect to yj with period ρj such that the two-valued function u associated with u0 has k-fold symmetry. We will deﬁne a 1,µ 1,µ map T : W (C) → W (C) by letting u0 = T v0 if, letting u and v denote the the two-valued functions associated with u0 and v0,

i n−2 (DPj A )(Dv)Diju + B(v, Dv) = 0 in C\{0} × R ,

u0 = ϕ0 on ∂C, (3.18) which we interpret as discussed in Section 1.3. For T to be well-deﬁned, there must 1,µ 2 n−2 exist a unique u0 ∈ V (C) ∩ V (C\{0} × R ) such that u0 and the two-valued function u associated with u0 satisfy (3.18). Uniqueness follows by maximum principle Theorem 8 from Section 2.2. Existence will follow from Lemma 10 below. Lemma 10 and the Schauder estimate in Theorem 12 from Section 2.3 shows that T is in fact a continuous map from W1,µ(C) into W1,τ (C) for τ ∈ (µ, 1/2), so T is compact. We 1,µ will need to show that if u0 ∈ W (C) and the two-valued function u associated with CHAPTER 3. EXISTENCE THEOREMS 74

u0 satisfy

i n−2 Di(A (Du)) + σB(u, Du) = 0 in C\{0} × R ,

u0 = σ0ϕ on ∂C, (3.19)

i then ku0kV1,µ(C) ≤ C for some µ ∈ (0, 1/2) and C > 0 depending on n, A , B, and kϕ0kV2(∂C). Then by the Leray-Schauder theory, there exists u0 and u satisfying (3.17). Hence u satisﬁes

i n−2 Qu = Di(A (Du)) + B(u, Du) = 0 in C\{0} × R , u = ϕ on ∂C,

in the sense discussed in Section 1.2 for two-valued functions. Note that u0 = T u0 ∈ V1,τ (C) for all τ ∈ (0, 1/2), so u ∈ C1,τ (C) for all τ ∈ (0, 1/2). The existence of T follows from the following lemma.

Lemma 10. Let 1 < µ ≤ τ < 1/2 and aij, g, and ϕ be two-valued functions associated ij 0,µ 0,µ 1,τ ij with a0 ∈ V (C), g0 ∈ V (C), and ϕ0 ∈ V (∂C). Suppose a0 , g0, and ϕ0 are ij lm i j periodic with respect to yj with period ρj, a (R2π/kX) = a (X)RlRm for all X ∈ C i where R2π/k = (Rj) is as in Section 1.1, and

ij 2 n ij a0 (X)ξiξj ≥ λ|ξ| for X ∈ C, ξ ∈ R , ka kC0,µ(C) ≤ Λ, (3.20)

0 1,τ 2 for some constants λ, Λ > 0. Then there exists a unique u0 ∈ V (C) ∩ V (C) ∩ V (C\ n−2 {0} × R ) such that if u is the two-valued function associated with u0 then u has k-fold symmetry, u0 is periodic with respect to yj with period ρj, and

ij n−2 a Diju = g in C\{0} × R ,

u0 = ϕ0 on ∂C, (3.21) which we interpret in the sense discussed in Section 1.3.

ij Proof. For now suppose a0 and ϕ0 are smooth. By extending ϕ0 to a smooth function CHAPTER 3. EXISTENCE THEOREMS 75

on C and replacing u0 with u0 − ϕ0, we may suppose ϕ = 0. We can re-write (3.21) as

ij ij n−2 Di(a Dju) − (Dia )Dju = g in C\{0} × R ,

u0 = 0 on ∂C. (3.22)

1,τ We will solve (3.22) using the method of continuity [3, Theorem 5.2]. Let W0 (C) 1,τ denote the space of u0 ∈ W (C) such that u0 = 0 on ∂C. Deﬁne the family {Lt}t∈[0,1] 1,τ of weak linear operators on W0 (∂C) by

ij ij Ltu = (1 − t)∆u + t(Di(a Dju) − (Dia )Dju) and consider

i n−2 Ltu = Dif + g in C\{0} × R ,

u0 = 0 on ∂C, (3.23)

i i 0,τ 0 where f and g are two-valued functions associated with f0 ∈ V (C) and g0 ∈ V (∂C) i such that f0 and g0 are are periodic with respect to yj with period ρj and

j j l f0 (R2π/kX) = Rl f (X), g(R2π/kX) = g(X) for all X ∈ C. Note that we interpret (3.23) as discussed in Section 1.3. By Theorem 17 from Section 3.2 and the maximum principle Theorem 9 from Section 2.2, for t = 0 we can ﬁnd a unique u0 such that u0 and the two-valued function u associated with u0 satisfy (3.23). Suppose we can ﬁnd a unique solution to (3.23) for t = s for some s ∈ [0, 1]. Then (3.23) can be rewritten as

n−2 Lsu = f + (Ls − Lt)u in C\{0} × R ,

u0 = 0 on ∂C.

1,τ 1,τ Deﬁne a map U : W (C) → W (C) by letting u0 = U(v0) if the two-valued functions CHAPTER 3. EXISTENCE THEOREMS 76

u and v associated with u0 and v0 satisfy

n−2 Lsu = f + (Ls − Lt)v in C\{0} × R ,

u0 = 0 on ∂C,

1,τ in the sense discussed in Section 1.3. Let u0,1, u0,2, v0,1, v0,2 ∈ W (C) and the two- valued functions u1, u2, v1 and v2 associated with u0,1, u0,2, v0,1, and v0,2 respectively. Since

Ls(u1 − u2) = (Ls − Lt)(v1 − v2) = (s − t)(L1 − L0)(v1 − v2) ij ij = (s − t)(−∆(v1 − v2) + Di(a Dj(v1 − v2)) − (Dia )Dj(v1 − v2)), by the Schauder estimate in Corollary 3 from Section 2.4

ku0,1 − u0,2kC1,τ (C) ij ij ≤ C|s − t| [D(v1 − v0)]τ,C + [a Dj(v1 − v0)]τ,C + k(Dia )Dj(v1 − v0)kC0(C)

≤ C|s − t|kv1 − v0kC1,τ (C),

ij where C > 0 depends only on n, τ, λ, Λ, supC |Da |, and ρ1, . . . , ρn−2. So if |s − t| < 1/2C, then U is a contraction mapping and we can solve (3.23) for t with |s − t| < 1/2C. By dividing [0, 1] into intervals of length less than 1/2C and applying this result we conclude that we can solve (3.23) for all t ∈ [0, 1], in particular 1,µ t = 1. This gives us a two-valued function u associated with u0 ∈ V (C) satisfying 2,µ n−2 (3.22). By elliptic regularity, u0 ∈ Vloc (C\{0}×R ) and thus u and u0 satisfy (3.21).

ij ij To get the result for general a0 , g0, and ϕ0, we will solve (3.21) with a , g, and ϕ ij ij ij ij replaced by smooth approximating functions. To approximate a , write a = aa +as ij ij ij ij where aa is the average part of a and as is the two-valued symmetric part of a . ij ij 0,µ Note that as is the two-valued function associated with some a0,s ∈ V (C). Using ij convolution, we can construct smooth single-valued functions am,a that have k-fold ij ij symmetry and are periodic with respect to yj with period ρj such that am,a → aa CHAPTER 3. EXISTENCE THEOREMS 77

ij ij uniformly and kam,akC0,µ(C) ≤ kam,akC0,µ(C). By Lemma 9, we can construct a smooth ij 0,µ ij ij functions a0,m,s ∈ V (C) such that the two-valued function am,s associated with a0,m,s has k-fold symmetry and are periodic with respect to yj with period ρj such that ij ij ij ij a0,m,s → a0,s uniformly and ka0,m,skV0,µ(C) ≤ Cka0,skV0,µ(C) for some C = C(n, µ) > ij ij ij ij 0,µ 0. Let am = am,a + am,s, which is associated with an element in a0,m ∈ V (C) ij ij ij ij ij so that a0,m → a0 uniformly and (3.20) hold when replace a0 with a0,m and a ij 0,µ with am. Similarly, construct g0,m ∈ V (C) such that g0,m → g0 uniformly and kg0,mkV0,µ(C) ≤ Ckg0kV0,µ(C) for some C = C(n, µ) > 0. Let gm denote the two- valued function associated with g0,m. Using convolution, we can construct smooth 0 two-functions ϕ0,m ∈ V (∂C) such that ϕ0,m is periodic with respect to yj with period

ρj, the two-valued function ϕm associated with ϕ0,m has k-fold symmetry, ϕ0,m → ϕ0 ij ij uniformly on ∂C and sup∂C |ϕm| ≤ sup∂C |ϕ|. Replacing a , g, and ϕ with am, gm, 1,τ and ϕm respectively, we can solve (3.21) for um ∈ C (C). Observe that if we let v ∈ C∞(Rn) be the single-valued function deﬁned by

kgkC0(C) v(X) = sup |ϕ| + (e2 − ex1+1) , ∂C λ where x1 is the coordinate of X such that X = (x1, x2, y1, . . . , yn−2), then

ij ij ij n−2 − a Dijv ≥ a Dijum ≥ a Dijv in C\{0} × R , −v ≤ um ≤ v on ∂C, so by maximum principle Theorem 8

C sup |um| ≤ sup |ϕ| + kgkC0(C) C ∂C λ for some constant C > 0. Thus by the Schauder estimate Theorem 12 from Section 1,τ 2.4, {u0,m} is uniformly bounded in V on compact subsets of C, so after passing 1 1,τ to a subsequence {u0,m} converges in Vloc(C) to u0 ∈ Vloc (C). Let u denote the two- valued function associated with u0. By the local Schauder estimates [3, Corollary

6.3], the single-valued branches of the um on compact simply-connected subsets of n−2 2,µ C\{0} × R are uniformly bounded in C , so after passing to a subsequence u0,m 2 n−2 ij converges in Vloc(C\{0} × R ) to u0 and a Diju = g. By local barriers [3, Section CHAPTER 3. EXISTENCE THEOREMS 78

6.3], we can establish continuity estimates for um at points on ∂C that are uniform independent of m, so u extends to a continuous function on C such u = ϕ on ∂C.

Suppose that u satisﬁes (3.19) and recall that we want to bound ku0kV1,µ(C) for some µ ∈ (0, 1/2). Using maximum principle Theorem 8 from Section 2.2 like in the proof of Theorem 10.3 in [3] and using (1.27),

sup |u| ≤ sup |ϕ| + Cβ2, C ∂C where C = C(β1) > 0 for β1 and β2 as in (1.27). By the standard argument involving local barriers [3, Corollary 14.3], which uses structure conditions (1.28) from the statement of Theorem 4, we can bound

sup |Du| ≤ M1, (3.24) ∂C where M1 > 0 depends on n, kϕ0kV2(∂C), β1, β2, and β3, where β3 is as in (1.28).

To get a global gradient estimate, we use the same approach as [3, Section 15.1].

In particular, apply the operator DluDl to Qu = 0 to get

i i i DPj A (Du)Dijv − 2DPj A (Du)DiluDjlu + DPj Pl A (Du)DijuDlv n−2 + DPl B(u, Du)Dlv + DZ B(u, Du)v = 0 on C\{0} × R , (3.25)

2 2 where v(X) = {|Du1(X)| , |Du2(X)| } if Du(X) = {Du1(X), Du2(X)}. Note that v is the two-valued function associated with some element of V0(C) since u is associated 1 with u0 ∈ V (C) and thus we interpret (3.25) as we discussed in Section 1.3. Since i DPj A (Du)DiluDjlu ≥ 0,

i i DPj A (Du)Dijv + (DPj Pl A (Du)Diju + DPl B(u, Du))Dlv + DZ B(u, Du)v ≥ 0.

By the strong maximum principle for single-valued functions [3, Theorem 3.5] applied locally on C like in the ﬁrst half of the proof of Theorem 8 in Section 2.1 and noting CHAPTER 3. EXISTENCE THEOREMS 79

that v = |Du|2, sup |Du|2 = sup |Du|2. C {0}×Rn−2∪∂C n−2 On {0} × R , Dxu = 0 since u has 3-fold symmetry, so

2 2 2 sup |Du| = sup |Dyu| ≤ sup |Dyϕ| {0}×Rn−2 {0}×Rn−2 ∂C

We claim that 2 2 sup |Dyu| ≤ sup |Dyϕ| {0}×Rn−2 ∂C from which it will follow that

sup |Du| ≤ sup |Du| ≤ M1, C ∂C

n−2 where M1 is the constant from (3.24). Recall from Section 1.3 that for h ∈ R n−2 nonzero and η ∈ S we have the diﬀerence quotient operator δh,η on two-valued 0 functions functions w associated with w0 ∈ V (C) deﬁned by δh,ηw being the two- valued function associated with

w (reiθ, y + hη) − w (reiθ, y) 0 0 . h

Applying δh,η to Qu = 0, we obtain

ij j j n−2 Di(a Djδh,ηu) + b Djδh,ηu(x, y) + c δh,ηu(x, y) = 0 on C\{0} × R , where

Z 1 ij i a = DPj A ((1 − t)Du(x, y + h) + tDu(x, y))dt, 0 Z 1 j b = DPj B((1 − t)u(x, y + h) + tu(x, y), (1 − t)Du(x, y + h) + tDu(x, y))dt, 0 Z 1

c = DZj B((1 − t)u(x, y + h) + tu(x, y), (1 − t)Du(x, y + h) + tDu(x, y))dt. 0 CHAPTER 3. EXISTENCE THEOREMS 80

By maximum principle Theorem 9 from Section 2.2,

Hence

sup |Dyu| ≤ sup |Dyϕ|. C ∂C

2,2 2 2 Next we want to show u ∈ Wloc (C) by obtaining a local L estimate on D u. Recall from Section 1.3 that Z i (A (Du)Diζ − B(u, Du)ζ) = 0 (3.26) C

1 n−2 for all two-valued functions ζ associated with some ζ0 ∈ Vc (C\{0} × R ), where we interpret (3.26) as discussed in Section 1.3. Replace ζ by Dlζ and some l and integrate by parts to get Z i DPj A (Du)DjluDiζ − DPj B(u, Du)Djluζ − DZ B(u, Du)Dluζ = 0. (3.27) C

so using Cauchy-Schwartz Z 2 2 |D u| χδ BR/2(X0) Z ¯−2 −2 i 2 2 i 2 2 2 ≤ Cλ R |DP A (Du)| |Du| + |DP A (Du)| |Du| |Dχδ| BR(X0) Z Z ¯−2 2 2 ¯−1 2 + Cλ |DP B(u, Du)| |Du| + Cλ |DZ B(u, Du)||Du| BR(X0) BR(X0)

¯ where λ = inf|P |≤M1 λ(P ) for M1 as in (3.24) and C = C(n) > 0. Since Z i 2 2 2 n−2 i 2 2 |DA (Du)| |Du| |Dχδ| ≤ CR sup |DA (Du)| |Du| , BR(x0,y0) C we have Z 2 2 ¯−2 n−2 i 2 2 |D u| χδ ≤ C λ R sup |DP A (Du)| |Du| BR/2(x0,y0) C ¯−2 n 2 2 ¯−1 n 2 +λ R sup |DP B(u, Du)| |Du| + λ R sup |DZ B(u, Du)||Du| C C

n−2 2 ¯−2 i 2 ¯−2 2 2 ≤ CR M1 λ sup |DP A (P )| + λ R sup |DP B(Z,P )| |P |≤M1 |Z|≤M0, |P |≤M1 ! ¯−1 2 +λ R sup |DZ B(Z,P )| , |Z|≤M0, |P |≤M1

where M0 = sup∂C |ϕ| and M1 is as in (3.24). Letting δ ↓ 0 yields

Z 2 2 n−2 2 ¯−2 i 2 |D u| ≤ CR M1 λ sup |DP A (P )| n−2 BR/2(x0,y0)\{0}×R |P |≤M1 ! ¯−2 2 2 ¯−1 2 +λ R sup |DP B(Z,P )| + λ R sup |DZ B(Z,P )| |Z|≤M0, |P |≤M1 |Z|≤M0, |P |≤M1

1,2 for C = C(n) > 0. Since Dlu ∈ Wloc (C) we can apply the boundary H¨oldercontinuity estimate Theorem 16 from Section 2.5 and the boundary H¨oldercontinuity estimates CHAPTER 3. EXISTENCE THEOREMS 82

for single-valued functions [3, Theorem 8.29] to (3.27) to obtain

[Du]µ,C ≤ C

¯ i for some constant C > 0 and µ ∈ (0, 1/2) depending on n, β1, β2, β3, λ, sup |DP A |, sup |DP B|, sup |DZ B|, and kϕ0kV2(∂C). Therefore we have shown that if u0 and u satisﬁes (3.19), then

ku0kV1,µ(C) ≤ C (3.28)

¯ i for some constant C > 0 and µ ∈ (0, 1/2) depending on n, β1, β2, β3, λ, sup |DP A |, sup |DP B|, sup |DZ B|, and kϕ0kV2(∂C).

To complete the proof of Theorem 4, we must solve (3.17) for ϕ0 that are not pe- 2 riodic with respect to the y variable. Consider any ϕ0 ∈ V (∂C) with sup∂C |ϕ0| < ∞ 2 and let ϕ be the two-valued function associated with ϕ0. Find a smooth ϕ0,m ∈ V (∂C) such that the two-valued function ϕm associated with ϕ0,m has k-fold symmetry,

ϕ0,m is periodic with respect to yj with period ρm for ρm → ∞, ϕ0,m → ϕ0 uniformly on compact subsets of ∂C and kϕ0,mkV2(∂C) ≤ Ckϕ0kV2(∂C) for some constant 1 C = C(n) > 0. We can ﬁnd u0,m ∈ V (C) such that the two-valued function um associated with u0,m satisﬁes (3.17) with ϕ0 replaced by ϕ0,m. Recall that by (3.28), 1,µ {u0,m} is bounded in V (C). After passing to a subsequence {u0,m} converges in 1 1,µ Vloc(C) to some u0 ∈ Vloc (C) such that, letting u denote the two-valued function associated with u0, u0 and u satisfy (3.17).

For Corollary 1, we can no longer apply (3.28) since the bound depends on kϕ0kV2(∂C). So instead we need to obtain an interior gradient estimate. We will do so by generalizing of an interior gradient estimate due to Simon [7, Theorem 1] to two-valued functions.

Lemma 11. Let Ai ∈ C2(Rm×n), B ∈ C1(Rm×n) be single-valued functions such that 0 1,µ (1.26), (1.31), (1.32), (1.33), (1.34) from Corollary 1 hold. Let u0 ∈ V (C) ∩ Vloc (C) CHAPTER 3. EXISTENCE THEOREMS 83

such that the two-valued function u associated with u0 satisﬁes

i n−2 DiA (Du) + B(u, Du) = 0 on C\{0} × R ,

which we interpret as discussed in Section 1.3. Then for each X0 ⊂ C,

|Du(X0)| ≤ C

for some C > 0 depending on n, γ1, γ2, γ3, γ4, supC |u|, dist(X, ∂C).

Proof. The proof is nearly identical to the proof in [7, Theorem 1]. Note that we will 2 1/2 ij −2 use the notation v = (1+|Du| ) and g = δij −v DiuDju. First we will establish (2.11) from [7] for two-valued functions associated with elements of V0(C); that is, we need to show Z 2 i 2 (1 − τ/v)C + vDPj A (Du)DiωDjω χ(v)ζ dµ {v≥τ} Z 2 2 2 ≤ C(1 + cχ) ζ + |Dζ| χ(v)dµ, (3.29) {v≥τ}

2 −1 i kl where C = v DPj A (Du)g DikuDjlu and ω = log(v), for any τ ≥ 2γ1, single- 1 n−2 0,1 valued function ζ ∈ Cc (C\{0} × R ), and single-valued function χ ∈ Cc (R) such 0 that χ ≥ 0 and 0 ≤ (v − τ)χ (v)/χ(v) ≤ cχ for v ≥ τ. Recall from Section 1.3 that Z i A (Du)Diφ − B(u, Du)φ = 0 (3.30) C

1 n−2 for all two-valued functions φ associated with some φ0 ∈ Cc (C\{0} × R ), which we interpret as discussed in Section 1.3. Replacing φ by Dlφ and integrating by parts yields Z i DPj A (Du)DjluDiφ + B(u, Du)Dlφ = 0. (3.31) C n−2 −1 To obtain (3.29) in the case that ζ = 0 near {0} × R , let φ = v Dluζ in (3.31) and argue as in the proof of (2.11) in [7, Theorem 1]. CHAPTER 3. EXISTENCE THEOREMS 84

Next we will establish (2.1) from [7] for two-valued functions associated with elements of V0(C); that is,

Z (n−2)/n Z 2n/(n−2) ij 2 2 h dµ ≤ C(n) g DihDjh + |H| h dµ (3.32) C C

1 n−2 where h is a two-valued function associated with some h0 ∈ Vc (C\{0} × R ) and H is the mean curvature vector of the graph of u. This is not immediate consequence of the standard Sobolev inequality for submanifolds (see [8, Theorem 18.6]) since the graph of u is immersed. Instead we can ﬁrst prove Z Z

divΣu Φ(X, u(X))dµ(X) = − H(X, u(X)) · Φ(X, u(X))dµ(X) (3.33) C C

1 n−2 n+1 for all two-valued functions Φ associated with some Φ0 ∈ Vc (C\{0} × R , R ), where Σu denotes the graph of u. Note that (3.33) makes sense since u, φ, and H(X, u) 1 1 n−2 n+1 are two-valued functions associated with elements in V (C), Vc (C\{0} × R , R ), 0 n−2 2 and V (C\{0}×R ) respectively. Take a partition of unity ψ1, ψ2,... of B1 (0)\{0} 2 with each ψj having support in a disk contained in B1 (0) \{0}. We can regard each iθ n−2 ψj as a function of (re , y) ∈ C \ {0} × R that is independent of y. By the ﬁrst variation formula for submanifolds, (3.33) holds if we replace ζ by ψjφ. By summing over j, we obtain (3.33). Now (3.32) follows from (3.33) using one of the usual proofs −1 −1 ij 2 −1 2 (such as for Theorem 18.6 in [8]). Note that since H = n v g Diju, H ≤ n C .

2 nq−n+2 2q Now let h = ((1 − τ/v)ζ) (log(v)) for q > 1 and τ ≥ 2γ1 and any single- 1 n−2 valued function ζ ∈ Cc (C\{0} × R ) in (3.32). Arguing as in the proof of Lemma

1 in [7], using (3.29) and letting w(v) = (1/2γ1) log(v), Λ(v) = γ4, and λ(v) = 1 (see the discussion in [7, Chapter 4]), we obtain

Z 1/κ (1 − τ/v)nζnw(v)2qκ γdµ {v≥τ, ζ>0} Z ≤ Cq4 (1 − τ/v)nζnw(v)2q |Dζ|2γdµ, (3.34) {v≥τ, ζ>0} CHAPTER 3. EXISTENCE THEOREMS 85

1 n−2 for any single-valued function ζ ∈ Cc (C\{0} × R ). We want to show that (3.34) 1 1 holds for ζ ∈ Cc (C). Letχ ˆδ ∈ C (C) be the logarithmic cutoﬀ function from (3.37) 1 and replace ζ with ζχˆδ for ζ ∈ Cc (C) in (3.34) to obtain

Z 1/κ (1 − τ/v)nζn(log(v))2qκ γdµ {v≥τ, ζ>0} Z 4 nq−n 2q 2 2 2 2 ≤ Cq ((1 − τ/v)ζ) (log(v)) |Dζ| χˆδ + ζ |Dχˆδ| dµ, {v≥τ}

1 Since v and ζ are bounded, we may let δ ↓ 0 to obtain (3.34) for ζ ∈ Cc (C). Let

Bρ(X0) ⊂⊂ C and let ζ = 1 in Bρ/8(X0), ζ = 0 on C\ Bρ/4(X0), and |Dζ| ≤ 12/ρ. Using (3.34) and arguing as in the proof of Lemma 1 [7], we obtain

Z sup (1 − τ/v)n(log(v))2 ≤ C (log(v))2dµ (3.35) Bρ/8(X0)∩{v≥τ} Bρ/4(X0)∩{v≥τ}

2 2 for some constant C = C(n, γ1, γ2, γ3, γ4, ρ) > 0. Next let φ = (log(v) −log(τ) )+(u−

2γ3(u−u(X0)) 2 1 n−2 u(X0))e ζ in (3.30), where ζ ∈ Cc (C\{0}×R ) is a single-valued function. Arguing as in the proof of Lemma 2 [7] with q = 2 and using (3.29), we obtain

Z 2 2 i w(v) ζ A (Du)Diu {v≥τ} Z 4γ3m 2 2 2 2 2 2 i ≤ Ce w(τ) ζ + m ζ + m |Dζ| A (Du)Diu (3.36) {v≥τ}

1 n−2 1 for any single-valued function ζ ∈ Cc (C\{0}×R ). To obtain (3.36) for ζ ∈ Cc (C), 1 letχ ˆδ ∈ C (C) be the logarithmic cutoﬀ function given by   0 if r ≤ δ2  iθ 2 2 χˆδ(re , y) = − log(r/δ )/ log(δ) if δ < r < δ (3.37)   1 if r ≥ δ. CHAPTER 3. EXISTENCE THEOREMS 86

1 Replace ζ by ζχˆδ in (3.36), where now ζ ∈ Cc (C), to obtain Z Z 2 2 i 2 4γ3m 2 2 2 2 log(v) ζ A (Du)Diuχˆδ ≤ Ce m ζ |Dχˆδ| {v≥τ} {v≥τ} Z 4γ3m 2 2 2 2 2 2 2 i ≤ Ce log(τ) ζ + m ζ + m |Dζ| χˆδA (Du)Diu {v≥τ}

1 Letting δ ↓ 0 yields (3.36) for ζ ∈ Cc (C). Let 0 ≤ ζ ≤ 1, ζ = 1 on Bρ/4(X0), ζ = 0 on

C\ Bρ/2(X0), |Dζ| ≤ 6/ρ and argue as in the proof of Lemma 2 [7] to obtain

Z 2 i log(v) A (Du)Diu Bρ/4(X0)∩{v≥τ} Z 4γ3m 2 2 i ≤ Ce log(τ1) + (m/ρ) A (Du)Diu (3.38) Bρ/2(X0)∩{v≥τ}

0,1 for some constant C = C(n, γ1, γ2, γ3, γ4, ρ) > 0. Finally, let ψ ∈ C (C) such that   0 on Bρ/2(X0),  −1 ψ = (2ρ |X − X0| − 1) oscBρ(X0) u on Bρ(X0) \ Bρ/2(X0),   oscBρ(X0) u on C\ Bρ(X0),

2γ3(u−u(X0)) and η = (u − infBρ(X0) u − ψ)+. Let φ = ηe in (3.30) and argue as in the proof of Lemma 3 [7] to obtain

Z i 4γ3M n A (Du)Diu ≤ C(1 + M)e ∆ρ , (3.39) Bρ/2(X0)∩{v≥τ}

where C = C(n, γ1, γ2, γ3, γ4, ρ) > 0, M = supBρ(X0) |u|,

i ∆ = sup (M/ρ)|A(P )| + M|B(Z,P )| + |A (P )Pi| , |Z|≤M, |P |≤τ2 and τ ≥ τ2 for τ2 chosen so that 4σ(τ2)M/ρ ≤ 1/2 where σ(v) = 2γ2/v (see the CHAPTER 3. EXISTENCE THEOREMS 87

discussion in [7, Chapter 4]). Now combining (3.35), (3.38), and (3.39),

n 2 4γ3m 2 2 4γ3M n sup (1 − τ/v) (log(v)) ≤ Ce (log(τ1) + (m/ρ) )(1 + M)e ∆ρ . Bρ/8(X0) for some constant C = C(n, γ1, γ2, γ3, γ4, ρ) > 0.

0 Now suppose ϕ0 ∈ V (∂C), ϕ is the two-valued function associated with ϕ0, and 1,µ sup∂C |ϕ| < ∞. Find a smooth function ϕ0,m ∈ V (∂C) such that the two-valued function ϕm associated with ϕ0,m has k-fold symmetry, ϕ0,m is periodic with respect to yj with period ρm for ρm → ∞, and ϕ0,m → ϕ0 uniformly on compact subsets 1 of ∂C. We can ﬁnd u0,m ∈ V (C) and the two-valued function um associated with u0,m such that u0,m and um satisfy (3.17) with ϕ replaced by ϕm. Using maximum principle Theorem 8 from Section 2.2 like in the proof of Theorem 10.3 [3],

sup |um| ≤ sup |ϕ| + Cβ2 C ∂C for C = C(β1) > 0 where β1 and β2 are as in (1.27). By Lemma 11, we have a uniform interior gradient estimate for um. Using the Theorem 16 from Section 2.5, 1,µ we can show that {um} is bounded in V on compact subsets of of C for some i µ ∈ (0, 1) depending on n, A , B, and sup∂C |ϕ|. After passing to a subsequence 1 1,µ {u0,m} converges in Vloc(C) to some u0 ∈ Vloc (C) such that the two-valued function i n−2 u associated with u0 satisﬁes DiA (Du) + B(u, Du) = 0 on C\{0} × R . Using local barriers [3, Chapter 14.5], we can show u extends continuously to a function on C such that u = ϕ on ∂C. Chapter 4

Regularity theorems

4.1 C1,µ regularity of C1 solutions

1 m In this section, we will consider two-valued solutions u ∈ C (B1(0), R ) satisfying the elliptic system

i Qκu = Di(Aκ(X, u, Du)) + Bκ(X, u, Du) = 0 on B1(0) \Bu (4.1)

i for κ = 1, . . . , m, where Aκ and Bκ are real analytic single-valued functions on B1(0)× Rm ×Rm×n. In this case,u ˆ(x, y) = u(x+g(y), y) is the two-valued function associated 1 m with someu ˆ0 ∈ V ({(x, y):(x + g(y), y) ∈ B1(0)}, R ) such thatu ˆ0 satisﬁes

2 X 2 k i l Dξk (4|ξ| Mi Aκ(X, uˆ0, (Dxu0 Dyg Dxl u0 + Dyu0)M)) i,k=1 n X 2 i l + Dyi (4|ξ| Aκ(X, uˆ0, (Dxu0 Dyg Dxl u0 + Dyu0)M)) i=3 n 2 X X 2 k j i l + Dξk (4|ξ| Mj Dig Aκ(X, uˆ0, (Dxu0 Dyg Dxl u0 + Dyu0)M)) i=3 j,k=1 2 l + 4|ξ| Bκ(X, uˆ0, {Dxu0, (Dxu0 Dyg Dxl u0 + Dyu0)M) = 0 (4.2)

88 CHAPTER 4. REGULARITY THEOREMS 89

iθ 1/2 iθ/2 n−2 weakly on {(re , y):(r e −g(y), y) ∈ B1(0)}\{0}×R for κ = 1, . . . , m, which 1/2 we obtain using the change of variables ξ1 + iξ2 = (x1 + ix2 − g1(y) − ig2(y)) . Here j 1/2 M = (Mi ) is the Jacobian matrix for the change of variable ξ1 + iξ2 = (x1 + ix2) from Section 1.3. Equivalently, we write Z i κ κ Aκ(X, u, Du)Diζ + Bκ(X, u, Du)ζ = 0 B1(0) for any two-valued function ζ such that ζ(x + g(y), y) is the two-valued function ˆ 1 m associated with some ζ0 ∈ Vc (B1(0), R ), which we interpret to mean

2 Z X 2 k i l ˆκ 4|ξ| Mi Aκ(X, uˆ0, (Dxu0 Dyg Dxl u0 + Dyu0)M)Dξk ζ0 i,k=1 Ω n Z X 2 i l ˆκ + 4|ξ| Aκ(X, uˆ0, (Dxu0 Dyg Dxl u0 + Dyu0)M)Dyi ζ0 i=3 Ω n 2 Z X X 2 k j i l ˆκ + 4|ξ| Mj Dig Aκ(X, uˆ0, (Dxu0 Dyg Dxl u0 + Dyu0)M)Dξk ζ0 i=3 j,k=1 Ω Z 2 l ˆκ − 4|ξ| Bκ(X, uˆ0, (Dxu0 Dyg Dxl u0 + Dyu0)M)ζ0 = 0. Ω

i Observe that since Aκ and Bκ are real analytic, there are precisely two choices foru ˆ0, where if one choice isu ˆ0(x, y) the other choice isu ˆ0(−x, y). To see this, recall that n−2 n−2 for every X0 ∈ Ω \{0} × R , on any ball BR(X0) ⊂ Ω \{0} × R , u = {u1, u2} 1 and Qu1 = Qu2 = 0 for single-valued functions u1, u2 ∈ C (BR(X0)). By elliptic regularity, u1 and u2 are real analytic and thus u = {u1, u2} on BR(X0) for unique real analytic single-valued functions u1 and u2. It follows from unique continuation that u is the two-valued function of the form

iθ 1/2 iθ/2 1/2 iθ/2 u(g(y) + re , y) = {uˆ0(r e , y), uˆ0(−r e , y)} (4.3)

1 for someu ˆ0 ∈ V (Ω) satisfying (4.2) and that there are precisely two such choices of uˆ0. In what follows, we will interpret (4.1) to mean thatu ˆ0 satisﬁes (4.2). CHAPTER 4. REGULARITY THEOREMS 90

Now we will show that in the case where m = 1, two-valued solutions u ∈ 1 1,µ C (B1(0)) to (4.1) are in C (B1(0)).

1 Theorem 18. Let u ∈ C (B1(0)) be a two-valued function such that Bu = graph g ∩ 1 n−2 2 B1(0) for some single-valued function g ∈ C (R , R ). Suppose u is a solution to the uniformly elliptic quasilinear equation

i DiA (X, u, Du) + B(X, u, Du) = 0 on B1(0) \Bu, (4.4)

i n where A and B are real analytic single-valued functions on B1(0) × R × R . Then i for some constants µ ∈ (0, 1/2) and C > 0 depending on n, kgkC1 , A , and B, 1,µ m u ∈ C (B1/2(0), R ) with

0,µ 2 kDukC (B1/2(0)) ≤ C kDukL (B1(0)) + 1 . (4.5)

Proof. Observe that v = Dku satisﬁes

i i i Di(DPj A (X, u, Du)Djv + DZ A (X, u, Du)v + DXk A (X, u, Du))

+ DPj B(X, u, Du)Djv + DZ B(X, u, Du)v + DXk B(X, u, Du) = 0

on B1(0) \Bu. Let Ω = {(x + g(y), y):(x, y) ∈ B1(0)}. Letv ˆ(x, y) = v(x + g(y), y) and observe thatv ˆ is a two-valued function associated with a function in V0(Ω) that satisﬁes

n n 2 n X i X X X j i j l Di(DPj A Djvˆ) − (Di(Dlg DPl A Djvˆ) + Dj(Dlg DPi A Divˆ)) i,j=1 i=1 j=1 l=3 2 n n X X i j l X i i + Di(Dlg Dmg DPm A Djvˆ) + Di(DZ A vˆ + DXk A ) i,j=1 l,m=3 i=1 2 n n 2 n X X i l l X X X j − Di(Dlg (DZ A vˆ + DXk A )) + DPj BDjvˆ − Dlg DPl BDjvˆ i=1 l=3 i=1 j=1 l=3

+ DZ B + DXk B = 0 (4.6) CHAPTER 4. REGULARITY THEOREMS 91

on Ω\{0}×Rn−2 in the sense that was discussed in Section 1.3, where the derivatives of i l A and B are evaluated at ((x+g(y), y), u(x+g(y), y), {Dxu(x+g(y), y),Dyg Dxl u(x+ g(y), y) + Dyu(x + g(y), y)}). It will be convenient to express (4.6) as

ij i j i n−2 Di(a Djvˆ + b vˆ) + c Djvˆ + dvˆ = Dif + g on Ω \{0} × R (4.7) where aij, bi, cj, d, f i, and g are two-valued functions associated with elements in V0(Ω) and we interpret (4.7) as discussed in Section 1.3. We claim thatv ˆ ∈ W 1,2(Ω). 1 Let ζ ∈ Cc (Ω) be a single-valued function. Then by (4.7), Z ij i i 2 j 2 a Djvˆ + b vˆ − f Di(ˆvζ ) − c Djvˆ + dvˆ − g vζˆ = 0. Ω

Rearranging terms, Z Z ij 2 ij i i 2 a DjvDˆ ivζˆ = 2a vDˆ jvζDˆ iζ − b vˆ − f Divζˆ + 2ˆvζDiζ Ω Ω Z j 2 + c Djvˆ + dvˆ − g vζˆ . Ω

Using Cauchy-Schwartz, Z Z |Dvˆ|2ζ2 ≤ C |Dv|2 + 1 ζ2 + |Dζ|2 Ω Ω

i 1 1 for some constant C > 0 depending on n, kukC (B1(0), kgkC , DA , and DB. Let

BR(X0) ⊂⊂ Ω and ζ = 1 on BR/2(X0), ζ = 0 on Ω \ BR(X0), and |Dζ| ≤ 3/R to obtain Z |Dvˆ|2 < ∞. BR/2(X0)

0,µ By Theorem 16 from Section 2.5,v ˆ ∈ C ({(x + g(y), y):(x, y) ∈ B1/2(0)}) with

0,µ 2 kDukC ({(x+g(y),y):(x,y)∈B1/2(0)}) ≤ C kDukL {(x+g(y),y):(x,y)∈B1(0)}) + 1 .

i 1 1 for some constants µ ∈ (0, 1/2) and C > 0 depending on n, kukC (B1(0), kgkC , DA , 1,µ and DB. It follows that u ∈ C (B1/2(0)) and (4.5) holds. CHAPTER 4. REGULARITY THEOREMS 92

Now we want to consider elliptic systems of the form

i n−2 Di(Aκ(Du)) = 0 on B1(0) \{0} × R for κ = 1, . . . , m, (4.8)

i m×n i ij where A is a real analytic single-valued function on with D λ A (0) = δ δκλ. κ R Pj κ By Taylor’s theorem, given Q ∈ Rm×n, we can write

i i i λ λ i A (P ) = A (Q) + D λ A (Q) · (P − Q ) + B (P ; Q) (4.9) κ κ Pj κ j j κ for all P ∈ Rm×n, where

i i |Bκ(P ; Q)| ≤ σ(|P | + |Q|)|P − Q|, |DP Bκ(P ; Q)| ≤ σ(|P | + |Q|) (4.10) for some function σ : (0, ∞) → (0, ∞) such that σ(t) → 0 as t ↓ 0. In particular, when Q = 0, i κ i Aκ(0) = Pi + Bκ(P ; 0).

Thus (4.8) satisﬁes the Legendre ellipticity condition

i κ λ i κ λ 1 2 1 2 DP λ Aκ(P )ξi ξj ≥ DP λ Aκ(0)ξi ξj − |ξ| = |ξ| (4.11) j j 2 2 for all ξ ∈ Rm×n provided σ(|P |) ≤ 1/2.

i 1 m×n Lemma 12. Let m ≥ 1 be an integer and let Aκ ∈ C (R ) be a single-valued func- i ij tion such that D λ A (0) = δ δκλ. There are constants γ > 0 and C > 0 depending Pj κ i n×m 1 on n, m, and Aκ such that if Λ ∈ R , u ∈ C (B1(0)) is a two-valued function such 1 n−2 2 that Bu = graph g ∩ B1(0) for some single-valued function g ∈ C (R , R ), and

sup |Du| ≤ γ, (4.12) B1(0)

2,2 then u ∈ W (B1/2(0)) with

2 2 2 RkD ukL (BR/2(0)) ≤ CkDu − ΛkL (BR(0)) CHAPTER 4. REGULARITY THEOREMS 93

Proof. By (4.8), Z i κ Aκ(Du)Diζ = 0, (4.13) B1(0) for every two-valued function ζκ such that ζκ(x + g(y), y) is the two-valued function κ 1 associated with some ζ0 ∈ Vc (B1(0) \Bu) of the form (4.3), where we interpret (4.13) κ κ as discussed at the beginning of this chapter. Replace ζ with Dkζ and integrate by parts to obtain Z i λ κ D λ A (Du)Djku Diζ = 0. (4.14) Pj κ B1(0) 1 Let χδ ∈ C (B1(0)) be a single-valued function such that 0 ≤ χδ ≤ 1, χδ = 1 outside a tubular neighborhood of graph g of radius δ, χδ = 0 on an open neighborhood of κ κ κ 2 2 1 graph g, and |Dχδ| ≤ 3/δ. Let ζ = (Dku − Λk)ζ χδ for ζ ∈ Cc (B1(0)) in (4.14) and sum over k = 1, . . . , n and κ = 1, . . . , m to obtain Z Z 2 2 2 2 2 2 2 |D u| ζ χδ ≤ C |Du − Λ||D u| |ζ||Dζ|χδ + ζ χδ|Dχδ| , (4.15) B1(0) B1(0) where we used the fact that (4.11) holds provided γ in (4.12) is suﬃciently small. By Cauchy-Schwartz, Z Z 2 2 2 2 2 2 2 2 |D u| ζ χδ ≤ C |Du − Λ| |Dζ| χδ + ζ |Dχδ| B1(0) B1(0) Z 2 2 2 2 2 ≤ C |Du − Λ| |Dζ| χδ + C sup |Du − Λ| ζ < ∞. B1(0) B1(0)

2,2 Hence u ∈ W (B1/2(0)). Letting δ ↓ 0 in (4.15),

Z Z |D2u|2ζ2 ≤ C |Du − Λ||D2u||ζ||Dζ|. B1(0) B1(0)

By Cauchy-Schwartz, Z Z |D2u|2ζ2 ≤ C |Du − Λ|2|Dζ|2. B1(0) B1(0) CHAPTER 4. REGULARITY THEOREMS 94

n Let 0 ≤ ζ ≤ 1, ζ = 1 on B1/2(0), ζ = 0 on R \ B1(0), and |Dζ| ≤ 6 to obtain Z Z |D2u|2 ≤ C |Du − Λ|2. B1/2(0) B1(0)

i 1 m×n Lemma 13. Let m ≥ 1 be an integer and Aκ ∈ C (R ) be a single-valued function i ij i such that D λ A (0) = δ δκλ. There is a constant γ > 0 depending on n, m, and A Pj κ κ m×n 1 m such that the following is true. Let Λ ∈ R with |Λ| ≤ γ. Let u ∈ C (B1(0), R ) 1 n−2 2 such that Bu = graph g ∩B1(0) for some single-valued function g ∈ C (R , R ) with |Dg| ≤ γ, u is a solution to (4.8), and u satisﬁes (4.12). Then there is a two-valued 2,2 function w ∈ W (B1/4(0)) of the form

iθ 1/2 iθ/2 1/2 iθ/2 w(re , y) = {w0(r e , y), w0(−r e , y)} (4.16)

1/2 iθ/2 for some measurable single-valued function w0 that is smooth on {(r e , y): iθ n−2 n−2 (re , y) ∈ B1(0) \{0} × R } such that w is harmonic on B1(0) \{0} × R ,

Z !1/2 2 2 G(u, w) ≤ kDu − ΛkL (B1(0)), (4.17) B1/4(0) where G is the metric on the space of unordered pairs from Section 1.2, and

2 2 kDw − ΛkL (B1/4(0)) ≤ kDu − ΛkL (B1(0)). (4.18)

m×n 1 m Proof. Suppose there exists Λk ∈ R and a two-valued function uk ∈ C (B1(0), R ) 1 n−2 2 such that Buk = graph gk ∩ B1(0) for some single-valued function gk ∈ C (R , R ) with |Dgk| ≤ 1/k, uk solves (4.8), and

sup |Duk| ≤ 1/k, |Λk| ≤ 1/k, (4.19) B1(0)

0,1 2,2 but for all two-valued functions w ∈ C (B1/4(0)) ∩ W (B1/4(0)) of the form (4.16) CHAPTER 4. REGULARITY THEOREMS 95

n−2 such that w is harmonic on B1/4(0) \{0} × R and

2 2 kDwkL (B1/4(0)) ≤ kDuk − ΛkkL (B1(0)), we have Z !1/2 2 2 G(uk, w) > kDuk − ΛkkL (B1(0)). (4.20) B1/4(0)

2 Letu ˆk = (uk − Λk · X)/kDuk − ΛkkL (B1(0)). By Lemma 12,

2 2 2 kDuˆkkL (B1(0)) = 1, kD uˆkkL (B1/2(0)) ≤ C (4.21)

i for some C > 0 depending on n, m, and Aκ.

By (4.8), Z i κ Aκ(Duk)Diζ = 0 (4.22) B1(0) for any two-valued function ζκ such that ζκ(x + g(y), y) is the two-valued function κ 1 associated with some ζ0 ∈ Vc (B1(0)), where we interpret (4.22) as discussed at the beginning of this chapter. Rewrite (4.22) using Taylor’s theorem (4.9) as

Z Z i λ κ κ i κ D λ A (Λk)(Dju − Λ )Diζ = − B (Duk;Λk)Diζ . Pj κ k i κ B1(0) B1(0)

2 Dividing by kDuk − ΛkkL (B1(0)) and using (4.10) and (4.21),

Z Z i κ κ κ D λ A (Λk)Diuˆ Diζ ≤ σ(1/k) |Duˆk||Dζ | ≤ σ(1/k) sup |Dζ| (4.23) Pj κ k B1(0) B1(0)

Letu ¯k(x, y) =u ˆk(x + gk(y), y) so that the domain {(x − gk(y), y):(x, y) ∈ B1(0)} ofu ¯k contains B3/4(0) for k suﬃciently large andu ¯k is associated with someu ¯0,k ∈ 0 V (B3/4(0)). By the Poincar´e’sinequality Theorem 8 from Section 2.5, for some constant `k ∈ R,

2 2 ku¯k − `kkL (B3/4(0)) ≤ CkDu¯kkL (B3/4(0)) ≤ C (4.24) CHAPTER 4. REGULARITY THEOREMS 96

for some constant C = C(n) > 0. Now we want to apply Rellich’s lemma with (4.21) 1,2 and (4.24) to get that after passing to a subsequence,u ˆk − `k → v in W (B1/4(0)) 2,2 strongly to some two-valued function v ∈ W (B1/4(0)) such that v has the form (4.16), and

2 2 2 kDvkL (B1/4(0)) ≤ 1, kD vkL (B1/4(0)) ≤ C (4.25)

by (4.20). However, theu ˆk are two-valued functions, so we cannot do this directly. (j) iθ 1/2 iθ/2 Restrictu ¯k to its single-valued branchesu ¯k (re , y) =u ¯0,k(r e , y) with |θ − iθ jπ/2| < π/2, j = 0, 1, 2,..., 7. Let Ωj = {(re , y) ∈ B1/2(0) : |θ − jπ/2| < π/2} for (j) j = 0, 1, 2,..., 7. By Rellich’s lemma with (4.21) and (4.24), {u¯k } converges to some (j) 2 1,2 function v strongly in L (Ωj) and weakly in W (Ωj). Piecing the single-valued branches together, we get a two-valued function v of the form (4.16) on B1/2(0) such that v has the single-valued branch v(j) for |θ − jπ/2| < π/2 for each j = 0, 1, 2,..., 7 2 and {u¯k} converges to v strongly in L (B1/2(0)). We want to show {uˆk} converges to 2 v strongly in L (B1/4(0)). We have

Z Z Z 2 2 2 G(ˆuk − `k, v) ≤ 2 G(ˆuk, u¯k) + 2 G(¯uk − `k, v) . (4.26) B1/4(0) B1/4(0) B1/4(0)

Since Z Z 2 2 G(ˆuk(x, y), u¯k(x, y)) = G(ˆuk(x, y), uˆk(x + gk(y), y)) B1/4(0) B1/4(0) Z Z 1 2 ≤ 2 G(Duˆk(x + tgk(y), y), 0) |gk(y)|dt B1/4(0) 0 Z Z 1 −2 2 ≤ 2k G(Duˆk(x + tgk(y), y), 0) dt B1/4(0) 0 −2 2 ≤ Ck kDuˆ k 2 k L (B1(0)) ≤ Ck−2 CHAPTER 4. REGULARITY THEOREMS 97

for some constant C = C(n) > 0, by (4.26),

Z 2 lim G(ˆuk − `k, v) = 0. (4.27) k→0 B1/4(0)

We can deﬁneu ¯i,k(x, y) = (Diuˆk)(x+gk(y), y) for i = 1, . . . , n and associate each two- 0 m valued functionu ¯i,k with au ¯0,i,k ∈ V (B3/4(0), R ). Applying Rellich’s lemma to the (j) iθ 1/2 iθ/2 single-valued branchesu ¯i,k (re , y) =u ¯0,i,k(r e , y) ofu ¯i,k with |θ − jπ/2| < π/2 (j) (j) 2 for j = 0, 1, 2,..., 7, {u¯k } converges to some function vi strongly in L (Ωj) and 1,2 weakly in W (Ωj) for i = 1, . . . , n and j = 0, 1, 2,..., 7. Since Diu¯k =u ¯i,k for i = 1, 2 1 2 and Diu¯k = Digku¯1,k + Digku¯2,k +u ¯i,k for i = 3, . . . , n, vi = Div and Diu¯k → Div 2 2 strongly in L (B1/4(0)) for i = 1, . . . , n. Sinceu ¯i,k → Div strongly in L (B1/4(0)) for i = 1, . . . , n, by (4.21),

Z Z n Z 2 X 2 2 |Dv| = lim |u¯i,k| ≤ lim |Duˆk| = 1. (4.28) k→0 k→0 B1/4(0) B1/4(0) i=1 B1(0)

(j) (j) 1,2 Sinceu ¯i,k → Div weakly in W (Ωj) for i = 1, . . . , n and j = 0, 1, 2,..., 7, by (4.21),

i 2,2 for some C > 0 depending on n, m, and Aκ, so v ∈ W (B1/4(0)). By (4.23) and the 2 fact thatu ¯i,k(x, y) = (Diuˆk)(x+gk(y), y) converges to Div strongly in L (B1/4(0)) for n−2 i = 1, . . . , n, w is harmonic on B1/4(0) \{0} × R .

By (4.27), Z 2 lim G(ˆuk, v + `k) < ε (4.29) k→0 B1/4(0)

2 for large k. Let w = Λk · X + kDuk − ΛkkL (B1(0))(v + `k) so that w is harmonic on n−2 B1/4(0) \{0} × R ,

2 2 kDw − ΛkkL (B1/4(0)) ≤ kDuk − ΛkkL (B1(0)) CHAPTER 4. REGULARITY THEOREMS 98

by (4.28), and Z 2 G(uk, w) < εkDuk − ΛkkL (B1(0)) B1/4(0) by (4.29), contradicting (4.20).

2,2 Now let w be a two-valued harmonic function w ∈ W (B1/4(0)) of the form n−2 (4.16) that is harmonic on B1/4(0) \{0} × R . Using unique continuation and the n−2 fact that on any ball BR(X0) ⊂ B1/4(0) \{0} × R we can write w = {w1, w2} 1 for single-valued harmonic functions w1, w2 ∈ C (BR(X0)), we can show that there are precisely two choices of w0 in (4.16); ﬁx one of those choices for w0. We want 1,1/2 to show that w ∈ C (B1/8(0)). This will follow from some basic facts about frequency functions. Write w = wa + ws where wa(X) = (w1(X) + w2(X))/2 and ws(X) = {±(w1(X) − w2(X))/2}. By elliptic regularity, wa is a smooth single-valued function on B1/8(0). Thus we may suppose w is two-valued symmetric. Recall from Section 1.2 that ˘ Zw = {X ∈ B1/4(0) : w(X) = 0}.

˘n−2 For Y ∈ Zw ∪ {0} × B1/4 (0), we deﬁne the frequency function of w at Y by

ρ2−n R |Dw|2 N (ρ) = Bρ(Y ) , w,Y ρ1−n R |w|2 ∂Bρ(Y )

˘ where ρ > 0 such that Bρ(Y ) ⊂ B1/4(0). In order to use frequency functions, we need to show that Nw,Y (ρ) is defined and satisfies a monotonicity formula. However, 2,2 n−2 w is only in W (B1/4(0)) with a singularities along {0} × B1/4 (0). Thus we use n−2 cutoff functions that vanish near {0} × B1/4 (0) and we also truncate w and its n−2 gradient, which may be becoming infinitely large near {0} × B1/4 (0). For δ > 0, let 1 χδ ∈ C (B1/4(0)) such that 0 ≤ χδ ≤ 1, χδ(x, y) = 0 for |x| ≤ δ/2, χδ(x, y) = 1 for 1 |x| ≥ δ, and |Dχδ| ≤ 3/δ. For k > 0, let γk ∈ C (R) be an odd function such that CHAPTER 4. REGULARITY THEOREMS 99

0 γk(t) = t for 0 ≤ t ≤ k − 1, γk(t) = k for t ≥ k + 1, and 0 ≤ γk(t) ≤ 1. Note that Z Z 2 2 (X−Y ) ρ γk(w) χδ = γk(w) χδ (X − Y ) · ρ ∂Bρ(Y ) ∂Bρ(Y ) Z 0 2 2 = 2rγk(w)γk(w)Drwχδ + nγk(w) χδ + rγk(w) Drχδ Bρ(Y )

˘ for Y ∈ B1/4(0) and ρ ∈ (0, 1/4 − |Y |), where r = |X − Y | for X ∈ B1/4(0). Since

γk(w) is bounded, letting δ ↓ 0 yields for all k > 0 that Z Z 2 0 2 ρ γk(w) = 2rγk(w)γk(w)Drw + nγk(w) . ∂Bρ(Y ) Bρ(Y )

1,2 Since w ∈ W (B1/4(0)), we can let k → ∞ to obtain

Z Z 2 2 ρ w = 2rwDrw + nw . ∂Bρ(Y ) Bρ(Y )

1/2 2 ˘ Thus ρ kwkL (∂Bρ(Y )) is a bounded, continuous function of Y ∈ B1/4(0) and ρ ∈ 2,2 (0, 1/4−|Y |). Note that by a similar argument using the fact that w ∈ W (B1/4(0)),

Z Z 2 2 ρ |Dw| = 2rDiwDrDiw + n|Dw| , ∂Bρ(Y ) Bρ(Y )

1/2 ˘ so ρ kDwk∂∂Bρ(Y ) also a bounded function of Y ∈ B1/4(0) and ρ ∈ (0, 1/4 − |Y |).

2,2 Lemma 14. For a two-valued harmonic function w ∈ W (B1/4(0)) of the form ˘n−2 (4.16) and Y ∈ Zw ∪{0}×B1/4 (0), Nw,Y (ρ) is monotone nondecreasing as a function of ρ ∈ (0, 1/4 − |Y |).

Proof. Let Z Z 2−n 2 1−n 2 D(ρ) = Dw,Y (ρ) = ρ |Dw| ,H(ρ) = Hw,Y (ρ) = ρ w , Bρ(Y ) ∂Bρ(Y ) so that Nw,Y (ρ) = D(ρ)/H(ρ). We want to show that ∂Nw,Y /∂ρ ≥ 0, i.e.

D0(ρ)H(ρ) − D(ρ)H0(ρ) ≥ 0. CHAPTER 4. REGULARITY THEOREMS 100

By the coarea formula, Z Z D0(ρ) = (2 − n)ρ1−n |Dw|2 + ρ2−n |Dw|2. (4.30) Bρ(Y ) ∂Bρ(Y )

1/2 1/2 2 2 Since ρ kwkL (∂Bρ(Y )) and ρ kDwkL (∂Bρ(Y )) are bounded,

Z ! Z 0 ∂ 1−n 2 H (ρ) = ρ w = 2wDrw. (4.31) ∂ρ ∂Bρ(Y ) ∂Bρ(Y )

We want to prove the following identities Z Z 2 |Dw| = wDrw, (4.32) Bρ(Y ) ∂Bρ(Y ) Z Z 2 2 2 (n − 2) |Dw| = ρ (|Dw| − 2|Drw| ). (4.33) Bρ(Y ) ∂Bρ(Y )

By integration by parts, for δ > 0 and k > 0, Z Z (Xi−Y i) γk(Diw)Diwχδ = wγk(Diw) ρ χδ Bρ(Y ) ∂Bρ(Y ) Z 0 − (γk(Diw)wDiiwχδ + γk(Diw)wDiχδ) Bρ(Y ) and Z 2 j j γk(Dlw) δij − 2γk(Diw)γk(Djw) χδDiζ + ζ Diχδ Bρ(Y ) Z 2 (Xi−Y i) j = γk(Dlw) δij − 2γk(Diw)γk(Djw) χδ ρ ζ ∂Bρ(Y ) Z 0 0 j + (2γk(Dlw)γk(Dlw)Djlw − 2γk(Diw)γk(Djw)Diiw) χδζ Bρ(Y ) Z 0 j − 2γk(Diw)γk(Djw)Dijwχδζ Bρ(Y ) CHAPTER 4. REGULARITY THEOREMS 101

1 n 1 n where ζ = (ζ , . . . , ζ ) ∈ C (Bρ(Y ), R ). Letting δ ↓ 0, Z Z Z (Xi−Y i) 0 γk(Diw)Diw = wγk(Diw) ρ χδ − γk(Diw)wDiiw Bρ(Y ) ∂Bρ(Y ) Bρ(Y ) and Z 2 j γk(Dlw) δij − 2γk(Diw)γk(Djw) Diζ Bρ(Y ) Z 2 (Xi−Y i) j = γk(Dlw) δij − 2γk(Diw)γk(Djw) ρ ζ ∂Bρ(Y ) Z 0 0 j + (2γk(Dlw)γk(Dlw)Djlw − 2γk(Diw)γk(Djw)Diiw) ζ Bρ(Y ) Z 0 j − 2γk(Diw)γk(Djw)Dijwζ . Bρ(Y )

2,2 1/2 1/2 2 2 Since w ∈ W (B1/4(0)) and ρ kwkL (∂Bρ(Y )) and ρ kDwkL (∂Bρ(Y )) are bounded functions of ρ ∈ (0, 1/4 − |Y |), we can let k → ∞ to obtain (4.32) and

Z Z 2 j 2 (Xi−Y i) j |Dw| δij − 2DiwDjw Diζ = |Dw| δij − 2DiwDjw ρ ζ . Bρ(Y ) ∂Bρ(Y )

Let ζj = Xj −Y j to get (4.33). Note that by (4.32), H(ρ) 6= 0 since otherwise Dw = 0 in Bρ(Y ) and thus w is constant in B1/4(0). Hence Nw,Y (ρ) is deﬁned for all ρ > 0 such that Bρ(Y ) ⊂ B1/4(0).

Now using (4.30), (4.31), (4.32), and (4.33), we compute

Z 0 1−n 2 H (ρ) = 2ρ wDrw = D(ρ) ∂Bρ(Y ) ρ Z 0 2−n 2 D (ρ) = 2ρ |Drw| . ∂Bρ(Y ) CHAPTER 4. REGULARITY THEOREMS 102

Thus, using Cauchy-Schwartz,

D0(ρ)H(ρ) − D(ρ)H0(ρ)   Z ! Z ! Z !2 3−2n 2 2 = 2ρ  Drw| |w| − wDrw  ≥ 0. ∂Bρ(Y ) ∂Bρ(Y ) ∂Bρ(Y )

Lemma 14 has following consequences. We can deﬁne the frequency Nw,Y of w at Y to be

Nw,Y = lim Nw,Y (ρ). ρ↓0

n ˘ If Yk → Y in R , using the fact that Nw,Y (ρ) is a continuous function of Y ∈ B1/4(0) and ρ ∈ (0, 1/4 − |Y |) and Nw,Y (ρ) is monotone nondecreasing, we can show that

Nw,Y ≥ lim sup Nw,Yk .

Since 0 ρHw,Y (ρ) Nw,Y (ρ) = , 2Hw,Y (ρ) it follows that

ρ 2Nw,Y (R) ρ 2Nw,Y H (R) ≤ H (ρ) ≤ H (R) (4.34) R w,Y w,Y R w,Y for 0 < ρ < R with ∂BR(Y ) ⊂ B1/4(0).

2,2 Lemma 15. For a two-valued harmonic function w ∈ W (B1/4(0)) of the form ˘n−2 0,1 ˘ (4.16) and Y ∈ Zw ∪ {0} × B1/4 (0), Nw,Y ≥ 1 and thus w ∈ Cloc (B1/4(0)). Proof. Recall from Section 1.2 that

˘ Kw = {X ∈ B1/4(0) : w(X) = 0, Dw(X) = 0}.

˘n−2 If Y ∈ Zw \ (Kw ∪ {0} × B1/4 (0)), then obviously Nw,Y ≥ 1. CHAPTER 4. REGULARITY THEOREMS 103

˘n−2 Suppose Y ∈ Kw \{0} × B1/4 (0) with Kw ∩ Bδ(Y ) = Zw ∩ Bδ(Y ) for some δ > 0.

Then we can ﬁnd a X1 ∈ Bδ(Y ) \Zw and ρ > 0 such that Bρ(X1) ∩ Zw consists of 1 a single point X2 ∈ Ku. We can write w = {+w1, −w1} for some nonnegative, C , single-valued function w1 on Bρ(X1). But w1(X2) = 0 and Dw1(X2) = 0, contradicting the Hopf boundary point lemma. Thus for every Y ∈ Kw, there is a sequence of points Yj ∈ Zw \Kw converging to Y . By the upper semi-continuity of frequency,

Nw,Y ≥ 1.

˘n−2 Suppose Y = (0, y) ∈ {0}×B1/4 (0) and δ ∈ (0, 1/4−|Y |). Since w has the special form (4.16), for every δ ∈ (0, 1/4 − |Y |), there exists a point in Bδ(Y ) ∩ Zw \{0} × ˘n−2 ˘n−2 B1/4 (0). Therefore there is a sequence of points Yj ∈ Zw \{0} × B1/4 (0) converging to Y . By the upper semi-continuity of frequency, Nw,Y ≥ 1.

0,1 ˘ The fact that w ∈ Cloc (B1/4(0)) follows from (4.34) and the Schauder theory.

1,µ We want to show that w ∈ C (B1/8(0)) for some µ ∈ (0, 1/2]. Fix Y ∈ {0} × ˘n−2 B1/4 (0). First we must show that for some σ0 ∈ (0, 1/4 − |Y |) depending on w and Y ,

2 −n/2 2 2 2 σkD wkL (Bσ(Y )) ≤ CkDwkL (Bσ(Y )), sup |Dw| ≤ Cσ kDwkL (Bσ(Y )), (4.35) Bσ(Y ) for σ ∈ (0, σ0] for some constant C = C(n, Nw,Y ) > 0. Since w is harmonic in n−2 B1/4(0) \{0} × R , Z DiwDiζ = 0 (4.36) B1/4(0)

1 n−2 for every two-valued function ζ associated with some ζ0 ∈ Vc (B1/4(0) \{0} × R ). ∞ For δ > 0, let χδ ∈ C (B1(0)) be a single-valued function such that χδ(x, y) = 1 if 2 2 |x| ≥ δ, χδ(x, y) = 0 if |x| ≤ δ/2, and |Dχδ| ≤ 3/δ. Letting ζ = wη χδ in (4.36), 1 where η ∈ Cc (B1/4(0)), Z Z 2 2 2 2 2 |Dw| η χδ ≤ 2 wDiw ηDiηχδ + η χδDiχδ . B1/4(0) B1/4(0) CHAPTER 4. REGULARITY THEOREMS 104

2 Since wDiw ∈ L (B1/4(0)), we can let δ ↓ 0 to obtain

Z Z 2 2 |Dw| η ≤ 2 wDiwηDiη. B1/4(0) B1/4(0)

By Cauchy-Schwartz, Z Z |Dw|2η2 ≤ C w2|Dη|2. B1/4(0) B1/4(0)

Letting 0 ≤ η ≤ 1, η = 1 on Bσ/2(Y ), η = 0 on B1/4(0) \ Bσ(Y ), and |Dη| ≤ 6/σ, we obtain Z Z σ2 |Dw|2 ≤ C w2 (4.37) Bσ/2(Y ) Bσ(Y )

0 1,2 for σ ∈ (0, 1/4 − |Y |). Similarly, since Dw ∈ C (B1/4(0)) ∩ W (B1/4(0)) and Dw is n−2 harmonic on B1/4(0) \{0} × R , Z Z σ2 |D2w|2 ≤ C |Dw|2 (4.38) Bσ/2(Y ) Bσ(Y ) for σ ∈ (0, 1/4 − |Y |). Also,

2 2 2 n−2 ∆|Dw| = 2|D w| ≥ 0 weakly on B1/4(0) \{0} × R ,

2 1,2 so |Dw| is a single-valued function in W (B1/4(0)) that is subharmonic on B1/4(0). By standard elliptic theory, Z sup |Dw|2 ≤ C |Dw|2 (4.39) Bσ/2(Y ) Bσ(Y ) for σ ∈ (0, 1/4 − |Y |). Now observe that for some σ0 ∈ (0, 1/4 − |Y |), Nw,Y (8σ0) ≤

2Nw,Y . By (4.34), Z ρ 4Nw,Y +n−1 Z w2 ≤ w2 (4.40) ∂Bρ(Y ) σ ∂Bσ(Y ) CHAPTER 4. REGULARITY THEOREMS 105

for 0 < ρ ≤ σ ≤ 8σ0. Applying integration to (4.40),

Z ρ 4Nw,Y +n Z w2 ≤ C w2 (4.41) Bρ(Y ) σ Bσ(Y ) for 0 < ρ ≤ σ ≤ 8σ0 for some constant C = C(n, Nw,Y ) > 0. By (4.37), (4.38), (4.39), and (4.41), we obtain (4.35).

Now suppose Nw,Y = 1 and let

w(Y + σX) wσ(X) = 1−n/2 2 σ kDwkL (Bσ(Y )) for Y = (0, y0) ∈ Zw, σ ∈ (0, 1/4 − |Y |], and X ∈ B1/4σ(0). Since w is the two-valued 0,1 function associated with some w0 ∈ V (B1/4(0)), wσ is the two-valued function associated with 1/2 1/2 iθ/2 iθ w0(σ r e , y0) w0,σ(re , y) = 1−n/2 2 σ kDwkL (Bσ(Y )) By (4.35),

2,2 kwσkW (B1(0)) ≤ C, sup |wσ| + sup |Dwσ| ≤ C B1(0) B1(0) for σ ∈ (0, σ0] for some constant C = C(n, Nw,Y ) > 0. It follows that for some σj ↓ 0, 0 0,1 {w0,σj } converges in V (B1(0)) to some non-zero function φ0 ∈ V (B1(0)) and {wσj } 1,2 converges in W (B1(0)) to the two-valued function φ associated with φ0. Moreover 2,2 n−2 φ ∈ W (B1(0)), φ is harmonic on B1(0) \{0} × R , and Nφ,0(ρ) = Nw,Y = 1 for all ρ ∈ (0, 1]. Since Nφ,0(ρ) = 1 for all ρ ∈ (0, 1], by the proof of Lemma iθ iθ 14, Drφ(re , y) = φ(re , y) on B1(0), so φ is homogeneous degree 1. Hence Dφ n−2 is homogeneous degree zero and is harmonic on B1(0) \{0} × R . For δ > 0, ∞ let χδ ∈ C (B1(0)) be a single-valued function such that χδ(x, y) = 1 if |x| ≥ δ,

χδ(x, y) = 0 if |x| ≤ δ/2, and |Dχδ| ≤ 3/δ for some constant C = C(n) > 0. By CHAPTER 4. REGULARITY THEOREMS 106

integration by parts Z Z 2 |∇Sn−1 Dφ| χδ = − (Diφ ∆Sn−1 Diφ χδ + Diφ ∇Sn−1 Diφ · ∇Sn−1 χδ) Sn−1 Sn−1 Z = − Diφ ∇Sn−1 Diφ · ∇Sn−1 χδ. Sn−1

2,2 Since φ ∈ W (B1(0)), we can let δ ↓ 0 to obtain Z 2 |∇Sn−1 Dφ| = 0. Sn−1

Hence φ is an aﬃne function, i.e. φ(X) = {`(X), −`(X)} for some single-valued aﬃne function `(X). But that contradicts the fact that φ is a two-valued function 1 n−2 associated with some φ0 ∈ V (B1(0) \{0} × B1 (0)). Hence Nw,Y > 1. By the gap lemma [11, Lemma 2.5 and Lemma 4.1], Nw,Y ≥ 3/2. By (4.34) and the Schauder 1,1/2 theory, w ∈ C (B1/8(0)).

i 1 m×n Lemma 16. Let m ≥ 1 be an integer, µ ∈ (0, 1/2) and Aκ ∈ C (R ) be a single- i ij valued function such that D λ A (0) = δ δκλ. There are constants γ > 0 and η ∈ Pj κ i (0, 1) depending on n, m, µ, and Aκ such that the following is true. Suppose u ∈ 1 m C (B1(0), R ) is a two-valued function such that Bu = graph g ∩ B1(0) for some single-valued function g ∈ C1(Rn−2, R2) with |Dg| ≤ γ, u satisﬁes (4.8), and u satisﬁes (4.12). Given Λ ∈ Rm×n, there is a Λ0 ∈ Rm×n such that Z Z η−n |Du − Λ0|2 ≤ η2µ |Du − Λ|2. (4.42) Bη(0) B1(0)

i Proof. Let w be as in Lemma 13. Let `(X) = w(0) + X Diw(0). By (4.17), Z (η/8)−n−2 |u − `|2 Bη/8(0) Z Z ≤ Cη−n−2ε |Du − Λ|2 + 2(η/8)−n−2 |w − `|2. (4.43) B1(0) Bη/8(0) CHAPTER 4. REGULARITY THEOREMS 107

Write w = wa + ws. By the standard estimates for single-valued harmonic functions, Z Z −n−2 j 2 −n+2 2 2 (η/8) |wa − `a| ≤ C(η/8) |D wa| Bη/8(0) Bη/8(0) Z 4 2 ≤ Cη |Dwa − Λ| . (4.44) B1/8(0) for some constant C = C(n) > 0. Also by Poincar´einequality Lemma 8, (4.37), (4.38), (4.39), and (4.41),

Z Z −n−2 j 2 3 2 (η/8) |ws − `s| ≤ Cη |Dws| . (4.45) Bη/8(0) B1/8(0) for some constant C = C(n) > 0 provided η < 1/8. Putting together (4.44) and (4.45),

Z Z Z (η/8)−n−2 |w − `|2 ≤ Cη3 |Dw − Λ|2 ≤ Cη3 |Du − Λ|2, Bη/8(0) B1/8(0) B1(0) so by (4.43),

Z Z (η/8)−n−2 |u − `|2 ≤ C(η−n−2ε + η3) |Du − Λ|2. (4.46) Bη/8(0) B1(0)

We need to show Z Z |Du − Dw(0)|2 ≤ CR−2 |u − `|2 (4.47) BR/2(0) BR(0) for R ∈ (0, 1/4). Write Dw(X) = {Dw1(X), Dw2(X)} at X ∈ B1/4(0) and observe that

2 2 2 2 n−2 ∆(|Dw1| + |Dw2| ) = 2|D w| ≥ 0 weakly on B1/4(0) \{0} × R ,

2 2 1,2 so |Dw1| + |Dw2| is a single-valued function in W (B1/4(0)) that is subharmonic CHAPTER 4. REGULARITY THEOREMS 108

on B1/4(0). Hence

2 2 |Dw(0)| ≤ sup |Dw| ≤ CkDwkL (B1/4(0)) ≤ CkDukL (B1(0)) ≤ Cγ. (4.48) B1/8(0)

Using Taylor’s theorem (4.9) write

i i i λ λ i A (P ) = A (Dw(0)) + D λ A (Dw(0)) · (P − Djw (0)) + B (P ; Dw(0)), κ κ Pj κ j κ

i where |Bκ(P ; Dw(0))| ≤ (1/2)|P − Dw(0)| provided γ is suﬃciently small by (4.48), and rewrite (4.22) as

Z Z i λ λ κ i κ D λ A (Dw(0))(Dju − Djw (0))Diζ = − B (Du; Dw(0))Diζ Pj κ κ B1/4(0) B1/4(0) for any two-valued function ζκ such that ζκ(x + g(y), y) is the two-valued function κ 1 κ κ κ 2 1 associated with some ζ0 ∈ Vc (B1(0)). Let ζ = (u − ` )ζ , where ζ ∈ Cc (B1(0)) is a single-valued function. By Cauchy-Schwartz Z Z |Du − Dw(0)|2ζ2 ≤ C |u − `|2|Dζ|2. B1/4(0) B1(0)

n Now let 0 ≤ ζ ≤ 1, ζ = 1 on BR/2(0), ζ = 0 on R \ BR(0), and |Dζ| ≤ 3/R to obtain (4.47). Combining (4.46) and (4.47),

Z Z (η/16)−n |Du − Dw(0)|2 ≤ C(η−n−2ε + η3) |Du − Λ|2 (4.49) Bη/16(0) B1(0) for some constant C = C(n, m) > 0. Let µ ∈ (0, 1/2) and choose ε and η such that Cη−n−2ε + Cη3/2 < (η/8)µ so that

Z Z η˜−n |Du − Dw(0)|2 ≤ η˜2µ |Du − Λ|2 (4.50) Bη˜(0) B1(0) forη ˜ = η/8. CHAPTER 4. REGULARITY THEOREMS 109

m×n Iterating (4.42), we obtain a sequence Λ0 = 0, Λ1, Λ2,... ∈ R such that Z Z −n 2 2µ 2 η |Du − Λj| ≤ η |Du − Λj−1| (4.51) Bηj (0) Bηj−1 (0) for j = 1, 2,.... By (4.51)

Z 2 2µj 2 |Λj − Λj−1| ≤ Cη |Du| (4.52) B1(0)

n×m for some C = C(n, µ, η) > 0, so {Λj} converges to some Λ in R . Moreover, by (4.51) and (4.52),

Z Z η−nj |Du − Λ|2 ≤ Cη2µj |Du|2

Bηj (0) B1(0) for all j = 1, 2,... for some C = C(n, µ, η) > 0, hence Λ = Du(0). Given R ∈ (0, 1), let j be the integer such that ηj+1 < R ≤ ηj to get Z Z R−n |Du − Du(0)|2 ≤ CR2µ |Du|2 (4.53) BR(0) B1(0) for some constant C = C(n, m, η) > 0. For Y ∈ B1/2(0) ∩ Bu, we can translate Y to the origin and rescale so that (4.53) yields

Z Z R−n |Du − Du(Y )|2 ≤ CR2µ |Du|2 (4.54) BR(Y ) B1(0)

i for R ∈ (0, 1/2] and some constant C > 0 depending on n, m, µ, and Aκ.

Similarly, we can show that for Y ∈ B1/2(0) \Bu and 0 < R < R0 = dist(Y, Bu),

Z Z −n 2 2µ −n 2 R G(Du, Du(Y )) ≤ C(R/R0) R0 |Du − Λ| (4.55) BR(Y ) BR0 (Y )

m×n i for all Λ ∈ R with |Λ| ≤ γ for some constant C > 0 depending on n, m, µ, and Aκ. 1 Write u as u = {u1, u2} on BR0 (Y ) for single-valued functions u1, u2 ∈ C (BR0 (Y )) CHAPTER 4. REGULARITY THEOREMS 110

i such that DiA (Dul) = 0 on BR0 (Y ) for l = 1, 2. (4.55) follows from showing Z Z −n 2 2µ −n 2 R |Dul − Dul(Y )| ≤ C(R/R0) R0 |Dul − Λ| for l = 1, 2 (4.56) BR(Y ) BR0 (Y ) for all Λ ∈ Rm×n with |Λ| ≤ γ for some constant C > 0 depending on n, m, µ, i and Aκ. (4.56) follows from an argument similar to the proof of (4.54) given above with slight modiﬁcations so that the argument applies to the case of single-valued functions. In particular, we use a standard harmonic approximation lemma for single- valued functions [8, Lemma 21.1] to more readily prove the analogue of Lemma 13 for single-valued functions. The resulting single-valued harmonic function w is smooth since single-valued harmonic functions in W 1,2 are always smooth by standard elliptic theory. We then repeat the proof of Theorem 16 and the iteration argument involving

(4.42) nearly identically, except for the iteration argument we choose Λ0 = Λ where

Λ is as in (4.56). Now suppose R0 ≤ 1/4 and let Z ∈ Bu such that |Y − Z| = R0. Combining (4.54) and (4.55), we get

i for C > 0 depending on n, m, µ, and Aκ. Therefore Z Z R−n G(Du, Du(Y ))2 ≤ CR2µ |Du|2 (4.57) BR(Y ) B1(0) for all Y ∈ B1/2(0) \Bu and 0 < R < dist(Y, Bu).

Now let X1,X2 ∈ B1/4(0) be distinct points. Let dl = dist(Xl,B1/4(0) ∩ Bu) for l = 1, 2 and suppose 0 ≤ d1 ≤ d2. If |X1 − X2| ≤ d2/2, then by (4.57) with CHAPTER 4. REGULARITY THEOREMS 111

Y = (X1 + X2)/2 and R = 3|X1 − X2|/4,

µ 2 G(Du(X1), Du(X2)) ≤ C|X1 − X2| kDukL (B1(0)) for some constants C = C(n, m) > 0, where G denotes the metric on the space of unordered pairs from Section 1.2. If d2/2 ≤ |X1 − X2| ≤ d2, by (4.54) with

Y ∈ Bu ∩ B1/4(0) such that d2 = |X2 − Y | and R = 2d2,

µ µ 2 2 G(Du(X1), Du(X2)) ≤ Cd2 kDukL (B1(0)) ≤ C|X1 − X2| kDukL (B1(0))

Finally we must suppose |X1 − X2| ≥ d2. First note that if d1 = d2 = 0, then by

(4.54) with Y ∈ Bu ∩B1/4(0) such that |Xl −Y | < 3|X1 −X2|/4 for l = 1, 2 (assuming

|Dg| ≤ γ for γ suﬃciently small) and R = 3|X1 − X2|/4,

µ G(Du(X1), Du(X2)) ≤ C oscB (0,y¯) Du ≤ C|X1 − X2| kDukL2(B (0)). 3|X1−X2|/4 1

Thus for |X1 − X2| ≥ d2, let Yl ∈ Bu ∩ B1/4(0) such that dl = |Xl − Y | for l = 1, 2 so that we obtain

G(Du(X1), Du(X2)) = G(Du(X1), Du(0, y1)) + G(Du(0, y1), Du(0, y2))

+G(u(0, y2), u(X2)) µ µ µ 2 ≤ C(d1 + |y1 − y2| + d2 )kDukL (B1(0)) µ 2 ≤ 3C|X1 − X2| kDukL (B1(0)) for some constant C = C(n, m, µ) > 0. Therefore

1,µ 2 kukC (B1/4(0)) ≤ CkDukL (B1(0)). CHAPTER 4. REGULARITY THEOREMS 112

4.2 Smoothness of two-valued solutions

In the remainder of this chapter, we will prove Theorem 5, which involves studying elliptic equations of the form

i ˘ n−2 Qu = Di(A (X, u, Du)) + B(X, u, Du) = 0 in B1(0) \{0} × R , (4.58)

1,µ ˘ where u ∈ Cloc (B1(0)), µ ∈ (0, 1), is a two-valued function of the form (1.14) and i ˘ n A (X,Z,P ) and B(X,Z,P ) are real analytic single-valued functions on B1(0)×R×R whose regularity will be given. Recall from the discussion in Section 1.3 that u is the 1 two-valued function associated with some u0 ∈ V (B1(0)) and that there are precisely two choices of u0, where if one choice is u0(x, y) the other is u0(−x, y). Fix one choice of u0. Our goal is to determine higher regularity of u(x, y) with respect to the y variable for x ﬁxed. Note that no higher regularity can be expected of u(x, y) with respect to the x variable since there are many examples of solutions u where D2u is singular along {0} × Rn−2. We will begin with the following result.

1,µ ˘ Theorem 19. Let u ∈ Cloc (B1(0)) be a two-valued function such that Bu = {0} × ˘n−2 B1 (0). Suppose u satisﬁes the non-linear, uniformly elliptic diﬀerential equation i ˘ n (4.58) where A and B are real analytic single-valued functions on B1(0) × R × R are single-valued functions such that

i 2 ˘ n n (DjA )(X,Z,P )ζiζj ≥ λ(X,Z,P )|ζ| for (X,Z,P ) ∈ B1(0)×R×R , ζ ∈ R , (4.59)

0 ˘ n γ for some positive single-valued function λ ∈ C (B1(0) × R × R ). Then Dy u ∈ 1,µ ˘ Cloc (B1(0)) for all γ.

γ 1,µ ˘ Proof. We shall prove that Dy u ∈ Cloc (B1(0)) using induction on |γ|. First we shall n−2 1,µ ˘ consider the case |γ| = 1. Let η ∈ R . We shall show that D(0,η)u ∈ Cloc (B1(0)).

Applying δh,η for h > 0 to (4.58), which we can do since u is the two-valued function 1,µ ˘ associated with some u0 ∈ Vloc (B1(0)),

ij i i j ˘ n−2 Di(ah Djδh,ηu + bhδh,ηu + fh) + chDjδh,ηu + dhδh,ηu + gh = 0 in B1−|hη|(0) \{0} × R , CHAPTER 4. REGULARITY THEOREMS 113

ij i j i provided |hη| < 1, where ah , bh, ch, dh, fh, and gh to be the two-valued functions ij i j i 1 ˘ associated with a0,h, b0,h, c0,h, d0,h, f0,h, g0,h ∈ V (B1−|hη|(0)) respectively, which are given by

Z 1 ij i a0,h(x, y) = (DPj A )(x, y + thη, u0,t(x, y), Du0,t(x, y)M)dt, 0 Z 1 i i b0,h(x, y) = (DZ A )(x, y + thη, u0,t(x, y), Du0,t(x, y)M)dt, 0 Z 1 j c0,h(x, y) = (DPj B)(x, y + thη, u0,t(x, y), Du0,t(x, y)M)dt, 0 Z 1 d0,h(x, y) = (DZ B)(x, y + thη, u0,t(x, y), Du0,t(x, y)M)dt, 0 n Z 1 i X i f0,h(x, y) = (DXj A )(x, y + thη, u0,t(x, y), Du0,t(x, y)M)dt · ηj, k=1 0 n X Z 1 g0,h(x, y) = (DXj B)(x, y + thη, u0,t(x, y), Du0,t(x, y)M)dt · ηj, k=1 0

1/2 where M is the Jacobian matrix for the change of variables ξ1 + iξ2 = (x1 + ix2) from Section 1.3 and

u0,t(x, y) = (1 − t)u0(x, y) + tu0(x, y + hη).

i ˘ n By (4.59) and our assumption that A and B are real analytic on B1(0) × R × R ,

ij iθ ¯ 2 2 2iθ ˘ n a0,h(re , y)ζiζj ≥ λ(X)|ζ| whenever (r e , y) ∈ B1−|hη|(0), ζ ∈ R , ij i i 0,µ ˘ j 0 ˘ ah , bh, fh ∈ C (B1−|hη|(0)), ch, dh, gh ∈ C (B1−|hη|(0)),

¯ 0 ˘ for some positive single-valued function λ ∈ C (R). Let BR(x0, y0) ⊂ B1(0) and CHAPTER 4. REGULARITY THEOREMS 114

suppose |hη| ≤ R/4. By the Schauder estimates Theorem 10 from Section 2.3,

0 RkDδ uk 0,µ h,η C (BR/4(x0,y0)) µ 2 0 0 ≤ C kδh,ηukC (BR/2(x0,y0)) + R [fγ]µ,BR/2(x0,y0) + R kgγkC (BR/2(x0,y0)) 0 ≤ C|η| kDyukC (BR(x0,y0)) + 1 for some constant C > 0 depending on n, µ, A, and B and independent of h. So given any sequence of hj → 0, we can pass to a subsequence {δhj0 ,ηu} that converges in 1 ˘ 1,µ ˘ Cloc(B1(0)) with a limit in Cloc (B1(0)). But δh,ηu → D(0,η)u pointwise, where D(0,η)u n denotes the derivative of u in the direction (0, η) ∈ R . So in fact δh,ηu → D(0,η)u in 1 ˘ 1,µ ˘ Cloc(B1(0)) and D(0,η)u ∈ Cloc (B1(0)).

γ 1,µ ˘ Now suppose for some p ≥ 1 that whenever |γ| < p, Dy u ∈ Cloc (B1(0)). Let γ be n−2 γ 1,µ multi-index with |γ| = p−1 and let η ∈ R . We shall show that D(0,η)Dy u ∈ C (Ω) ˘ for every closed subset Ω of B1(0). Write

γ i i γ i k Dy (A (X, u, Du)) = (DPj A )(X, u, Du)DjDy u + Fγ(X, u, {DDy u}k≤p−2), γ γ k Dy (B(X, u, Du)) = (DPj B)(X, u, Du)DjDy u + Gγ(X, u, {DDy u}k≤p−2),

i for some functions Fγ and Gγ. To simplify notation, let

ij i i i k a = (DPj A )(X, u, Du), fγ = Fγ(X, u, {DDy u}k≤|γ|−1), j k b = (DPj B)(X, u, Du), gγ = Gγ(X, u, {DDy u}k≤|γ|−1). (4.60)

i Note that we can express fγ as

i X α i Y βZ,l Y βP,l fγ = cα,j,β(D(y,Z,P )A )(X, u, Du) · Dy u · Dy Djl u (4.61)

l≤|αZ | l≤|αP |

α αy αZ αP where α = (αy, αZ , αP ) and D(y,Z,P ) = Dy D D to condense notation and where the sum is taken over nonzero multi-induces α, βZ,k, and βP,l and 1 ≤ jk ≤ n such CHAPTER 4. REGULARITY THEOREMS 115

that X X αy + βZ,l + βP,l = γ (4.62)

l≤|αZ | l≤|αP | and |βP,l| < p and the coeﬃcients cα,j,β are positive integers depending on α, j1, . . . , j|α|, and β1, . . . , β|α|. Note that in (4.61) and (4.62) assume the convention that sums over l ≤ 0 equal zero and products over l ≤ 0 equal one. It follows from (4.61), our i ˘ n assumption that A is real analytic on B1(0) × R × R , and our induction hypothesis i i 0,µ ˘ on the regularity of u that fγ,Dyfγ ∈ Cloc (B1(0)). We can write a similar expres- 0 ˘ sion for gγ and by an analogous argument deduce gγ,Dygγ ∈ C (B1(0)). Moreover, i ˘ n by our our assumption that A and B are real analytic on B1(0) × R × R and our ij ij 0,µ ˘ induction hypothesis on the regularity of u, we have a ,Dya ∈ C (B1(0)) and j j 0 ˘ γ b ,Dyb ∈ C (B1(0)). Applying Dy to (4.58),

ij γ j γ i ˘ n−2 Di(a DjDy u) + b DjDy u = Difγ + gγ in B1(0) \{0} × R , (4.63)

Then applying δh,η to (4.63)

ij γ ij γ j γ Di(a (X + hη)Djδh,ηDy u(X) + δh,ηa (X)DjDy u(X)) + b (X + hη)Djδh,ηDy u(X) j γ i ˘ n−2 + δh,ηb (X)DjDy u(X) = Diδh,ηfγ(X) + δh,ηgγ(X) in B1(0) \{0} × R ,

1,µ ˘ which makes sense since u is the two-valued function associated with u0 ∈ Vloc (B1(0)) ij j i and a , b , fγ, and gγ are the two-valued functions associated with single-valued 0 ˘ functions in V (B1(0)). By (4.59),

ij 2 ˘ n a (X)ζiζj ≥ λ(X, u(X), Du(X))|ζ| for X ∈ B1(0), ζ ∈ R . CHAPTER 4. REGULARITY THEOREMS 116

By the Schauder estimates Theorem 10 from Section 2.3 and Lemma 3 and the prop- ˘ erties of hµ,BR/2 (0) from Section 1.3, if BR(x0, y0) ⊂ B1(0) and |hη| ≤ R/4,

γ 0 RkDδ D uk 0,µ h,η y C (BR/4(x0,y0)) γ ij 0 γ 0 ≤ C kδ D uk 0 + Rkδ a k 0,µ kDD uk 0,µ h,η y C (BR/2(x0,y0)) h,η C (BR/2(x0,y0)) y C (BR/2(x0,y0+hη)) 2 j γ 0 0 + R kδh,ηb kC (BR/2(x0,y0))kDDy ukC (BR/2(x0,y0+hη)) 0 2 +Rkδ f k 0,µ + R kδ g k 0 h,η γ C (BR/2(x0,y0)) h,η γ C (BR/2(x0,y0))

γ ij 0 γ 0 ≤ C|η| kD D uk 0 + RkD a k 0,µ kDD uk 0,µ y y C (BR(x0,y0)) y C (BR(x0,y0)) y C (BR(x0,y0)) 2 j γ 0 0 + R kDyb kC (BR(x0,y0))kDDy ukC (BR(x0,y0)) 0 2 +RkD f k 0,µ + R kD g k 0 y γ C (BR(x0,y0)) y γ C (BR(x0,y0)) for some constant C > 0 depending on n, µ, A, and B and independent of h. So given γ any sequence of hj → 0, we can pass to a subsequence {δhj0 ,ηDy u} that converges in 1 ˘ 1,µ ˘ γ γ Cloc(B1(0)) with a limit in Cloc (B1(0)). But δh,ηDy u → D(0,η)Dy u pointwise, where γ γ n D(0,η)Dy u denotes the derivative of Dy u in the direction (0, η) ∈ R . So in fact γ γ 1 ˘ γ 1,µ ˘ δh,ηDy u → D(0,η)Dy u in Cloc(B1(0)) and D(0,η)Dy u ∈ Cloc (B1(0)).

We have a similar result for linear systems. The proof is the same as for Theorem 19, except we use Theorem 11 instead of Theorem 10.

1,µ ˘ m Theorem 20. Let u ∈ C (B1(0), R ) be a two-valued function of the form (1.14). Suppose u is a solution to the non-linear, uniformly elliptic diﬀerential equation

i ˘ n−2 Di(Aκ(X, u, Du)) + Bκ(X, u, Du) = 0 in B1(0) \{0} × R , κ = 1, . . . , l

i ˘ m m×n where m ≥ 2 and Aκ and Bκ are real analytic functions on B1(0) × R × R such that

i ij ˘ m l×n |(D λ A )(X,Z,P ) − δ δκλ| < ε for (X,Z,P ) ∈ B1(0) × × , Pj κ R R

γ 1,µ ˘ m where ε > 0 is as in Theorem 11. Then Dy u ∈ Cloc (B1(0), R ) for all γ. CHAPTER 4. REGULARITY THEOREMS 117

Now we want to prove Theorem 5, which shows that the solution u(x, y) is real analytic with respect to y. One approach to showing a single-valued solution to an elliptic equation is real analytic is an approach by Morrey [5, Sections 5.8 and 6.7] [6] of using integral kernels to show u extends to a holomorphic function on some domain in Cn. However, we cannot use integral kernels for two-valued functions. So instead we take another approach of inductively using the Schauder estimates. For example, in the special case that u is two-valued function that is harmonic on ˘ n−2 B1(0) \{0} × R , we can show that

p γ Cp G(D u(X), 0) ≤ kuk 0 y R C (BR(X0))

˘ for all multi-induces γ with |γ| = p if BR(X) ⊂ BR0 (X0) ⊂ B1(0), where G denotes the metric on the space of unordered pairs from Section 1.2. The induction step involves γ 1,µ observing that Dy u is a two-valued function in C (BR0 (X0)) that is harmonic on n−2 BR0 (X0) \{0} × R , so by the Schauder estimates,

γ γ G(DyDy u(X), 0) ≤ sup G(DyDy u, 0) BR/2(p+1)(X)

C(p + 1) γ ≤ sup G(Dy u(X1), 0) R X1∈BR/(p+1)(X) C(p + 1)p+1 ≤ kukC0(B (X )) R R0 0 since if X1 ∈ BR/(p+1)(X0) then BpR/(p+1)(X1) ⊂ BR(X0). Now consider the case of Theorem 5 where u satisﬁes (4.58). Recall from the proof of Theorem 19 that

ij γ j γ i ˘ n−2 Di(a DjDy u) + b DjDy u = Difγ + gγ in B1(0) \{0} × R ,

ij i for all multi-values γ, where a , b , fγ, and gγ are given by (4.60). Hence by Schauder CHAPTER 4. REGULARITY THEOREMS 118

˘ estimate Theorem 10, for every ball BR(X1) ⊂ B1(0),

γ 0 γ 1+µ kDD uk 1,µ ≤ C kD uk 0 + (R/p) [f ] y C (BR/2p(X1)) y C (BR/p(X1)) γ µ,BR/p(X1)

2 0 +(R/p) kgγkC (BR/p(X1)) .

i Recall that fγ and gγ can be expressed in terms of the functions u, Du, DDyu, . . . , p−1 γ 0 DD u. Thus we can inductively compute bounds on kDD uk 0,µ . The y y C (BR/2p(X0))

0 diﬃculty is bounding [fγ]µ,BR/p(X1) and kgγkC (BR/p(X1)). We accomplish this using a modiﬁed version of a technique due to Friedman in [2]. Since the estimate on

0 kgγkC (BR/p(X1)) is easier to obtain, we will show that ﬁrst in Section 4.3, and then we will get the estimate on [fγ]µ,BR/p(X1) in Section 4.4.

4.3 Bounding gγ

Lemma 17. Let p ≥ 5 be a positive integer and K0,K,H0 ≥ 1 be constants. Let ˘ BR(X1) ⊂ B1(0). For some constants C > 0 and H ≥ 1 depending on n, K0, K, and

H0 and independent of p, if for any multi-index α = (αX , αZ , αP ) with |α| = k,

α k −1−|αX |−|αZ | |D(X,Z,P )B(X,Z,P )| ≤ K0K R for k = 2, 3,

α k −1−|αX |−|αZ | |D(X,Z,P )B(X,Z,P )| ≤ (k − 3)!K0K R for 4 ≤ k ≤ p,

0 for X ∈ B (X ) and |Z| + R|P | ≤ kuk 1 and for any multi-index β with R 1 C (BR(X1)) |β| = q < p and µ ∈ (0, 1],

−1 β 0 −q (R/q) kD uk 1 ≤ H R for q = 0, 1, 2, y C (BR/p(X1)) 0 −1 β 0 q−3 −q (R/q) kD uk 1 ≤ (q − 2)!H H R for 3 ≤ q < p, y C (BR/p(X1)) 0 then p−3 −p 0 RkgγkC (BR/p(X1)) ≤ C(p − 2)!H R .

Proof. Suppose we had a function Ψ(y1, . . . , yn−2,Z,P1,...,Pn) such that Ψ(0, 0, 0) = CHAPTER 4. REGULARITY THEOREMS 119

0 and

|Dαy DαZ DαP B(X,Z,P )| ≤ Dαy DαZ DαP Ψ(0, 0, 0) (4.64) y Z P y z0 (z1,...,zn)

0 for X ∈ B (X ) and |Z| + R|P | ≤ kuk 1 and for all nonnegative integers α R 1 C (BR(X1)) Z and multi-induces αy and αP such that 1 ≤ |αy| + αZ + |αP | ≤ p. Further suppose we had functions vj(y1, . . . , yn−2), j = 0, 1, . . . , n, such that v0(0) = 0 and

β β 0 kDy ukC (BR/p(X1)) ≤ Dy v0(0) (4.65)

γ for 0 ≤ |β| ≤ p and for j = 1, . . . , n, vj(0) = 0, Dy vj(0) ≥ 0, and

β β 0 kDy DjukC (BR/p(X1)) ≤ Dy vj(0) (4.66)

for 0 ≤ |β| < p. We will later construct appropriate functions Ψ and vj, j = 0, 1, . . . , n. Recall from the proof of Theorem 19 that

X α Y βZ,1 Y βP,l gγ = cα,j,β(D(y,Z,P )B)(X, u, Du) · Dy u · Dy Djl u (4.67)

l≤|αZ | l≤|αP | where the sum is taken over α = (αy, αZ , αP ), βZ,k, βP,l, and 1 ≤ jk ≤ n such that

(4.62) holds and |βP,l| < p, and the cαy,αZ ,αP ,j,β are positive integers. We also have

γ X α Y βZ,1 Y βP,l Dy (Ψ(y, v)) = cα,j,β(D(y,Z,P )Ψ)(y, v) · Dy v0 · Dy vjl (4.68)

l≤|αZ | l≤|αP | where the sum is taken over (4.62) and the coeﬃcients cα,j,β are the same as above.

Here v = (v0, v1, . . . , vn) and Ψ(y, v) = Ψ(y1, . . . , yn−2, v0, v1, . . . , vn−2). Comparing (4.67) and (4.68) and using (4.64), (4.65), and (4.66), we obtain

γ 0 kgγkC (BR/p(X1)) ≤ Dy (Ψ(y, v(y)))|y=0.

For simplicity, we may suppose v0(y) = Rv(y1 + ··· + yn−2) and v1(y) = ... = vn(y) = v(y1 + ··· + yn−2) for some function v(ξ) where ξ = y1 + ··· + yn−2 and CHAPTER 4. REGULARITY THEOREMS 120

replace Ψ(y1, . . . , yn−2,Z,P1,...,Pn) with Ψ(ξ, ζ) where ξ = y1 + ··· + yn−2 and −1 ζ = R Z + P1 + ··· + Pn so that Ψ(y, v0, v1, . . . , vn) is replaced with Ψ(y, (1 + n)v). Under this simpliﬁed setup, we require that Ψ(0, 0) = 0,

|αZ | αy αZ αP |αy| αZ +|αP | R |Dy DZ DP B(X,Z,P )| ≤ Dξ Dζ Ψ(0, 0)

0 for X ∈ B (X ), |Z| + R|P | ≤ kuk 1 , and 1 ≤ |α | + α + |α | ≤ p, v(0) = 0, R 1 C (BR(X1)) y Z P

−1 β |β| 0 R kDy ukC (BR/p(X1)) ≤ Dy v(0) for 0 < |β| ≤ p, and β |β| 0 kDy DjukC (BR/p(X1)) ≤ Dy v(0) for j = 1, . . . , n and 0 < |β| < p.

We can choose

2 p X 1 X 1 Ψ(ξ, ζ) = K KkR−1(R−1ξ + ζ)k + K KkR−1(R−1ξ + ζ)k, k! 0 k(k − 1)(k − 2) 0 k=1 k=3 p 1 X 1 v(ξ) = H R−1ξ + H R−2ξ2 + H Hq−3R−qξq. 0 2 0 q(q − 1) 0 q=3

q q For functions f(ξ) and g(ξ), let f p g denote that |Dξ f(0, 0)| ≤ Dξ g(0, 0) for 1 ≤ q ≤ p. We claim that for k = 1, 2, . . . , p

−1 k k −k k (R ξ + (1 + n)v(ξ)) p (1 + (1 + n)H0) R ξ p X ck−1 + ck−1(1 + (1 + n)H )kR−k−1ξk+1 + (1 + (1 + n)H )kHq−k−2R−qξq. 0 (q − k)2 0 q=k+2 for some constant c ≥ 1 independent of k. This follows by induction with the induction step given by the following computation with k ≥ 2: CHAPTER 4. REGULARITY THEOREMS 121

(R−1ξ + (1 + n)v(ξ))k = (R−1ξ + (1 + n)v(ξ)) · (R−1ξ + (1 + n)v(ξ))k−1 1 (1 + (1 + n)H )kR−kξk + + ck−2 (1 + (1 + n)H )kR−k−1ξk+1 p 0 2 0 p X 6 + ck−2(1 + (1 + n)H )kHq−k−2R−qξq (q − k)2 0 q=k+2 p q−k−1 X X ck−2 + (1 + (1 + n)H )kHq−k−4R−qξq (j − 1)2(q − j − k + 1)2 0 q=k+4 j=3

Since

q−k q−k 2 X 1 X 1 1 1 = + (j − 1)2(q − j − k + 1)2 (q − k)2 j − 1 q − j − k + 1 j=2 j=2 q−k X 1 4 2π2 ≤ ≤ (4.69) (q − k)2 (j − 1)2 3(q − k)2 j=2 we have

(R−1ξ + (1 + n)v(ξ))k 3 (1 + (1 + n)H )kR−kξk + ck−2(1 + (1 + n)H )kR−k−1ξk+1 p 0 2 0 p X (6 + 2π2/3)ck−2 + (1 + (1 + n)H )kHq−k−2R−qξq, (q − k)2 0 q=k+2 so the claim follows if c = 6 + 2π2/3. CHAPTER 4. REGULARITY THEOREMS 122

Now for p ≥ 5,

2 X 1 RΨ(ξ, (n + 1)v(ξ)) K Kk (1 + (1 + n)H )kR−kξk p k! 0 0 k=1 p ! X ck−1 +ck−1(1 + (1 + n)H )kR−k−1ξk+1 + (1 + (1 + n)H )kHq−k−2R−qξq 0 (q − k)2 0 q=k+2 p X 1 + K Kk (1 + (1 + n)H )kR−kξk + ck−1(1 + (1 + n)H )kR−k−1ξk+1 (k − 2)3 0 0 0 k=3 p−2 p ! X 1 X ck−1 + + K Kk (1 + (1 + n)H )kHq−k−2R−qξq (k − 2)3 0 (q − k)2 0 k=3 q=k+2

It follows that

p 1 ∂ 1 p−3 −p R p Ψ(ξ, (n + 1)v(ξ)) ≤ 2 K0K(1 + (1 + n)H0)H R p! ∂ξ ξ=0 (p − 1) c 1 + K K2(1 + (1 + n)H )2Hp−4R−p + K Kp(1 + (1 + n)H )pR−p 2(p − 2)2 0 0 (p − 2)3 0 0 1 + cp−2K Kp−1(1 + (1 + n)H )p−1R−p (p − 3)3 0 0 p−2 X ck−1 + K Kk(1 + (1 + n)H )kHp−k−2R−p (k − 2)2(p − k)2 0 0 k=3

Suppose H satisﬁes cK(1 + (1 + n)H0) ≤ H so that, using a computation similar to (4.69), we have

p 1 ∂ C 3 3 p−3 −p R p Ψ(ξ, (n + 1)v(ξ)) ≤ 2 K0K (1 + (1 + n)H0) H R p! ∂ξ ξ=0 (p − 2) for some constant C > 0 independent of p. Thus

p−3 −p 0 RkgγkC (BR/p(X1)) ≤ C(p − 2)!H R

for some constant C = C(n, K0,K,H0) > 0 independent of p. CHAPTER 4. REGULARITY THEOREMS 123

i 4.4 Bounding the H¨oldercoeﬃcient of fγ

Lemma 18. Let p ≥ 5 be a positive integer and K0,K,H0 ≥ 1 be constants. Let ˘ BR(X1) ⊂ B1(0). For some constants C > 0 and H ≥ 1 depending on n, K0, K, and

H0 and independent of p, if for any multi-index α = (αX , αZ , αP ) with |α| = k,

α i k −|αX |−|αZ | |D(X,Z,P )A (X,Z,P )| ≤ K0K R for k = 2, 3,

α i k −|αX |−|αZ | |D(X,Z,P )A (X,Z,P )| ≤ (k − 3)!K0K R for 4 ≤ k ≤ p + 1,

0 for X ∈ B (X ) and |Z| + R|P | ≤ kuk 1 and for any multi-index β with R 1 C (BR(X1)) |β| = q < p and µ ∈ (0, 1],

−1 β 0 −q (R/q) kD uk 1,µ ≤ H R for q = 1, 2, y C (BR/p(X1)) 0 −1 β 0 q−3 −q (R/q) kD uk 1,µ ≤ (q − 2)!H H R for 3 ≤ q < p, y C (BR/p(X1)) 0 then µ i p−3 1−p (R/p) [fγ]µ,BR/p(X1) ≤ C(p − 2)!H R .

Proof. We will use a similar argument as for Lemma 17, except now we need to compute a Höldercoefficient. To this we will introduce an auxiliary parameter t such that derivatives of Ψ and v with respect to t bound to Höldercoefficients of expressions involving Ai and u. The basic idea is that the sum, product, and chain rules for computing derivatives with respect to t are similar to sum, product, and composition rules for computing Höldercoefficients.

Suppose we had a function Ψ(y1, . . . , yn−2, t, Z, P1,...,Pn) such that Ψ(0, 0, 0, 0) = 0 and

|Dαy DαZ DαP Ai(X,Z,P )| ≤ Dαy DαZ DαP Ψ(0, 0, 0, 0), y Z P y z0 (z1,...,zn) (2R/p)1−µ|D Dαy DαZ DαP Ai(X,Z,P )| ≤ D Dαy DαZ DαP Ψ(0, 0, 0, 0), (4.70) y y Z P t y z0 (z1,...,zn)

0 for X ∈ B (X ), |Z| + R|P | ≤ kuk 1 , and 1 ≤ |α | + α + |α | ≤ p and we R 1 C (BR(X1)) y Z P CHAPTER 4. REGULARITY THEOREMS 124

γ had functions vj(y1, . . . , yn−2, t), j = 0, 1, . . . , n, such that vj(0, 0) = 0 and Dy vj(0, 0) for j = 0, 1, . . . , n,

β β 0 kDy ukC (BR/p(X1)) ≤ Dy v0(0, 0) for 0 < |β| ≤ p, β β [Dy u]µ,BR/p(X1) ≤ DtDy v0(0, 0) for 0 ≤ |β| ≤ p, (4.71) and

β β 0 kDy DjukC (BR/p(X1)) ≤ Dy vj(0, 0) for 0 < |β| < p, β β [Dy Dju]µ,BR/p(X1) ≤ DtDy v0(0, 0) for 0 ≤ |β| < p, (4.72)

for j = 1, . . . , n. We will later construct appropriate functions Ψ and vj. Recall from the proof of Theorem 19 that

i X α i Y βZ,1 Y βP,l fγ = cα,j,β(D(y,Z,P )A )(X, u, Du) · Dy u · Dy Djl u

l≤|αZ | l≤|αP | where the sum is taken over α = (αy, αZ , αP ), βZ,k, βP,l, and 1 ≤ jk ≤ n such that

(4.62) holds and |βP,l| < p, and the cαy,αZ ,αP ,j,β are positive integers. Using Lemma 3 and the properties of hµ,BR/p(X1) from Section 1.3,

1 [f i ] 2 γ µ,BR/p(X1) 

X α i Y βZ,1 Y βP,l 0 0 ≤ cα,j,β [(D(y,Z,P )A )(X, u, Du)]µ,BR/p(X1) kDy ukC · kDy Djl ukC

l≤|αZ | l≤|αP |

α i X βZ,1 Y βZ,m Y βP,l 0 0 0 + kD(X,Z,P )A kC · [Dy u]µ kDy ukC · kDy Djl ukC

l≤|αZ | m6=l l≤|αP | 

α i Y βZ,1 X βP,l Y βP,l 0 0 0 +kD(X,Z,P )A kC · kDy ukC · [Dy Djl u]µ kDy Djl ukC  , (4.73)

l≤|αZ | l≤|αZ | m6=l

i where the norms of derivatives of A are taken over X ∈ BR(X1) and |Z| + R|P | ≤ 0 kuk 1 and the norms and H¨oldercoeﬃcients of derivatives of u are taken over C (BR(X1)) CHAPTER 4. REGULARITY THEOREMS 125

BR/p(X1), and

α i [(D(y,Z,P )A )(X, u, Du)]µ,BR/p(X1) 1−µ α i 0 ≤ (2R/p) k(DyD(y,Z,P )A )(X, u, Du)kC (BR/p(X1)) α i 0 + k(DZ D(y,Z,P )A )(X, u, Du)kC (BR/p(X1))[u]µ,BR/p(X1) n X α i 0 + k(DPl D(y,Z,P )A )(X, u, Du)kC (BR/p(X1))[Dlu]µ,BR/p(X1) l=1

We also have

γ X α Y βZ,1 Y βP,l Dy (Ψ(y, t, v0, v1, . . . , vn)) = cα,j,β(D(y,Z,P )Ψ)(y, v) · Dy v0 · Dy vjl

l≤|αZ | l≤|αP | where the sum is taken over (4.62) and the coeﬃcients cα,j,β are the same as before.

Here v = (v0, v1, . . . , vn) and Ψ(y, t, v) = Ψ(y1, . . . , yn−2, t, v0, v1, . . . , vn−2). Thus

γ DtDy (Ψ(y, t, v(y, t))) 

X α Y βZ,1 Y βP,l = cα,j,β Dt((D(y,Z,P )Ψ)(y, t, v)) Dy v0 · Dy vjl

l≤|αZ | l≤|αP |

α X βZ,1 Y βZ,m Y βP,l + (D(y,Z,P )Ψ)(y, t, v)) · DtDy v0 Dy v0 · Dy vjl

l≤|αZ | m6=l l≤|αP | 

α Y βZ,1 X βP,l Y βP,l +(D(y,Z,P )Ψ)(y, t, v)) · Dy v0 · DtDy vjl Dy vjl  (4.74)

l≤|αZ | l≤|αZ | m6=l where

α α Dt((D(y,Z,P )Ψ)(y, t, v)) ≤ (DtD(y,Z,P )Ψ)(y, t, v)) n α X α + (D(y,Z,P )Ψ)(y, t, v))Dtv0 + (DPl D(y,Z,P )Ψ)(y, t, v))Dtvl. l=1 CHAPTER 4. REGULARITY THEOREMS 126

Comparing (4.73) and (4.74) and using (4.64), (4.65), and (4.66), we obtain

[f i ] ≤ 2D ((Dα Ψ)(y, t, v(t, y))) . γ µ,BR/p(X1) t (y,Z,P ) y=0,t=0

For simplicity, we may suppose v0(y, t) = Rv(y1 + ··· + yn−2, t) and v1(y, t) = ... = vn(y, t) = v(y1 + ··· + yn−2, t) for some function v(ξ, t) where ξ = y1 + ··· + yn−2 and replace Ψ(y1, . . . , yn−2, t, Z, P1,...,Pn) with Ψ(ξ, t, ζ) where ξ = y1 + ··· + yn−2 and −1 ζ = R Z +P1 +···+Pn so that Ψ(y, t, v0, v1, . . . , vn) is replaced with Ψ(y, t, (1+n)v). Under this simpliﬁed setup, we require that Ψ(0, 0, 0) = 0,

|αZ | αy αZ αP i |αy| αZ +|αP | R |Dy DZ DP A (X,Z,P )| ≤ Dξ Dζ Ψ(0, 0, 0),

1−µ 1−µ+|αZ | αy αZ αP i |αy| αZ +|αP | (2/p) R |DyDy DZ DP A (X,Z,P )| ≤ DtDξ Dζ Ψ(0, 0, 0),

0 for X ∈ B (X ), |Z|+R|P | ≤ kuk 1 , and 1 ≤ |α |+α +|α | ≤ p, v(0, 0) = 0, R 1 C (BR(X1)) y Z P

−1 β |β| 0 R kDy ukC (BR/p(X1)) ≤ Dy v(0, 0) for 0 < |β| ≤ p, −µ −1+µ β |β| p R [Dy u]µ,BR/p(X1) ≤ DtDy v(0, 0) for 0 ≤ |β| ≤ p, and

β |β| 0 kDy DjukC (BR/p(X1)) ≤ Dy v(0, 0) for 0 < |β| < p, µ β |β| (R/p) [Dy Dju]µ,BR/p(X1) ≤ DtDy v(0, 0) for 0 ≤ |β| < p, for j = 1, . . . , n. CHAPTER 4. REGULARITY THEOREMS 127

We can choose

Ψ(ξ, t, ζ) = (R−1ξ + ζ + (R/2p)−µt) p+1 ! 1 X 1 · K K + K K2(R−1ξ + ζ) + K Kk(R−1ξ + ζ)k−1 , 0 2 0 k(k − 1)(k − 2) 0 k=3 p ! 1 X 1 v(ξ, t) = (1 + (R/p)−µt) H R−1ξ + H R−2ξ2 + H Hq−3R−qξq . 0 2 0 q(q − 1) 0 q=3

q q or functions f(ξ, t) and g(ξ, t), let f p,1 g denote that |Dξ f(0, 0)| ≤ Dξ g(0, 0) and q q |DtDξ f(0, 0)| ≤ DtDξ g(0, 0) for 1 ≤ q ≤ p. Since

−1 −µ −1 R ξ + (1 + n)v(ξ, t) p,1 (1 + (R/p) t)R ξ + (1 + n)v(ξ, t) p ! 1 X 1 (1 + (R/p)−µt)(1 + (n + 1)H ) R−1ξ + R−2ξ2 + Hq−3R−qξq p,1 0 2 q(q − 1) q=3

By the computation of (R−1ξ+(1+n)v(ξ))k in the previous section, for k = 1, 2, . . . , p,

−1 k −µ k −k k (R ξ + (1 + n)v(ξ, t)) p,1 (1 + k(R/p) t) (1 + (1 + n)H0) R ξ p ! X ck−1 +ck−1(1 + (1 + n)H )kR−k−1ξk+1 + (1 + (1 + n)H )kHq−k−2R−qξq , 0 (q − k)2 0 q=k+2 CHAPTER 4. REGULARITY THEOREMS 128

¯ where c ≥ 1 is constant and is independent of k. Let H0 = 1 + (n + 1)H0. For p ≥ 5,

p ! 1 X 1 Ψ(ξ, t, ζ) K K(1 + (R/p)−µt)H¯ R−1ξ + R−2ξ2 + Hq−3R−qξq p,1 0 0 2 (q − 1)2 q=3 p ! 1 X c + K K2(1 + 2(R/p)−µt)H¯ 2 R−2ξ2 + cR−3ξ3 + Hq−4R−qξq 2 0 0 (q − 2)2 q=4 p X 1 + K Kk(1 + k(R/p)−µt)H¯ k R−kξk + ck−1R−k−1ξk+1 (k − 2)3 0 0 k=3 p−2 p X 1 X ck−1 + K Kk(1 + k(R/p)−µt)H¯ k Hq−k−2R−qξq (k − 2)3 0 0 (q − k)2 k=3 q=k+2 p ! 1 1 X 1 + (R/2p)−µt K K + K K2H¯ R−1ξ + R−2ξ2 + Hq−3R−qξq 0 2 0 0 2 (q − 1)2 q=3 p+1 X 1 + K KkH¯ k−1 R1−kξk−1 + ck−2R−kξk (k − 2)3 0 0 k=3 p−1 p ! X 1 X ck−2 + K KkH¯ k−1 Hq−k−1R−qξq (k − 2)3 0 0 (q − k + 1)2 k=3 q=k+1

It follows that

p+1 µ 1 ∂ (R/p) p Ψ(ξ, (n + 1)v(ξ)) p! ∂t∂ξ ξ=0,t=0 2 c ≤ K K(1 + (1 + n)H )Hp−3R−p + K K2(1 + (1 + n)H )2Hp−4R−p (p − 1)2 0 0 (p − 2)2 0 0 3 5cp−2 + K Kp+1(1 + (1 + n)H )pR−p + K Kp(1 + (1 + n)H )p−1R−p (p − 1)2 0 0 (p − 3)2 0 0 p−1 X 4ck−1 + K Kk(1 + (1 + n)H )kHp−k−2R−p (k − 2)2(p − k + 1)2 0 0 k=3

Suppose H satisﬁes cK(1 + (1 + n)H0) ≤ H so that, using a computation similar to CHAPTER 4. REGULARITY THEOREMS 129

(4.69), we have

p+1 µ 1 ∂ C 4 3 p−3 −p (R/p) p Ψ(ξ, (n + 1)v(ξ)) ≤ 2 K0K (1 + (1 + n)H0) H R p! ∂t∂ξ ξ=0,t=0 (p − 3) for some constant C > 0 independent of p. Thus

µ i 3 p−3 −p (R/p) [fγ]µ,BR/p(X1) ≤ C(p − 2)!H0 H R

for some constant C = C(n, K0,K,H0) > 0 independent of p.

4.5 Analyticity of two-valued solutions

Theorem 5 will be a consequence of the following lemma.

Lemma 19. Assume the hypotheses of Theorem 5 and let K0,K,H0 ≥ 1 and ˘ BR0 (X0) ⊂ B1(0). For some H = H(n, K0,K,H0) ≥ 1the following holds. Suppose whenever BR(X1) ⊆ BR0 (X0), for every multi-index α = (αX , αZ , αP ) with |α| = k

α α k −|αX |−|αZ | |D(X,Z,P )A| + R|D(X,Z,P )B| ≤ K0K R for k = 2, 3,

α α k −|αX |−|αZ | |D(X,Z,P )A| + R|D(X,Z,P )B| ≤ (k − 3)!K0K R for 4 ≤ k < p,

0 for X ∈ B (X ) and |Z| + R|P | ≤ kuk 1 , and for every multi-index β with R 1 C (BR(X1)) |β| = q

−1 β 0 −q (R/q) kD uk 1,µ ≤ H R for q = 1, 2, y C (BR/2(X1)) 0 −1 β 0 q−2 −q (R/q) kD uk 1,µ ≤ (q − 2)!H H R for 3 ≤ q < p. y C (BR/2q(X1)) 0

Then for every BR(X1) ⊆ BR0 (X0) and multi-index γ with |γ| = q,

−1 γ 0 p−2 1−p (R/p) kD uk 1,µ ≤ (p − 2)!H H R . y C (BR/2p(X1)) 0 CHAPTER 4. REGULARITY THEOREMS 130

γ Proof. Recall from Section 4.2 that by applying Dy to (4.58), we obtain

ij γ j γ i ˘ n−2 Di(a DjDy u) + b DjDy u = Difγ + gγ in B1(0) \{0} × R ,

ij i , where a , b , fγ, and gγ are given by (4.60). By Schauder estimate Theorem 10 from Section 2.3, for every ball BR(X1) ⊆ BR0 (X0), we have

−1 γ 0 −1 γ (R/p) kD uk 1,µ ≤ C (R/p) kD uk 0 y C (BR/2p(X1)) y C (BR/p(X1))

µ 0 +(R/p) [fγ]µ,BR/p(X1) + (R/p)kgγkC (BR/p(X1)) . where C = C(n, µ, λ, Λ) > 0 is a constant.

For X ∈ BR/p(X1), B(p−1)R/p(X) ⊆ BR0 (X0) and q ≥ 3. Then

q q q q G(D u(X), 0) ≤ kD uk 0 R y R y C (B(p−1)R/2pq(X)) p q e ≤ (q − 2)!H ≤ (q − 2)!H , 0 (p − 1)R 0 Rq where G denotes the metric on the space of unordered pairs from Section 1.2, and similarly q −q G(DDyu(X), 0) ≤ (q − 2)!H0eR .

0 0 ˆ Now let X,X ∈ BR/p(X1) and q ≥ 3. If |X − X | < (p − 1)R/2pq, then X = 0 0 ˆ (X + X )/2 ∈ BR/2pq(X1), X,X ∈ B(p−1)R/2pq(X) ⊂ BR0 (X0), so

µ q q 0 µ R G(DDyu(X),DDyu(X )) R q ≤ [DD u] ˆ q |X − X0|µ q y µ,B(p−1)R/2p(X) p q+µ 2µe ≤ (q − 2)!2µH R−q−µ ≤ (q − 2)!H . 0 (p − 1) 0 Rq+µ CHAPTER 4. REGULARITY THEOREMS 131

If |(x, y) − (x0, y0)| ≥ (p − 1)R/2pq, then

µ q q 0 µ q R G(DD u(X),DD u(X )) R 2kDD ukC0(B (x ,y )) y y ≤ y R/p 1 1 q |X − X0|µ q |X − X0|µ 2eH /Rq 21+µeH ≤ (q − 2)! 0 ≤ (q − 2)! 0 , ((p − 1)R/p)µ Rq+µ

Therefore −1 q 0 1+µ −q (R/q) kD uk 1,µ ≤ (q − 2)!2 eH R . y C (BR/p(X1)) 0 Similarly for q = 0, 1, 2,

−1 q 0 1+µ −q R kD uk 1,µ ≤ 2 eH R . y C (BR/p(X1)) 0

By Lemma 17 and Lemma 18,

µ i 3 p−3 −p 0 (R/p) [fγ]µ,BR/p(X1) + (R/p)kgγkC (BR/p(X1)) ≤ C(p − 2)!H0 H R .

for some constant C = C(n, µ, K0,K) > 0. Thus

−1 γ 0 p−3 −p p−3 −p (R/p) kD uk 1,µ ≤ C p(p − 3)!H H R + (p − 2)!H R y C (BR/2p(X1)) 0 p−2 −p p 1 ≤ C(p − 2)!H0H R + (p − 2)H H0H p−2 −p ≤ (p − 2)!H0H R

where the C > 0 are constants that depend on n, µ, λ(X,Z,P ), K0, K, and H0 and are independent of p and H is large enough that Lemmas 17 and 18 hold and

H ≥ max{4C, 2C/H0}. ˘ To complete the proof of Theorem 5, let B2R(x0, y0) ⊂ B1(0). Since A and B are real analytic and Du(x, y) is smooth in y, there are constants K,K0,H0 ≥ 1 such CHAPTER 4. REGULARITY THEOREMS 132

that for any multi-index α = (αX , αZ , αP ) with |α| = k

α α k −|αX |−|αZ | |D(X,Z,P )A| + R|D(X,Z,P )B| ≤ K0K R for k = 2, 3,

α α k −|αX |−|αZ | |D(X,Z,P )A| + R|D(X,Z,P )B| ≤ (k − 3)!K0K R for 4 ≤ k < p,

0 for X ∈ B (x , y ) and |Z| + R|P | ≤ kuk 1 and for any multi-index β with 2R 0 0 C (B2R(X0)) |β| = q −1 β 0 −q (R/q) kD uk 1,µ ≤ H R for q = 1, 2. y C (B2R(x0,y0)) 0

If (x, y) ∈ BR(x0, y0), then BR(x, y) ⊆ B2R(x0, y0). By the Lemma 19 and induction, for some H suﬃciently large,

β p−2 −p G(Dy Du(X), 0) ≤ (p − 2)!H0H R for |β| = p ≥ 3, where G denotes the metric on the space of unordered pairs from Section 1.2. By replacing Theorem 10 with Theorem 11, we obtain Theorem 6. Bibliography

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