Optimal Regularity and Free Boundary Regularity for the Signorini Problem

Algebra i analiz St. Petersburg Math. J. Tom 24 (2012), 3 Vol. 24 (2013), No. 3, Pages 371–386 S 1061-0022(2013)01244-1 Article electronically published on March 21, 2013

OPTIMAL REGULARITY AND FREE BOUNDARY REGULARITY FOR THE SIGNORINI PROBLEM

J. ANDERSSON

Abstract. A proof of the optimal regularity and free boundary regularity is an- nounced and informally discussed for the Signorini problem for the Lamésystem. The result, which is the first of its kind for a system of equations, states that if u =(u1,u2,u3) ∈ W 1,2(B+ : R3) minimizes 1 ⊥ J(u)= |∇u + ∇ u|2 + λ div(u) 2 + B1 in the convex set K = u =(u1,u2,u3) ∈ W 1,2(B+ : R3); u3 ≥ 0onΠ, 1 ∞ + u = f ∈ C (∂B1)on(∂B1) , ≥ ∈ 1,1/2 + where, say, λ 0, then u C (B1/2). Moreover, the free boundary, given by 3 1,α Γu = ∂{x; u (x)=0,x3 =0}∩B1, will be a C -graph close to points where u is nondegenerate. Historically, the problem is of some interest in that it is the first formulation of a variational inequality. A detailed version of this paper will appear in the near future.

§1. Introduction The concept of a variational inequality was born by the Italian mathematical physicist Antonio Signorini during a talk in 1932 (published in 1933, [25]). It naturally arises in continuum mechanics in the presence of obstacles. Signorini derived the constitutive equations for an elastic body resting on a rigid surface without friction. If we assume that the elastic material is modeled by the Lam´e system, then the problem can be formulated as follows: minimize 2 (1) J(u)= |∇u + ∇⊥u|2 + λ div(u) + B1 on the set 1 2 3 ∈ 1,2 + R3 3 ≥ ∈ ∞ + K = u =(u ,u ,u ) W (B1 : ); u 0onΠ, u = f C (∂B1)on(∂B1) . ∇⊥ ∇ T + Here u =( u) is the transpose of the gradient matrix. We think of B1 as some elastic body laying on a rigid surface given by {x3 =0}. The body is subject to some prescribed deformation f on the curved part of its boundary. The function u(x) is seen as the displacement of the point x, that is, when we apply the deformation f on the boundary then the point x moves to x + u(x). A simple perturbation argument can be used to show regularity for more general smooth domains and obstacles.

2010 Mathematics Subject Classiﬁcation. Primary 49J40, 49N60. Key words and phrases. Free boundary regularity, Signorini problem, optimal regularity, system of equations.

c 2013 American Mathematical Society 371

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What turns the problem into a variational inequality is the constraint u3 ≥ 0on {x3 =0}, which should be interpreted as that the elastic body cannot penetrate the ∈ ∞ + rigid surface. For any compactly supported vector-valued function v(x) C (B1 )such 3 that v ≥ 0on{x3 =0} for t ≥ 0, we have J(u + tv) ≥ J(u) because u is a minimizer. The minimizer u satisﬁes the following Euler–Lagrange equations 2+λ (2) Lu ≡ Δu + ∇ div(u)=0 in R3 , 2 + 3 (3) u =0 on Λu, ∂u3 λ (4) + div(u)=0 on Π\ Λu, ∂x3 4 ∂ui ∂u3 (5) + = 0 on Π for i =1, 2, ∂x3 ∂xi (6) u3 ≥ 0onΠ, ∂u3 λ (7) + div(u) ≤ 0onΠ. ∂x3 4 3 Here we have used the notation Π = {x; x3 =0} and Λu = {x ∈ Π; u (x)=0}. It is important to note that this problem is highly nonlinear, because the set Λu is not known apriori. The major diﬃculty in analyzing the regularity of this problem consists in understanding not only the behavior of the solution, but also the unknown set Λu. This paper contains a somewhat informal description of the proof of the optimal regularity (C1,1/2) for the Signorini problem and also a satisfactory theory for the regularity of the free boundary Γu = Ωsu ∩ Λsu. The main theorem of this paper is as follows.

+ The Main Regularity Theorem. If u is a solution of the Signorini problem in B1 , then

u 1,1/2 + ≤ Cu 2 + . C (B1/2) L (B1 ) 0 Furthermore, if x ∈ Γu and

ln u 2 + 0 lim Lr (Br (x )) < 2, r→0 ln(r)

0 1,α then there exists s>0 such that Γu ∩ Bs(x ) is a C -manifold. Remark 1. The free boundary regularity result in the second part of the theorem can be formulated in several ways, and it is equivalent to the best regularity results for scalar equations (see [5]). It states that if u is worse than C1,1 at a free boundary point, then the free boundary is C1,α around that point. There are also quantiﬁable versions of the statement that if u is -close, with a suﬃciently small , to a two-dimensional solution 1,α 1,α in B1, then the free boundary is C in B1/2,withtheC -norm controlled by C. Remark 2. We state the theorem only for n = 3. By changing minor details in the proof we could have proved it in any dimension. Sometimes, in fairly technical computations, we shall employ the curl operator, which has a much simpler form in R3; therefore, we formulated the theorem in 3 dimensions. The reader should keep in mind that the theorem is also true in Rn and the proof in n dimensions is almost the same.

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To frame the current result, we start by outlining the history of free boundary problems (used here more or less synonymously to variational inequalities) since Signorini’s talk in 1932. This will also draw attention to the paucity of regularity results for free boundaries arising from systems of equations. The rest of the paper is dedicated to description of the proof of the regularity theorem. The full proofs, with all their mind- numbing technicalities, will be provided in a future publication. Here we will skip details and several entire lemmas in order to focus on the aspects that the author finds most important. It was not until the development of direct methods in the calculus of variations that the existence of solutions of the Signorini problem was shown by G. Fichera [16] in 1963. A general existence theory for variational inequalities was later developed by Lions and Stampacchia [22]. A few years later, results on the regularity of solutions for free boundary problems started to appear. The first results were obtained for scalar equations (the obstacle problem) by Lewy and Stampacchia [21]. Later, in 1974, Brezis and Kinderlehrer proved optimal W 2,∞-regularity for the obstacle problem [8], still for the scalar case. The first results stating regularity theory for the free boundary in two dimensions arose in the same period and are due to Lewy and Stampacchia [21]. In higher dimensions, a beautiful theory was developed by Caffarelli starting with [10]; see also [11]. The approach of Caffarelli was dependent on a very skilled usage of the maximum principle methods. Using the Haranck inequality, Athanasopoulos and Caffarelli improved regularity to C1,α (almost everywhere) for the obstacle problem [6]. By using the methods developed by Caffarelli in an ingenuous way, Shahgholian and Uraltseva managed to extend the interior free boundary regularity results of Caffarelli up to the boundary of the domain [24]. The Harnack inequality was also vital in the development of regularity theory for two phase problems; see [12, 14] and [13]. The development of the regularity theory of free boundary problems for systems of equations has been much slower and not as satisfactory. The first regularity results for the Signorini problem are due to Kinderlehrer, who proved that in two dimensions the solutions are C1,β [20]. In 1986, Arkhipova and Uraltseva [3] proved that solutions to Signorini like problems are C1,β in n>2 dimensions, provided that the system of partial differential equations is diagonal. In 1989, Schumann [23] proved that solutions of the Signorini problem are C1,β for some β>0. The lack of maximum principle methods and Harnack inequalities for systems has however stifled further development of regularity theory for free boundary systems. The only optimal regularity results and results on the regularity of the free boundary for systems of equations were related to systems that can be reduced to scalar equations, as in [18]. In the last decade the investigation of the Signorini problem for scalar equations, which I shall call the thin obstacle problem, has risen to prominence. The thin obstacle problem is the minimization of the Dirichlet energy

(8) |∇u|2 + B1

on the set

1,2 + (9) K = u ∈ W ; u ≥ 0onΠ,u= f on (∂B1) .

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In 2004, Athahasopoulos and Caffarelli proved optimal C1,1/2-regularity for the thin obstacle problem (see [4]) by using maximum principle methods together with a monotonicity formula. Together with Salsa, they also showed partial regularity for the free boundary in [5], by employing the Harnack inequality methods of [6]. This brief historical description of free boundary problems clearly indicates the im- portance of maximum-principle/Harnack-inequality methods in the development of the regularity theory for free boundaries. This article constitutes the authorsmodestat- tempt to develop a theory independent of those methods. In this paper there are several advancements made in the regularity theory for free boundaries. First, this is the first time that optimal regularity is proved for a free boundary for a genuine system of partial differential equations. It is also the only paper that shows regularity for the free boundary for a system of PDEs. In order to achieve this, it has been necessary to develop some new techniques that will also prove useful for other free boundary problems. In particular, the techniques used here are sufficiently strong to prove some regularity for the set where a free boundary touches the fixed boundary, thus extending the analysis of [24]. Second, there are many other and important problems that arise in applied science whose regularity cannot be proved by maximum principle methods. For instance, such is the biharmonic obstacle problem that models the bending of a plate in the presence of an obstacle. Other interesting questions arise in elastic plastic problems in continuum mechanics. The proof presented in this paper consists of two parts. In the first part, we show that around the points where the solution of the Signorini problem (2)–(7) is not C1,1, this solution is asymptotically two-dimensional. The second part utilizes the first part to linearize the problem around points where it is not C1,1. It is the second part that is the most difficult and also the most interesting. The first part employs the fact that the set of solutions of (2)–(5) in the entire half-space with growth strictly less than linear at infinity is one-dimensional, which shows, in particular, that the tangential derivatives of the solution are multiples of each other. This implies that the solution only depends on two directions. The proof is rather elementary and specific for the Lamésystem.In§2 we shall describe this procedure. The two-dimensional solutions can be calculated explicitly, see Lemma 1. Since we know the profile of the solution at points where it is not C1,1, we are in a good position to linearize the problem. This linearization will be described in §3. This is the heart of the paper, and the techniques developed here are directly applicable to other problems such as the biharmonic obstacle problem or the description of the set where the free boundary meets the fixed boundary for the obstacle problem. I intend to return to those problems in later publications, where an appropriate technique will be developed to substitute the theory in §2.

Notation. Sometimes, we write n for the dimension. But some proofs in the paper are written only for n = 3. This is for simplicity, because the curl operator is more explicit in R3. The pedantic (we mean no negative connotation) reader can always think that n =3. { } Rn Π= x; xn =0 is the boundary of + . We use bold face letters u, v, w, p, etc. to denote vector-valued functions u = (u1,u2,u3,...,un), v =(v1,v2,...,vn)etc. For a continuous function u =(u1,u2,...,un), we set n Λu =Λ={x; xn =0,u (x)=0}. 1 2 n For a continuous function u =(u ,u ,...,u ), we set Ωu =Ω=Π\ Λu. For a continuous function u, we deﬁne its free boundary Γu =Γ=ΛĎu ∩ ΩĎu.

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∇u is the following matrix: ⎡ ⎤ ∂u1 ∂u1 ··· ∂u1 ⎢ ∂x1 ∂x2 ∂xn ⎥ ⎢ ∂u2 ∂u2 ··· ∂u2 ⎥ ∇ ⎢ ∂x1 ∂x2 ∂xn ⎥ u = ⎢ . . . . ⎥ . ⎣ . . .. . ⎦ ∂un ∂un ··· ∂un ∂x1 ∂x2 ∂xn ∇⊥u is the transpose of the matrix ∇u: ⎡ ⎤ ∂u1 ∂u2 ··· ∂un ∂x1 ∂x1 ∂x1 ⎢ 1 2 n ⎥ ⎢ ∂u ∂u ··· ∂u ⎥ ∇⊥ ⎢ ∂x2 ∂x2 ∂x2 ⎥ u = ⎢ . . . . ⎥ . ⎣ . . .. . ⎦ ∂u1 ∂u2 ··· ∂un ∂xn ∂xn ∂xn Often, we use a prime to indicate the projection of ann-dimensional vector into an ∂ ∂ ∂ (n − 1)-dimensional vector: x =(x1,x2,...,xn−1), ∇ = , ,..., ,etc.At ∂x1 ∂x1 ∂xn−1 ∇ times, we shall slightly abuse notation and write x =(x1,x2,...,xn−1, 0) and = ∂ , ∂ ,..., ∂ , 0 . It will always be clear from the context what we mean. ∂x1 ∂x2 ∂xn−1 We use the notation ∇ = 0, ∂ , ∂ ,..., ∂ , 0 . More generally, for any vector ∂x2 ∂x3 ∂xn−1 ∂ 2 ξ ∈ Πwewrite∇ = ∇−en −ξw/|ξw| (ξw ·∇), which is simply the gradient restricted ξ ∂xn to the subspace orthogonal to en and ξ. By W k,p(D) we mean the usual Sobolev space. We shall often be quite informal when assigning vector-valued functions to this space, writing (u1,u2,u3,...,un) ∈ W k,p instead of (u1,u2,u3,...,un) ∈ (W k,p)n,etc. By u we mean the norm Lrp(Ω) 1 1/p u = |u|p , Lrp(Ω) | | Ω Ω as defined in Definition 1. ξ ξ The homogeneous solutions p1/2, p3/2, p1/2, p3/2, pr3/2,... of the Lamésystemare defined in Definition 2.

§2. Asymptotic flatness of the free boundary In nonlinear partial differential equations, the usual way to derive regularity is by linearization. That is, to consider some sequence of solutions uj , or rescalings of the j 0 j same solution still denoted by u , subtract the limit u ≡ limj→∞ u , and normalize in some appropriate way, uj − u0 (10) → v. j Heuristically, v should solve some linear version of the same system of partial differential equations as uj . Using the regularity theory for linear systems, one may deduce that v is regular and satisfies some good estimates. If the convergence in (10) is sufficiently strong, then uj will inherit those estimates, in a slightly weaker form, for j sufficiently large. This method has been used extensively in the calculus of variations (see for instance [15] or [19]) in particular, and for nonlinear PDE in general (see [9] for an application to the Navier–Stokes equations). This strategy involves two fundamental difficulties that need to be resolved. The first difficulty is to identify the limit u0. In general, it is not possible to identify the limit, and therefore, regularity can only be shown at points where we assume that u0

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has some specific form. Most common, in the regularity theory for systems, it is assumed that u0 is an affine function, which leads to partial regularity results for systems of partial differential equations (see [15] or [19]). In very rare cases it is however possible to identify the limit u0, which leads to a fuller regularity theory (see, e.g., [1]). In the context of the Signorini problem, we are not aiming at classifying all the possible blow-up limits u0 (though that is probably possible). Instead, we classify all the blow-up limitsatpointswherethesolutionisnotC1,α for all α<1. This suffices to derive a regularity theory because by assumption, at all the other points of the free boundary the solution is C1,α for all α<1. In this section we shall describe this classification. The second problem in linearizing a system of equations is to show that the convergence in (10) is sufficiently strong to derive regularity about uj from the regularity of v.This will be discussed in the next section. The first observation we make regarding the regularity of solutions of the Signorini problem is that we can make a difference quotient argument to show that the solutions are in fact W 2,2 and satisfy the estimate

u 2,2 + ≤ Cu 2 + . W (B1/2) L (B3/4)

∂u 1,2 This implies in particular that ui ≡ ∈ W ,andfori = 1 or 2 the function ui ∂xi solves system (2)–(5). There is a beautiful and simple argument due to Apushkinskaya, Shahgholian, and Uraltseva [2] that shows that if we have a function f(x) ∈ L∞ satisfying ln sup |f| lim inf Br =1+α, r→0 ln(r)

then there exists a sequence rj → 0 such that f(r x) j → f 0 sup |f| Brj in some appropriate sense, where f 0 satisfies sup |f 0|≤1+R1+α. BR This idea has been used in different contexts, together with monotonicity formulas, the Hopf lemma, or assumptions on the density for the contact set on obstacle problems, to show optimal regularity. In the context of the Signorini problem we lack the first two technical tools and the third is not relevant. We shall still rely on this idea in order to control the growth of solutions of different blow-up sequences. But lacking the technical tools available for scalar equations, we shall have to transform and complicate the idea significantly in order to gain the knowledge of the second term in the asymptotic expansion of u. We shall also develop machinery to control the growth of blow-ups of u both at infinity and at the origin simultaneously. 2 This argument will be applied to the L -norm of u in Br. To that end, we define a normalized version of the L2 norm that scales better.

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If the origin is a free boundary point and

ln u 2 + lim inf Lr (Br ) =1+α<2, r→0 ln(r) then we can use the idea of Apushkinskaya, Shahgholian, and Uraltseva on the function → f(r)= u 2 + to find a sequence rj 0 such that Lr (Br ) u(r x) j → u0, u 2 + Lr (Brj ) where u0 is a solution of (2)–(7) and 0 1+α (11) u 2 + ≤ 1+R . Lr (BR ) 0 ≡ ∂u0 But this means that ui ∂x is a solution of (2)–(5) for i =1, 2, and i 0 ≤ α ui 2 + C 1+R . Lr (BR ) 0 In some sense, ui is the first eigenfunction for system (2)–(5). To the author sgreat regret, to see that the first eigenfunction is unique, we need to resort to some theory of harmonic functions. The argument is quite simple and involves the observation that div(u0)andcurl(u0) are harmonic functions. That, together with the growth condition (11) and after several pages of quite trivial calculations, allows us to reduce the function to the following form: 0 λ +2 ∂τ λ +3 ∂τ u = ∇ x3 − τ + e3 , 2(λ +4)∂x3 λ +4 ∂x3

where e3 is the unit vector pointing in the x3-direction and τ is a harmonic function. This reduction of u0 in terms of a harmonic function is only possible because u0 is defined Rn in the entire half-space + and has bounded growth (11). That τ is harmonic makes it possible to apply the stronger theory for eigenfunctions developed in [17] and [7], which applies to nonconical domains. In particular, we can use the uniqueness of the first eigenvalue to deduce that there are constants a1 and a2 0 0 0 − such that a1u1 = a2u2. Therefore, u is independent of the (a2, a1, 0)-direction. By a 0 0 rotation of the coordinates, we may assume that u (x)=u (x1,x3). In particular, the free boundary consists of parallel hyperplanes. A simple argument also shows that the free boundary of a global solution in two dimensions consists of at most one point. This part of the argument relies on the theory of harmonic functions and seems to be rather specific to the Lamé system. It is satisfactory only to the extent that it is needed to prove regularity for the Signorini problem. In the next section, when we linearize the system, we shall not rely on any theory for harmonic functions; therefore, this material is directly applicable to other systems of equations or higher order equations where the maximum principle methods are not available. It should be noted that in other interesting applications, such as the biharmonic obstacle problem, there are other arguments (of measure theoretic nature, not yet published) to show that the blow-up of the free boundary at a free boundary point becomes a hyperplane in an appropriate a.e. sense. Before we state the main proposition of this section, we need to make a simplifying assumption that we shall adhere to for the rest of this paper. If x0 is a free boundary point of u “under consideration”, then we assume that ∇u(x0) = 0. Since u is C1,we can always subtract a linear function from u to obtain this situation. But it will only

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complicate the exposition to do that. With this harmless assumption in mind, we state the ﬁrst proposition of this paper.

+ Proposition 1. Let u be a solution of the Signorini problem in B1 , and assume that 0 ∈ Γu. Assume, furthermore, that

ln u 2 + (12) lim inf Lr (Br ) < 2. r→0 ln(r) 2 Then there exists a subsequence rj → 0 such that in L u(r x) j → u0, u 2 + Lr (Brj ) where u0 is a global solution of the Signorini problem and furthermore, after a rotation, 0 0 u (x)=u (x1,x3) and Λu0 = {x ∈ Π; x1 ≥ 0}. Homogeneous solutions of the Signorini problem in two dimensions solve an ordinary diﬀerential equation and can thus be calculated explicitly. Lemma 1. Let u(x, y)= u(x, y),v(x, y) be a global homogeneous solution of order − R2 1+α>0 to the Lam´esystemwithλ = 2 in +: 2+λ ∂ div(u) Δu + =0 in R2 , 2 ∂x + 2+λ ∂ div(u) Δv + =0 in R2 , 2 ∂y + v(x, 0) = 0 on {x>0}, ∂u (x, 0) = 0 on {x>0}, ∂y ∂u ∂v (x, 0) + (x, 0) = 0 on {x<0}, ∂y ∂x ∂v λ (x, 0) + div(u)(x, 0) = 0 on {x<0}, ∂y 4 ∈ 2,2 R2 u Wloc ( +). ∈ N ∈ N 1 1 3 5 Then α or α + 2 = 2 , 2 , 2 ,... , and in polar coordinates, x = r cos(φ) and y = r sin(φ),forsomea ∈ R we have 10+3λ + α(2 + λ) 2+λ (1) u(r, φ)=r1+α cos (1 + α)φ − cos (1 − α)φ a, 8(1 + α) 8 2 − λ − α(2 + λ) 2+λ (r, φ)=r1+α sin (1 + α)φ − sin (1 − α)φ a 8(1 + α) 8 if α ∈ N,and 6+λ + α(2 + λ) 2+λ (2) u(r, φ)=r1+α cos (1 + α)φ − cos (1 − α)φ a, 8(1 + α) 8 6+λ − α(2 + λ) 2+λ v(r, φ)=r1+α sin (1 + α)φ − sin (1 − α)φ a 8(1 + α) 8 ∈ N 1 if α + 2 . Lemma 1 and Proposition 1 together give an explicit form of u0.

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Deﬁnition 2. We denote by p1/2(x1,x3), p1(x1,x3), p3/2(x1,x3), etc. the homogeneous solutions of order 1/2, 1, 3/2, etc. (as speciﬁed in Lemma 1) of the Signorini problem in R2 + with p1+α L2(B+) =1. r 1 ξ | | ξ · ∈ We shall also use the notation pk+1/2(x)= ξ pk+1/2 |ξ| x ,x3 ,whereξ Πand

∂pk+3/2(x1,x3) prk+1/2(x1,x3)= . ∂x3 In terms of this deﬁnition, we have shown that an L2-convergent blow-up of u at free 0 boundary points where (12) is satisﬁed is given by u = p3/2.

§3. Linearization of solutions at asymptotically flat points As stated before, we want to linearize the Signorini problem. For this, we consider a free boundary point x0 such that

ln u 2 + 0 lim inf Lr (Br (x )) < 2; r→0 ln(r) we may assume that x0 = 0. By the discussion in the previous section we know that, locally in L2, u(rjx) → p3/2. u 2 + Lr (Brj (0)) Therefore, we need to investigate limits of the form (13) lim vj = v, j→∞ where − j u(rj x) p3/2 v = − . u p3/2 2 + Lr (Brj (0)) 2 The limit exists, maybe after taking a subsequence, in the weak L -sense. And clearly v + \ 2 2 ≤ will solve (2)–(5) in the set B1 x; x1 + x3 t for any t>0. j However, v satisfies neither Dirichlet nor Neumann conditions in Λu(rj x)ΔΛp3/2 (here, by shamelessly abusing notation, we denote the symmetric difference by Δ). In the j − → definition of v we divide by u p3/2 2 + 0; therefore, we need to be very Lr (Brj (0)) careful to make sure that the boundary data for vj does not concentrate on the line x3 = x1 = 0, in order to be able to deduce that the convergence in (13) is actually strong. We shall split the proof in two cases.

3.1. The tangential derivatives. Since the convergence issue is much simpler in the x2-direction, we treat that separately. We slightly change the notation and assume that j + for some sequence of solutions u in B1 we have j − ξ j − (14) inf u p 2 + = u p3/2 2 + , ξ∈Π 3/2 Lr (B1 ) Lr (B1 )

+ which means that among all rotations of p3/2 it is p3/2 that best approximates u in B1 . Then we consider the linearization uj (x) − p vj = 3/2 j − u p3/2 2 + Lr (B1 (0))

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j → for some sequence u p3/2. This is slightly diﬀerent from what we described above, j j because now we are not subtracting the limit u (rx)/u 2 + but the best approxi- Lr (Br ) j + mation of u in B1 . We have 1 ∂vj 1 ∂uj = ∈ W 1,2. ∂vj ∂uj 2 + ∂x2 2 + ∂x2 ∂x2 Lr (B1 ) ∂x2 Lr (B1 (0)) 2 So, the L -convergence is not that diﬃcult for the derivatives in the x2-direction since here we have W 1,2-regularity. Skipping some technical details, it is possible to show that 1 ∂vj (15) → v0, ∂vj 2 + ∂x2 ∂x2 Lr (B1 ) where v0 solves 2+λ (16) Δv0 + ∇ div(v0)=0 in B+, 2 1 0 (17) v · e3 =0 on {x1 > 0}∩{x3 =0}, · 0 ∂e3 v λ 0 (18) + div(v )=0 on Π \{x1 > 0} ∩{x3 =0}, ∂x3 4 ∂v0 · e ∂v0 · e (19) i + 3 = 0 on Π for i =1, 2. ∂x3 ∂xi

Notice that this system is not dependent on Λv0 and thus is linear. Since v0 solves a linear system, we can use classical methods to show that ∞ 0 v = Hk(x), k=1

where Hk is a solution that is homogeneous of order k/2. We need to show that H1 = 0. If not, then it is easy to show that H1 is a multiple of p1/2, which implies that for some small κ we have

j u = p3/2 + κx2p1/2 +rest. But this directly contradicts the choice of coordinates in (14). Therefore, the ﬁrst nonzero 0 term Hk is for k =2.Thatimpliesthatv decays like a linear function at the origin, j and strong convergence in (15) implies that the x2-derivatives of u decayalmostata linear rate for j suﬃciently large. We can prove the following lemma.

+ Lemma 2. If u solves the Signorini problem in B1 and 0 <γ<1/2, then there exists δγ > 0 such that if − ξ − ≤ (20) inf u p 2 + = u p3/2 2 + δγ ξ∈Π 3/2 Lr (B1 ) Lr (B1 ) and ∇ ≤ u 2 + δγ , Lr (B1 )

then there exists 0

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3.2. Decay of the solution. In order to prove regularity for u we would want to prove something similar to Lemma 2 for u and not only for its x2-derivatives. We are able to prove the following proposition. + Proposition 2. If u solves the Signorini problem in B1 and 0 <γ<1/4, then there exists δγ > 0 such that if − ξ − ≤ inf u p 2 + = u p3/2 2 + δγ ξ∈Π 3/2 Lr (B1 ) Lr (B1 ) and ∇ ≤ u 2 + δγ , Lr (B1 ) then there exists sγ such that −(1/2+γ) −(3/2+γ) ξ max s ∇ u 2 + ,s u − p + γ Lr (Bs ) γ 3/2 L2(B ) (21) γ r sγ ≤ ∇ − max u 2 + , u p3/2 2 + , Lr (B1 ) Lr (B1 ) − ξ where ξ is the vector that minimizes infξ∈Π u p 2 + . 3/2 Lr (B1 ) The proof is very technical but is based on ideas in [2]. In particular, to get regularity for the solution it suffices to show that − ξ ≤ 3/2+γ (22) inf u p 2 + s , ξ 3/2 Lr (Bs ) which is a (simplified) continuous version of the conclusion of the proposition. We know u(r x)−pξ j j 3/2 from [2] that if (22) fails then we can make a blow-up of the type v = − ξ and u(rj x) p3/2 get a global solution that grows slower than r3/2+γ at infinity. The fact that we subtract ξ different p3/2 for different s naturally creates technical difficulties. There are other technical complications that are new in this context. The idea is that if (22) is not true for some γ<1/2, then the blow-up limit must be some combination of the first and second eigenfunctions (that is, p1/2 and p3/2) of the linearized problem, because any other eigenfunction would have faster growth at infinity than r3/2+γ .But ξ since we subtract p3/2 in the definition of the blow-up sequence, the blow-up limit can only be p1/2. Going for a contradiction, we have to show that this cannot happen. In particular, we need to control the growth of the blow-up limit at the origin, which we do in Lemmas 4 and 3. − ξ u(rj x) p3/2 Finally, we need to show that the blow-up converges strongly. Since − ξ does u(rj x) p3/2 not solve any nice boundary-value problem, there might be concentration phenomena at the free boundary. This difficulty is due to the fact that we subtract the limit. We develop some new estimates that shows that the convergence is in fact strong. Let us sketch some details. Our first result, out of many, needed to prove Proposition 2 is as follows. Lemma 3. (i) Let u be a global solution of the Signorini problem. Suppose 0 ∈ Γ and u − ξ ≤ inf u p3/2 2 + μ u L2(B+) ξ∈Π Lr (BR ) r R ≥ ξ ∈ for all R 1.Ifμ is sufficiently small, then u = p3/2 for some ξ Π. + (ii) Let u be a solution of the Signorini problem in B1 .Foreach>0,thereexists μ > 0 such that if − ξ ≤ ≤ inf u p 2 + μ u 2 + for all r 1, ξ∈Π 3/2 Lr (Br ) Lr (Br )

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then for r ≤ 1/2 we have 3/2+ 3/2− r ≤u 2 + ≤ r . Lr (Br ) The proof of this lemma is rather technical but straightforward. If μ is small, then 2 + 2 + the Lr -norm of u in B2−k cannot be too different from the Lr -norm of u in B2−k+1 .An iteration proves the lemma. Lemma 4. As in Proposition 2,letu correspond to a sufficiently small δ depending on γ.If0 ∈ Γ, then there exists a modulus of continuity σ such that − ξ ≤ (23) inf u p3/2 2 + σ(δ) u L2(B+) ξ∈Π Lr (Bt ) r t for each t<1. In particular, by Lemma 3, for each u we have − ξ infξ∈Π u p3/2 2 + (24) lim Lr (Bs ) =0. → s 0 δu 2 + Lr (Bs ) The proof is straightforward; we argue by contradiction. We consider a sequence of δ u corresponding to δ → 0 that violates (23) for some rδ. By making a blow-up and using the fact that the limit must be p3/2, we get a contradiction. The second part is a direct consequence of Lemma 3 together with Proposition 1. In what follows we are going to skip some technicalities related to the maximum on both sides of the inequality in the conclusion of Proposition 2. For the purposes of this exposition we shall assume, with some justification, that the conclusion of Lemma 2 gives enough control of the derivatives in the x2 direction. With this simplification in mind, we want to show that, for any s ∈ (0, 1] and each γ<1/2, − ξ ≤ γ − (25) inf u p 2 + Cγ s u p3/2 L2 + u L2 + . ξ 3/2 Lr (Bs ) r (B1 ) r (Bs ) If this is not true, then a continuity argument shows that we can find solutions uj , j δ = u − p + → 0, C →∞, s → 0, and γ < 1/2 such that j 3/2 Lr2(B ) j j j 1 j − ξ infξ∈Π u p3/2 2 + Lr (Bs ) ≤ C sγj if s~~s , j j j δj u 2 + Lr (Bs )~~

with equality for s = sj . We shall show that if Cj →∞, then any convergent subsequence of γj converges to something equal to at least 1/2. In particular if we ﬁx γj = γ<1/2, then Cj must be uniformly bounded, say by C0, in order for (26) to be true for some sj . Choosing − 1 sγ

~~where ξj is the minimizer of j − inf u p3/2 L2(B+ ). ξ r sj~~

~~Since Cj →∞, it follows that sj → 0. We are going to show that γj → 1/2. The deﬁnition of vj is quite complicated, but we can prove the following lemma.~~

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j Lemma 5. Let v be as in (27),andletsj , ξj , cj ,andδj be chosen as in the discussion leading up to (27); in particular, we assume that (26) is true. Also, let C>1. Then for each r there exist jr such that 3/2 ≤ j Cr if r 1, v 2 + ≤ Lr (Br ) Cr3/2+γj if r>1,

for j>jr. The proof is very technical and involves estimates of different projections of uj (rx) { ξ ∈ } onto the space p3/2 : ξ Π . Once Lemma 5 has been proved, we are almost done. From Lemma 5 we know that if we choose t small, then j 2 ≤ 6 (28) v 2 + Ct L (Bt ) when j is sufficiently large. We also know from Lemma 2 that the derivatives of vj are 2 small in the x2-direction, at least in the L -norm. But the following corollary to Lemma 4 and Proposition 1 show that the domain is sufficiently good for (L2 −Cα)-estimates, and α the derivatives in the x2-direction are actually small in the C -norm.

~~Corollary 1. If u satisfies the assumptions of Proposition 2 with δγ sufficiently small, then Λu ∩ B1/2 is a Reifenberg flat set in the plane {xn =0}.~~

~~The fact that the x2-derivatives are small together with (28) implies that vj 2 √ ≤ 2 3 2 { 2 2 }∩ + Ct R , L ( x; x1+x31,~~

where γ0 = limj→∞ γj. Since (16)–(19) is a linear system, we can use the standard linear 0 ∞ theory to write v = k=0 Hk(x), where Hk is a solution that is homogeneous of order k/2. If γ0 < 1/2, then from (29) we can directly conclude that Hk =0forallk except ξj j k =3.ButH3 must be zero because we subtracted p3/2 in the deﬁnition of v . It follows 0 j 0 0 that v = 0. But the strong convergence v → v implies that v 2 + =1,which Lr (B1 ) leads to a contradiction. We can therefore conclude that γ0 ≥ 1/2. As explained before, for any γ<1/2 we may ﬁnd a constant Cγ such that there is no sj for which (26) is true. That is, − ξ infξ∈Π u p 2 + 3/2 Lr (Bs ) γ

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≤ Choosing s = sγ suﬃciently small and noticing that, by Lemmas 4 and 5, u 2 + Lr (Br ) r3/2− if u satisﬁes the condition of Proposition 2, we see that − ξ ≤ 3/2+γ/2 3/2+γ/2 − inf u p3/2 2 + sγ δ = sγ u p3/2 L2(B+). ξ∈Π Lr (Bsγ ) r 1

§4. Putting it all together It is quite standard to show that Proposition 2 implies regularity. We shall sketch some details, but in order to avoid technical complications we assume that ∇ ≤ −1 − ξr u 2 + r u p 2 + ξr Lr (Br ) 3/2 Lr (Br )

for all r,whereξr is the vector minimizing the term to the right. 3/2 − 3/2 ≤ 3/2 If for some r we have u 2 + = C0r and u p 2 + = δr δ r Lr (Br ) 3/2 Lr (Br ) 1/8 0 3/2 for δ1/8 as in Proposition 2, then u (x)=u(rx)/(C0r ) satisﬁes the conditions of the proposition. By a standard iteration it follows that j − ξ ≤ j/8 (30) inf u (x) p3/2 2 + s1/8δ, ξ∈R3 Lr (B1 ) where u(rsj x) uj (x)= 1/8 . 3j/2 3/2 s1/8 r An application of the triangle inequality implies ξj − ξk ≤ k/8 (31) p p 2 + Cδs 3/2 3/2 Lr (B1 ) 1/8 j − ξ for k

~~for all s~~

~~+ Lemma 6. Let u solve the Signorini problem in B and let u 2 + =1. For every 1 L (B1 ) δ>0,thereexistsCδ such that if for some r we have~~

~~u 2 + u 2 + Lr (Br ) Lr (Bs ) ≤ ≥ 3/2 =1 and 3/2 1 for s r, Cδr Cδs then u(rx) (32) inf − pξ <δ ∈ 3/2 3/2 ξ Π C r 2 δ Lr (B1)~~

~~and, assuming that the minimizing ξ is equal to |ξ|e1, we also have u(rx) (33) ∇ <δ. 3/2 Cδr 2 + Lr (B1 )~~

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~~Using this lemma with suﬃciently small δ, we see that either~~

~~u 2 + Lr (Bs ) < 1 3/2 Cδs~~

for all s, or there is a largest s,callits0, such that the assumption in the lemma is fulﬁlled with r = s0. But then the argument at the beginning of this section implies that ≤ 3/2 u 2 + 2Cδ + Cδ s Lr (Bs ) for all s

ln u 2 + 0 lim Lr (Br (x )) < 2. r→0 ln(r) But this implies that the normal of the free boundary at x0 is well deﬁned, namely, it 0 is ξ 0 . Using this together with the continuity of u 2 0 at x and the geometric x Lr (Br (x )) decay proved in Proposition 2, we see that the normal of the free boundary is Cα-close to x0. The argument is fairly standard.

~~References~~

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~~Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom E-mail address: [email protected] Received 1/NOV/2011 Originally published in English~~

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