A Class of Functions That Are Quasiconvex but Not Polyconvex

University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange

Masters Theses Graduate School

12-2003

A Class of Functions That Are Quasiconvex But Not Polyconvex

Catherine S. Remus University of Tennessee - Knoxville

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Recommended Citation Remus, Catherine S., "A Class of Functions That Are Quasiconvex But Not Polyconvex. " Master's Thesis, University of Tennessee, 2003. https://trace.tennessee.edu/utk_gradthes/2219

This Thesis is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Masters Theses by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council:

I am submitting herewith a thesis written by Catherine S. Remus entitled "A Class of Functions That Are Quasiconvex But Not Polyconvex." I have examined the final electronic copy of this thesis for form and content and recommend that it be accepted in partial fulfillment of the requirements for the degree of Master of Science, with a major in Mathematics.

Henry C. Simpson, Major Professor

We have read this thesis and recommend its acceptance:

Charles Collins, G. Samuel Jordan

Accepted for the Council: Carolyn R. Hodges

Vice Provost and Dean of the Graduate School

(Original signatures are on file with official studentecor r ds.) To the Graduate Council:

I am submitting herewith a thesis written by Catherine S. Remus entitled “A Class of

Functions That Are Quasiconvex But Not Polyconvex”. I have examined the final electronic copy of this thesis for form and content and recommend that it be accepted in partial fulfillment of the requirements for the degree of Master of Science, with a major in Mathematics.

Henry C. Simpson______

Major Professor

We have read this thesis and recommend its acceptance:

Charles Collins______

G. Samuel Jordan______

Acceptance for the Council:

Anne Mayhew______

Vice Provost and Dean of Graduate

Studies

(Original signatures are on file with official student records.) A Class of Functions That Are Quasiconvex But Not Polyconvex

A Thesis Presented for the Masters of Science Degree The University of Tennessee, Knoxville

Catherine S. Remus December 2003 Acknowledgements

I would like to thank everyone who helped me earn the Master of Science degree and master the discipline of Mathematics. I wish to thank Dr. Simpson for sharing his vast knowledge of Convex Analysis with me, but mostly for opening my eyes to the beauty of theoretical Mathematics and for drawing out my analytic abilities. I want to thank Dr.

Jordan for pulling together all the mathematical concepts and tools I had learned into one integrated body of knowledge and for introducing me to the Calculus of Variations that led me to my thesis topic. I would also like to thank Dr. Collins for serving on my committee.

I wish to thank my mother and son for their unflagging support and love during this journey of discovery I have taken us on. Even when I doubted, they never did. Finally, I thank God for all His Grace in guiding me and wish always to follow where He leads.

ii Abstract

In 1991 V. Sverak [11] gave an example of a function that was invariant and quasiconvex but not polyconvex. We have generalized this example to a wide class of functions that meet certain ellipticity and growth conditions. Quasiconvexity is one necessary and sufficient condition for the existence of solutions to the minimization problem in elliptic P.D.E. theory. Invariance is frequently a requirement of the stored energy function in Calculus of Variation approaches to elasticity problems.

iii Table of Contents

Chapter Page

I Introduction ………………………………………………………………….. 1

II Background ……………………………………………………………...…… 3

III Theorem 1 ………………………………………………………………...…. 12

IV Theorem 2 ………………………………………………………………….... 16

V Theorem 3 ………………………………………………………………….... 17

VI Examples ……………………………………………………………….……. 21

List of References ………………………………………………………….. 27

Vita ……………………………………………………………………………. 30

iv Chapter I

Introduction

In this thesis a class of functions which are continuous, subject to certain growth and ellipticity conditions, and quasiconvex are shown not to be polyconvex. The study of convex analysis concerns itself with four types of convexity: convex functions, polyconvex functions, quasiconvex functions, and rank-one convex functions. The easiest way to visualize the relationships between these types of convexity is the analogy of nested sets where convex functions is the smallest, most stringent, set and rank-one convex functions is the largest, least stringent, set. The “proper subset” relationship of these convexity conditions has been a source of study for some years. In applications, quasiconvexity is very difficult to determine whereas polyconvexity is easier to show and the two concepts are near each other. V. Sverak [11] gave a new example of a function that is quasiconvex but not polyconvex in 1991. Other examples have been given in the past, but his new example was suggestive of an entire class of such functions. We construct a wide class of functions that are quasiconvex but not polyconvex. Sverak’s example is a special case of our examples.

Quasiconvexity is a necessary and sufficient condition (along with others) to ensure the existence of solutions to the minimization problem in elliptic P.D.E. theory.

Techniques of variational calculus have long been utilized in elasticity problems and the stored energy function, our f , is often required to be objective, i.e., frame indifferent or

Galilean invariant, and isotropic, i.e., rotationally invariant (see Dacorogna [5]).

1 In Chapter II we give background information such as definitions, notation, and a lemma needed in the proof of Theorem 1. Chapter III states and proves a needed preliminary theorem, Theorem 1, guaranteeing specific growth and quasiconvexity conditions on our class of functions. In Chapter IV we have the primary emphasis of this thesis, Theorem 2. This theorem deals with the nonpolyconvexity of a class of quasiconvex functions subject to subquadratic growth and ellipticity conditions. Chapter

V presents Theorem 3 that addresses the invariance of functions in our class of quasiconvex functions. Finally, in Chapter VI we give four examples of functions in this class of functions we have constructed. Three of the four examples meet the criteria of all three of our theorems, but Example 2 is convex and therefore polyconvex and does not meet the criteria for Theorem 2. This alleged failure highlights the difficulty in constructing nontrivial examples of functions in this class satisfying all three theorems.

2 Chapter II

Background

In this thesis, + is the set of positive real numbers and M n×n is the space of all n× n real matrices, n ≥1, and ⋅ is any norm on this space. The 4-tensor, C , is a linear

n transformation M n×n → M n×n and is of the general form C [ X ] = C X , ij ijkl kl k,l=1

n×n n ∞ n

¡ ¡ ∈ ¡ ∈ Ω ⊂ φ ∈ (Ω)

Cijkl , for all X M . For , C with values in , we denote £

¢ ∂φ ∂φ

¤ ¥

1 ( x) ¨ 1 ( x) ¥ ¤ ∂ ∂

x1 xn n

¤ ¥

∂φ

k ¥ ∇φ ( x) = ¤ , so that C ∇φ ( x) = C (x) .

ij ijkl ∂ ¥ ¤ x

∂φ ∂φ k,l=1 l

¤ ¥

n ( ) ¨ n ( ) § ¦ x x ∂ ∂ x1 xn

n×n

For this work we will be considering functions f : M → ¡ having various convexity properties. These are seen in physical theories such as elasticity, phase transition and many other applications that are considered the minimization of energy functionals, f (∇φ ( x))dx . The function f represents a potential energy function and Ω

φ is the deformation variable. There exist many theorems discussing this type of function, f , and all require some kind of convexity and associated conditions, e.g., growth and coercivity estimates (see Dacorogna [5]).

3 Definitions and Notation

n > ∀ ∈ n { } C is strongly elliptic if and only if Cijkl aibjakbl 0 a,b \ 0 i, j,k,l=1

(Note this is a strengthened Legendre-Hadamard condition; the Legendre-Hadamard condition has ≥ in place of >.) (see Gelfand-Fomin [7], Wan [12], Morrey [9]).

Let K ⊂ ¡ . K is a convex set if, for any two points x, y ∈ K and t ∈[0,1] the point z = tx + (1− t) y is also in K .

⊂ n = Let K ¡ . The convex hull of K , co(K ) : the smallest convex set that

contains K = {tx + (1− t) y | x, y ∈ K, t ∈[0,1]} .

A function g : n → is convex if and only if, for all x, y ∈ n and for all t ∈[0,1], g (tx + (1− t) y) ≤ tg ( x) + (1− t) g ( y) .

For the following several definitions, let f : M n×n → and let Ω be the open unit ball in n .

The function f is rank-one convex if and only if, for every A∈ M n×n and for all

∈ n ¡ + ⊗ ∈ ⊗ a,b the mapping t f ( A ta b) , t ¢ is convex. Here, a b is the element of

M n×n with components (a ⊗b) = a b , i.e., a ⊗b = a bT ; this is the general rank one ij i j n× n matrix (provided either a or b is nonzero).

The function f is quasiconvex if and only if, for every A∈ M n×n and for every

φ ∈ ∞ (Ω) ( ) ≤ + ∇φ ( ) C0 , f A dx f ( A x )dx is true. Ω Ω

4 The function f is polyconvex if and only if, for every A∈ M n×n , there exists a

n → = ¢ convex function g mapping ¢ , such that f ( A) g (T ( A)) where

× τ

n n → n =

¢ ( ) ( ) T : M and T is defined by T A : A, adj2 A, , adjn A . Here, adjs A is the

n × ∈ n×n ≤ ≤ τ = σ adjugate matrix of all s s minors of the matrix A M , 2 s n , and n s

s=1

¢ £

£ ¢ £

¢ 2

¤ ¥

¥ ¤ ¥

¤ n n ¥

σ = = ¤ n!

¦ § ¦ § § where ¦ . s s s s!(n − s)!

n×n n

The function f is convex as defined above but with M replacing ¢ .

The above definitions of convexity can be found in Dacorogna [5] and Ball [4]. ¡ It can be proven (see Dacorogna [5]) that f convex ¡ f polyconvex f quasiconvex ¡ f rank-one convex. It can also be shown that in the definition of

Ω n quasiconvex, may be replaced with any bounded open set in ¢ and the resulting definition of quasiconvexity is equivalent to the one given above.

Next, we relate rank-one convexity to strong ellipticity as follows.

2 × × ∂ f ( A) Assuming f ∈C2 , define a 4-tensor C : M n n → M n n by C := . Again, it can ijkl ∂ ∂ Aij Akl be proven that if C is strongly elliptic for all A∈ M n×n , then f is rank-one convex.

Conversely, if f is rank-one convex, then C satisfies the Legendre-Hadamard condition for all A∈ M n×n .

n×n n ¢ Let f : M → ¢ , f ≥ 0 , and let Ω be the open unit ball in .

The convex envelope or convexification of f is Cf := sup{g ≤ f | g is convex},

n×n i.e., Cf ( x) = sup{g ( ) | g ≤ f on Ω, g convex on Ω} , x ∈Ω ; Cf : M → ¢ is convex.

The function Cf is the largest convex function below f , and f is convex if and only if f = Cf .

The quasiconvex envelope or quasiconvexification of f is

= ≤ n×n → Qf : sup{g f | g is quasiconvex} ; Qf : M ¢ is the largest quasiconvex function

below f , and f is quasiconvex if and only if f = Qf . It can be shown that

¢ ¢ ¤

( ) = £ 1 ( + ∇φ ( )) ∈ n×n φ ∈ ∞ (Ω) ¢

Qf A : inf ¢ f A x dx | A M , C0 (2.1) ¦ ¥ Ω Ω and Cf ≤ Qf ≤ f (see Dacorogna [5]); also, if f ≥ c , a constant, on M n×n then Cf ≥ c on M n×n . Here, Ω is the volume of Ω .

Ω ⊂ n We will need the following result from linear elliptic theory. Let ¢ be an open and bounded set and 1 < p < ∞ .

n n×n → n×n = ∈ Let C : M M be a 4-tensor, C [ X ] C X , C ¢ , for all ij ijkl kl ijkl k,l=1

∂2φ ∈ n×n φ ∈ ∞ Ω ∇φ = k ∈ n X M . We define, for C ( ) , divC ( x) : C ( x) , x § , i ijkl ∂x ∂x j,k,l=1 l j i =1, 2,..., n .

6 p n

As usual, we define, L (Ω) := the class of all measurable functions ψ : Ω → ¢ ¡

1 ¢

¢ p ¤

ψ p ψ = £ ψ p < ∞ ¢

such that is integrable with the corresponding norm, : ¢ dx . ¦ p ¥ Ω

We also define W 1, p (Ω) := {ψ ∈ Lp (Ω) | ∇ψ ∈ Lp (Ω)}with its norm,

1 p p p 1, p ψ 1, p := { ψ p + ∇ψ p } . The space W (Ω) is a Banach space. W (Ω) L (Ω) L (Ω)

1, p (Ω) = ψ ∈ 1, p (Ω) ψ = ∂Ω 1, p (Ω) Define W0 : { W | 0 on } ; W0 is a closed subspace

of W 1, p (Ω) and is therefore a Banach space with the same norm.

¢ £ ¥

−1,q (Ω) ¤ 1 + 1 = 1, p (Ω) § We define W , ¦ 1 , to be the dual space to W , i.e., p q 0

−1,q (Ω) = 1, p (Ω)* 1, p (Ω) W : W0 (= space of bounded linear functionals on W0 ), with the

corresponding norm ψ − := sup u,ψ where u,ψ = u ( x)ψ ( x) dx . W 1,q (Ω) ∈ 1, p (Ω) u W0 Ω u ≤1 W1, p (Ω)

p −1,q If ψ ∈ L (Ω) , then ∇ψ ∈W (Ω) and also ∇ψ − ≤ ψ (see Adams W 1,q (Ω) Lp (Ω)

[1]). In particular,

divC [∇φ] − ≤ C [∇φ] . (2.2) W 1,q (Ω) Lp (Ω)

If we suppose that C is strongly elliptic, then it can be easily shown that the partial differential operator divC [∇φ ] is elliptic. To do this, we have that

n 2

∂ φ ∂ k n divC ∇φ ( x) := C ( x) , x ∈ § . If we replace with ξ where i ijkl ∂x ∂x ∂x j j,k,l=1 l j j

n n×n

ξ = (ξ ξ ξ ) ∈ § { } ∈ 1, 2 , , n \ 0 , we arrive at the matrix M M , where

n n = ξ ξ η η > Mik Cijkl j l . But M is positive definite, i.e., Mik i k 0 for all j,l=1 i,k=1

n ¡

η = (η η η ) ∈ § { } η η > η ≠

1, 2 , , n \ 0 , i.e., M 0 for all 0, since C is strongly

¢ £ ¥ elliptic. Then M is nonsingular. Thus, divC ¤ ∇φ ( x) is elliptic (see Agmon, Douglis,

Nirenberg [3] or Morrey [9]). Then we have the following result.

Morrey’s Lemma

Ω ⊂ n < < ∞ Let ¢ be open and bounded, 1 p , and let C be a strongly elliptic 4-

1, p tensor. Then ∃ c > 0 such that c φ 1, p ≤ divC [∇φ ] − ∀ φ ∈W (Ω) , W (Ω) W 1,q (Ω) 0

1 1 + =1. In particular, p q

c φ 1, p ≤ C [∇φ] (2.3) W (Ω) Lp (Ω)

∀ φ ∈ 1, p (Ω) W0 .

Proof

The Lemma is essentially in Agmon, Douglis, Nirenberg [2] or Morrey [9] (see also Simpson [10]).

In Agmon, Douglis, Nirenberg [2] see Theorem 15.1 page 707. We will write the

n ∂ operator in “integral form” Lφ = divC [∇φ ] = f where f = C [∇φ ] . In that ∂x j j ij j=1 j

8 = = = Ω Ω ⊂ paper, l 1, m n , any bounded open set containing such that ¡ , and

¢ = Ω . Then the theorem in Agmon, Douglis, Nirenberg [2] states that

¢ £

n p n ¥

∂φ ¤ p p ¥

dx ≤ constant ¤ fβ + φ dx § ∂ ¦

xµ £ µ=1 Ω β =1

φ ∈ 1, p for all W ( ¤ ) . This yields

φ ≤ ∇φ + φ [ ] ¥ c 1, p C ¥ p (2.4) 1 W (Ω) Lp ( ) L ( )

1, p for all φ ∈W ( ¤ ) . To prove (2.3) from (2.4), assume for a contradiction, that (2.3)

∞ fails. Then, there exists a sequence, {φ } ⊂ W 1, p (Ω) , such that φ =1 for all j

j j=1 0 j W1, p (Ω)

¦ § © ¨ ∇φ → → ∞ 1, p (Ω) → p (Ω) and C j 0 as j . By Rellich compactness W0 L is a Lp (Ω) compact imbedding (see Adams [1]). So there exists a subsequence, {φ }, that converges ji strongly in Lp (Ω) . We will relabel φ as φ . Let φ → φ strongly in Lp (Ω) . From ji j j

(2.4) we have

φ −φ ≤ ∇φ − [∇φ ] + φ −φ c1 j k 1, p C j C k j k p

W (Ω) Lp (Ω) L (Ω)

≤ ∇φ + [∇φ ] + φ −φ C j C k p j k p (by Triangle Inequality). Lp (Ω) L (Ω) L (Ω) φ 1, p (Ω) φ → φ 1, p (Ω) φ ∈ 1, p (Ω) Then, { j} is Cauchy in W . So j in W0 where W0 . Thus,

C [∇φ ] = 0 a.e. in Ω . Therefore, divC [∇φ ] = 0 a.e. in Ω ( weak solution). By elliptic regularity theory (see Morrey [9]), φ ∈C ∞ (Ω) .

9 n

We now prove that φ ≡ 0 in Ω . First extend φ to be identically 0 on § \ Ω , so

1,2 n

φ ∈W ( § ) . The Fourier Transform of φ is

φ ξ = 1 −i( x ξ )φ ξ ∈ n ∇φ ( ) e ( x) dx, x, § . Then, the Fourier Transform of is

π n 2 ( ) ¡

2 n ¢ ∇φ (ξ ) = 1 −i(x £ ξ )∇φ ( ) = φ (ξ ) ⊗ξ n 2 e x dx i (by integration by parts) . Then, (2π ) ¤ n

n ¦

∂φ ¥ ¨ ∇φ = k = § φˆ ξ ⊗ξ = ξ ∈ n C [ ] C 0 yields C ( ) 0 a.e. for ¢ . Further, ij ijkl ∂x k,l=1 l

¥ ¦

ˆ n strong ellipticity hypothesis for C , φ (ξ ) = 0 a.e. for ξ ∈ ¢ . Thus, by Fourier Transform

Theory, φ ≡ 0 a.e. in ¢ . But φ =1 φ 1, p =1. Since, φ ≡ 0 , we have the j W1, p (Ω) W (Ω) desired contradiction and, therefore, (2.3) holds.

Alternatively, in Morrey [9], see Theorem 6.4.4 page 246. There, υ = N = n ,

= = = ρ = = = − tk 2, s j 0 , m j 1, jk 1, and h h0 1. Let B denote any open ball containing

Ω ∈ 1, p ( ) αβ = α = β = αβ = α = and let u W0 B . Also, a jk C jαkβ for 1, a jk 0 for 0 or

β = 0 ; f α = C [∇u] for α =1, f α = 0 for α = 0 . This is a solution to the weak

j jα j

¥ ¦

N n

α αβ β k α 1, p

υ − = ∀ υ ∈ ( )

formulation D j a jk D u f j dx 0 W0 B (see

§ ¨ B j=1 |α|≤1 k=1 |β|≤1

equation (6.4.1) on page 242 Morrey [9]). Then, by the theorem cited, we get the estimate

u ≤ c f + u

W1, p (B) Lp (B) L1(B)

≤ c′ f + u . Lp (B) Lp (B)

Then this yields (2.4) again when identifying ¢ with B . Therefore, as before, (2.3)

follows. £

11 Chapter III

Theorem 1

Purpose

We wish to construct a class of quasiconvex functions on M 2×2 which are not polyconvex. In order to achieve this we need the preliminary Theorem 1, ensuring certain specific conditions on this class of functions. First we look at a class of functions

n×n → f : M ¢ which are bounded above and below in a particular way. Then we look at f ’s quasiconvexification, Qf .

Theorem 1

< < ∞ n×n → Let 1 p and f : M , be continuous, such that

p ≤ ≤ + p ∀ ∈ n×n ∈ + C [ X ] f ( X ) c1 c2 X X M where c1,c2 ¡ and C is a strongly

× × elliptic 4-tensor. Let L = {X ∈ M n n | C [ X ] = 0} and K = {X ∈ M n n | f ( X ) = 0};

Ω ⊂ n let ¡ be the open unit ball. Then the quasiconvexification of f , Qf , satisfies

× 0 < Qf ( X ) for any X ∈ M n n \ K.

Proof

Note that K ⊂ L . Using the relationship, Cf ≤ Qf (see Dacorogna [5]) we first

× p show Qf is positive on M n n \ L . To do this, it suffices to demonstrate that C [ X ] is

p × convex to get 0 < C [ X ] ≤ Cf ( X ) ≤ Qf ( X ) ∀ X ∈ M n n \ L .

12 p ×

To show C [ X ] is convex, let X ,Y ∈ M n n, t ∈[0,1]. Then, ¦

¥ p ¨ § p C tX + (1−t)Y = tC [ X ] + (1− t)C [Y ]

= + p = = ( − )

t1C [ X ] t2C [Y ] , t1 t and t2 1 t ¦

¥ p (3.1) ¨ ≤ § +

t1 C [ X ] t2 C [Y ] , by the Triangle Inequality

¥ ¦

¡ ¡ p

1 1 1 1

£ ¢ £

q ¢ p + q p > 1 + 1 =

¥ ¤ ¥

¤ [ ] [ ] ¨ = § t t C X t t C Y , p,q 1 s.t. 1. 1 1 2 2 p q

Holder’s Inequality states

1 1 1 1 a b + a b ≤ (a q + a q ) q (b p + b p ) p , where p, q > 1 s.t. + =1. If we let 1 1 2 2 1 2 1 2 p q

1 1 1 1 = q = p = q = p

a1 t1 , b1 t1 C [X ] , a2 t2 , and b2 t2 C [Y ] , then

¦ § ¦ §

1 1 1 1 1

q p + q p ≤ + q +

( ) t1 t1 C [ X ] t2 t2 C [Y ] t1 t2 (t1 C [ X ] t2 C [Y ] ) . (3.2)

Using (3.2) in (3.1) we have

¥ ¦

¡ ¡

p ¦

¥ 1 1 1 1

£ ¢ £

p ¢ ¨

§ + − ≤ q p + q p

¥ ¤ ¥

( ) ¤ [ ] [ ] ¨

C tX 1 t Y § t1 t1 C X t2 t2 C Y

¥ ¦

1 p

1 p p p

≤ + q +

(t t ) t C [ X ] t C [Y ] ¨

§ 1 2 ( 1 2 )

¥ ¦

1 p

1 p p

= + ( − ) q + ( − ) p

(t 1 t ) t C [ X ] 1 t C [Y ] ¨

§ ( )

¥ ¦

1 p

= q p + − p p

1 t C [ X ] (1 t) C [Y ] ¨ § ( )

p p = t C [ X ] + (1−t) C [Y ] .

13 ¦

¥ p ¨ § p p p Therefore C tX + (1−t)Y ≤ t C [ X ] + (1− t) C [Y ] which gives us that C [ X ] is

× convex ∀ X ∈ M n×n . Thus, ∀ X ∈ M n n \ L, 0 < Qf ( X ) is true.

It now remains to be shown that 0 < Qf ( X ) ∀ X ∈ L \ K. Suppose, for a contradiction, that Qf ( A) = 0 for some A∈ L \ K. By (2.1) ∃ a sequence

φ ∈ ∞ (Ω) 1 + ∇φ ( ) = j C0 such that lim f ( A j x )dx 0. We know that j→∞ Ω Ω

∀ X ∈ M n×n \ K C [ X ] ≤ Cf ( X ) ≤ Qf ( X ) ≤ f ( X ). Then, ¦

¥ p ¨ 1 § + ∇φ ( ) ≤ 1 + ∇φ ( ) = lim C A j x dx lim f ( A j x )dx 0 which implies j→∞ Ω j→∞ Ω

Ω Ω ¦

¥ p ¨ 1 § + ∇φ ( ) = lim C A j x dx 0. j→∞ Ω

Ω

¡ ¡ ¡

£ ¢ £ ¢ £ ∈ ¢ + ∇φ = + ∇φ = ∇φ

Recall, since A L , C A j ( x) C [A] C j ( x) C j ( x) . ¦

¥ p ¨ § ∇φ ( ) = Therefore, lim C j x dx 0. By Morrey’s Lemma (see Chapter II), j→∞ Ω

φ = φ φ lim j 1, p 0 . There is a subsequence of j 's which we will relabel as j itself j→∞ W (Ω)

φ → Ω such that j 0 a.e. in (see Hewitt and Stromberg [8, Theorem 13.11]). { } Now we use Fatou’s Lemma, which states, let gn be a sequence of nonnegative

Ω ( ) = ( ) ≤ integrable functions over and let g x liminf g j x ; then, g dx liminf g j dx j→∞ j→∞ Ω Ω

14 ¦

¥ p ¨ = + ∇φ ≤ § ∇φ ≤ + ∇φ (see Friedman [6]). Now, if g j f ( A j ) , then 0 C j f ( A j )

≤ + ∇φ ≤ + + ∇φ p ≤ + p + ∇φ p implies 0 g j . Also, f ( A j ) c1 c2 A j c1 c3 ( A j ) for some

> Ω

constant, c3 0 implies g j is integrable over and f continuous implies

¢ £ ¥ + ∇φ = ¤ + ∇φ = ( ) Ω = ( ) lim f ( A j ) f A lim j f A a.e. in , which implies g f A . j→∞ j→∞

By Fatou’s Lemma, we have

1 + ∇φ ≥ 1 + ∇φ liminf f ( A j )dx lim f ( A j )dx j→∞ Ω Ω j→∞ Ω Ω

1 = f ( A)dx Ω Ω

= f ( A).

= 1 + ∇φ ≥ ( ) By our hypothesis, 0 lim f ( A j )dx , so we have 0 f A . However, j→∞ Ω Ω

0 = C [ A] ≤ f ( A) 0 ≤ f ( A). Thus, 0 ≤ f ( A) ≤ 0 , so that f ( A) = 0 and A∈ K which is the desired contradiction. Thus, Qf ( X ) > 0 ∀ X ∈ L \ K. We conclude Qf > 0

n×n on M \ K . £

15 Chapter IV

Theorem 2

Purpose

Now that we have established the needed preliminary theorem for quasiconvex functions on M n×n and for 1 < p < ∞ , we will address the issue of the polyconvexity of

Qf on M 2×2 but with 1 < p < 2 .

Theorem 2

2×2

Let 1 < p < 2, f : M → ¡ be continuous and satisfy

p ≤ ≤ + p ∀ ∈ 2×2 ∈ + C [ X ] f ( X ) c1 c2 X X M where c1,c2 and C is a strongly

× elliptic 4-tensor. Assume K = {X ∈ M 2 2 | f ( X ) = 0} is nonconvex. Then the quasiconvexification of f , Qf , is quasiconvex but not polyconvex.

Proof

Qf is quasiconvex by definition. Suppose, for a contradiction, that Qf is polyconvex. Since Qf ≤ f , then Qf has sub-quadratic growth at ∞ , i.e.,

( ) ≤ + p ∀ ∈ 2×2 2×2 Qf X c1 c2 X X M . It is known that every polyconvex function in M with subquadratic growth at ∞ is convex (see Sverak [11] or Simpson [10]). Next,

0 ≤ Qf ≤ f implies Qf = 0 on K . Therefore, Qf = 0 on co(K ) = convex hull of K .

But K is nonconvex, so there are points in co(K ) that are not in K . We now have a contradiction to Theorem 1, and therefore, the desired contradiction, i.e., Qf is not

polyconvex. £

16 Chapter V

Theorem 3

Purpose

We will now discuss the invariance of this class of quasiconvex, but nonpolyconvex functions. This class of functions is important to applications involving elasticity of materials, phase transitions, and other applications. Since polyconvexity implies quasiconvexity and polyconvexity is easier to verify and the gap between the two is slight, our construction of this class of functions that are quasiconvex but not polyconvex is mathematically important. Recall, O(2) is the set of orthogonal 2× 2 matrices, R , such that det R = ±1 and SO(2) is the set of orthogonal 2× 2 matrices, R+ , such that det R+ = +1where R+ denotes a proper rotation.

Theorem 3

2×2

Let 1 < p < 2, f : M → ¡ be continuous and satisfy

p ≤ ≤ + p ∀ ∈ 2×2 ∈ + C [ X ] f ( X ) c1 c2 X X M where c1,c2 and C is a strongly elliptic 4-tensor. Assume f is invariant with respect to O(2) , i.e., f (RXR ) = f ( X )

× ∀ X ∈ M 2×2 and ∀ R ∈O(2) , and assume K = {X ∈ M 2 2 | f ( X ) = 0} is nonconvex.

Then the quasiconvexification of f , Qf , is invariant with respect to O(2) and

+ quasiconvex but not polyconvex. Similarly, suppose f (R X ) = f ( X ) ∀ X ∈ M 2×2 and

17 + × ∀ R ∈ SO(2) , and assume K = {X ∈ M 2 2 | f ( X ) = 0} is nonconvex. Then the quasiconvexification of f , Qf , is invariant with respect to SO(2) and quasiconvex but not polyconvex.

Proof

Ω ⊂ 2 = ∀ ∈ 2×2 We suppose is the unit ball and f (RXR ) f ( X ) X M and

∀ R ∈O(2) . Recall, O(2) is the set of orthogonal 2× 2 matrices, R , such that

det R = ±1. We show that Qf is invariant, i.e., Qf (RXR ) = Qf ( X ) . Fix R ∈O(2) and

¡ ¢

£ £

¤ ¥ ¨

∈ 2×2 ¨ = 1 + ∇φ φ ∈ ∞ Ω £

( ) £ ( ( )) ( ) § fix X M . We have Qf RXR inf ¦ f RXR x dx | C . Ω 0

Ω

invariance of f . This implies

¡ ¢

£ £

¤ ¥ ©

( ) £ ( ( ) ) ( ) §

Qf RXR inf ¦ f X R x R dx | C . (5.1) Ω 0 Ω

We make a change of variable in the last integral and let x = Rx . This transformation

does not affect Ω . Then, ¡ ∂x

dx = dx ∂ ∂ £ ¢ x1 x1

∂x

¢ £

∂ ∂

∂x x1 x2

¢ £

= det R dx , = = R £

¢ ∂ ∂ ∂

x x2 x2

¤ ¥

= ±1 dx = dx

∂ ∂ x1 x2

and equation (5.1) becomes

¡ ¢

£ £

¤ ¥ ©

( ) £ ( ( ) ) ( ) § Qf RXR inf ¦ f X R Rx R dx | C . We now let Ω 0

Ω

£ ¡

¡ ∞

ψ ( ) = φ ( ) ∈Ω ψ ∈ (Ω) ∇ψ ( ) = ∇φ ( ) ∇ x R Rx , x , C0 . Next, we show x R Rx R where

is the matrix of partial derivatives with respect to x . In fact,

¥ ¦ ¤

∂ψ ( x) ¨ § ∇ψ = i (x)

ij ∂

x© j

∂

= φ R (Rx)

∂ ik k x j k=1 2

∂φ ∂x = k l R ik ∂ ∂ k=1 xl x j 2

∂φ = R k R ik ∂ lj

k=1 xl

¥ ¦

¨ = § R ∇φ (Rx) R .

= 1 + ∇ψ ψ ∈ ∞ Ω =

( ) ( ( )) ( ) ( ) We now have Qf RXR inf f X x dx | C Qf X . Ω 0 Ω

Thus, Qf is invariant as desired. Then we have, by Theorem 2, Qf is quasiconvex, invariant but not polyconvex.

+ + Now, suppose f (R X ) = f ( X ) ∀ X ∈ M 2×2 and ∀ R ∈ SO(2) . We again recall, SO(2) = the set of orthogonal 2× 2 matrices, R+ , such that det R+ = +1where R+ denotes a proper rotation. We show that Qf is invariant, i.e., Qf (R+ X ) = Qf ( X ). Fix

R ∈ SO(2) and fix X ∈ M 2×2 . We then have

+ = 1 + + ∇φ φ ∈ ∞ Ω

( ) ( ( )) ( ) Qf R X inf f R X x dx | C . Note that, Ω 0

Ω

f (R+ X + ∇φ ( x)) = f (R+ ( X + (R+ ) ∇φ (x))) = f ( X + (R+ ) ∇φ ( x)) as before. We ¡ ψ = + φ ∇ψ = ∇ + ¡ φ = + ∇φ

now let ( x) (R ) ( x) , then ( x) ((R ) ( x)) (R ) ( x). This yields

¢ £

¤ ¤ ¦

+ = ¥ 1 + ∇ψ ψ ∈ ∞ Ω ¤

( ) ¤ ( ( )) ( ) ¨ Qf R X inf § f X x dx | C Ω 0 Ω = Qf ( X ).

Thus, Qf is invariant with respect to SO(2) and is quasiconvex but not

polyconvex. £

In the theory of hyperelastic materials the total stored energy, W (∇u)dx , is to Ω

Ω → n be minimized. Here, u : © is the deformation of the material and is assumed locally invertible, det ∇u ( x) > 0 ∀ x∈Ω . The stored energy function W is assumed to be

n×n objective, i.e., W (QF ) = W (F ) ∀ Q ∈ SO(n) and for all F ∈ © + ( = the set of n× n matrices with det > 0 ). Often W is required to be isotropic, i.e., W (QFQ ¡ ) = W (F )

n×n

∀ F,Q as above. Here, f = W and we do not restrict to © + but the notions of quasiconvexity, polyconvexity and rank-one convexity still apply. Thus Theorem 3 provides examples of objective and isotropic quasiconvex functions that are not polyconvex.

20 Chapter VI

Examples

Construction of non-trivial examples of functions satisfying the hypotheses in the

2 previous theorems is a non-trivial exercise. We let Ω ⊂ be the unit ball, 1 < p < ∞ for the hypotheses in Theorem 1 and 1 < p < 2 for the hypotheses in theorems 2 and 3,

∈ + where c1,c2 and C is a strongly elliptic 4-tensor in all of the following examples.

Example 1

One example of f ( X ) that is in our class of functions is Sverak’s [11] example where C is the Lamé tensor, C [ X ] = µ ( X + X ¡ ) + λ (trX ) I , µ and λ are Lamé

1 constants with the bulk modulus, µ = = −λ . It is known that C is elliptic if and only if 2

µ > 0 and 2µ + λ > 0 . This problem occurs in linear elasticity and C [X ] represents the

stress on a material and ∇φ ( x) is the deformation gradient. This gives

¡ x − x x + x

= 1 + − 1 = 1 11 22 12 21 2×2 → ¦ £

( ) ¢ ¥ C [ X ] ( X X ) trX I ¤ . In his example, f : M , + − 2 2 2 x12 x21 x22 x11

1 < p < 2 , L is a 2-dimensional affine subset of M 2×2 which does not contain any rank- one direction, i.e., rank ( A− B) ≥ 2 for any 2 distinct A, B ∈ L , and K ⊂ L is a closed

p set. In his work, ⋅ is the Euclidean norm and f ( X ) = (dist ( X , K )) , where dist ( X , K ) = inf X −Y . His choice of f satisfies the hypotheses of our theorems as Y∈K

21 p

follows: In Theorem 1, f ( X ) = (dist ( X , K )) is clearly continuous. He lets

¡ £ ¢ − s t ¦

= ∈ ≥ £

¢ ( ) ( ) ¤ ¥ ¥ L ¤ , s,t which implies dist X , K dist X , L . It can be easily 0 t s 0

1 2 2 shown that dist ( X , L ) = C [ X ] = ( x − x ) + ( x + x ) . Notice L corresponds 0 2 11 22 12 21 0

× to our L = {X ∈ M n n | C [ X ] = 0} . This then establishes the left inequality in the hypothesis for f in the statement of Theorem 1. Sverak does not state an upper bound on

∈ f , however it is easily shown that his f satisfies our upper bound. Fix any X 0 K and

1 1 ∈ 2×2 ( ) ≤ − ≤ + ≤ q p + p p any X M ; then dist X , K X X 0 X X 0 2 ( X X 0 ) by Triangle

Inequality and Holder’s Inequality. Therefore, ¨ § p

1 1

( ) ( ) f X (dist X , K ) 2 ( X X 0 )

p = q p + p 2 ( X X 0 ) ; = p−1 p + p−1 p 2 X 0 2 X

= p−1 p = p−1 if we let c1 2 X 0 and c2 2 , we obtain the desired full inequality,

p ≤ ≤ + p

C [ X ] f ( X ) c1 c2 X , or in Sverak’s terms, ¨ § p

1 1

( ) ( ) (dist X , L0 ) (dist X , K ) 2 ( X X 0 ) . As in Sverak, if K is invariant, then f and Qf are also. Thus the hypotheses for our theorems are satisfied and Sverak’s results follow from our Theorems 1, 2, and 3.

22 Example 2

= p = µ + + λ Another example is f ( X ) C [X ] where C [ X ] ( X X ¦ ) (trX ) I , the general Lamé tensor, 1 < p < ∞ , µ > 0 , 2µ + λ > 0 . Here,

× × L = {X ∈ M n n | C [ X ] = 0} = K = {X ∈ M n n | f ( X ) = 0} . Theorem 1 then follows easily as in Example 1. But L = K is convex and thus Theorem 2 does not apply.

However, we can show the invariance of f with respect to O(2) . C [X ] is invariant

with respect to O(2) , i.e.,

¡ ¤

¤ ¤ ¤ ¤ £

C ¢ RXR = µ (RXR + (RXR ) ) + λ (tr (RXR )) I

¤¥¤ ¤

= µ (RXR ¤ + RX R ) + λ (tr (RXR )) I

¤ ¤

= µ (R( X + X ¤ ) R ) + λ (R (trX ) R ) I . ¤ = R (µ ( X + X ¤ ) + λ (trX ) I ) R

= RC [ X ]R ¤

p But C [ X ] is convex (see proof of Theorem 1), which implies that f ( X ) is convex.

Example 3

[ ] = µ + ¤ − ( ) = In this example we let C X ( X X trX I ) , L L0 as in Example 1,

K ⊂ L , K closed, nonconvex, and f ( X ) = g (d ) where d = dist ( X , K ) , g is

p ≤ ≤ + p ∈ + < < ( ) © continuous and satisfies c3d g d c1 c2d , c1,c2 ,c3 , 1 p 2 . We further assume that g (0) = 0; then K is as in Theorem 1. To show this choice of f satisfies the

[ ] p ≤ p hypotheses of Theorem 1, we need C X c3d . As in Example 1,

23 ¡

p £

p ¢ 2 2 ¥ [ ] = ¤ µ ( − ) + ( + ) C X 2 x11 x22 x12 x21 . The above inequality is true if

1 1 p p µ = ( ) p = µ ( ) ≤ + c3 or c3 ( 2 ) . The growth estimate, f X c1 c2 X , is as in 2

Example 1 and Theorems 1, 2, and 3 apply. Note that f is invariant if K is invariant.

Example 4

Here we let g :[0,∞) → ¦ , be continuous, invariant, and satisfy

< < ( ) = ⊂ hypotheses and 1 p 2 for Theorems 2 and 3. Also, g 0 0. Let K L0 , K closed ( ) and nonconvex with L0 and C as in Example 1. We give two versions of f X ,

Version A and Version B.

Version A

2×2 = ⊕ 2×2 We write M L0 N where N is a 2-dimensional subspace of M and

≥ 2×2 = −1 ( ) [ ] ≤ ( ) ≤ + ∀ ∈ 2×2 continuous, h 0 on M , K h 0 , and c4 P X h X c5 c6 X , X M

+ 2×2

Now we verify the hypotheses of Theorem 1: by Holder's Inequality ,

24 ( ) = ( ) ≤ + ( ) p f X g (h X ) c1 c2 (h X ) ≤ + + p

c1 c2 (c5 c6 X ) ¡ p

1 1 £

≤ + ¢ q + q q p + p p ¥ c1 c2 ¤ (c5 c6 ) (1 X )

p p = + ( ) q + c1 c2 c7 (1 X )

= c + c (c (1+ X p )) 1 2 8 = + p = + = c9 c10 X , c9 c1 c2c8 , c10 c2c8.

Thus the upper bound on f is satisfied. Now,

( ) = ( ) ≥ ( ) p ≥ [ ] p ≥ [ ] p f X g (h X ) c3 (h X ) c3 (c4 P X ) c11 C X by utilizing the following facts:

[ ] ≤ ∀ ∈ 2×2

i) C Y c12 Y Y M ,

¦ §

ii) C [ X ] C ¨ P[ X ] X M X M . ¡

− + £ ¢ y y y y

= 1 11 22 12 21 ¥ Property i) follows from C [Y ] ¤ and property ii) follows from + − 2 y12 y21 y22 y11

∈ 2×2 = ( − )[ ]+ [ ] ( − )[ ]∈

writing X M in the form X I P X P X , so I P X L0 . Therefore,

§ ¦ § ¦ § ¦ §

Note that f −1 (0) = K = h−1 (0) is nonconvex, therefore Theorem 2 applies. Similarly,

Theorem 3 applies if K is invariant, and therefore f is also invariant. We also note that h( X ) = dist ( X , K ) satisfies all of the above hypotheses on h and then Sverak’s example is a special case of Version A where g ( y) = y p .

Version B

≤ ≤ + ∀ ∈ 2×2 ∈ + ( ) ( ) © c11dist X , K h X c5 c6 X , X M where c11,c5 ,c6 . Let f ( X ) = g (h( X )) ∀ X ∈ M 2×2 . Verification of the upper bound on f is the same as in

Version A. For the lower bound we have

( ) = ( ) ≥ ( ) p f X g (h X ) c3 (h X ) ≥ ( ) p c3 (c11dist X , K ) = ( ) p ( ) p c3 c11 (dist X , K ) ≥ ( ) p = ( ) p c12 (dist X , L0 ) , where c12 c3 c11 = [ ] p c12 C X .

From this the three Theorems apply.

List of References

27 List of References

1 R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.

2 S. Agmon, A. Douglas, L. Nirenberg, Estimates near the boundary for

solutions of elliptic partial differential equations satisfying general boundary

conditions. I, Comm. Pure Appl. Math., Vol. 12 (1959), pp. 623 – 727.

3 S. Agmon, A. Douglas, L. Nirenberg, Estimates near the boundary for

solutions of elliptic partial differential equations satisfying general boundary

conditions. II, Comm. Pure Appl. Math., Vol. 17 (1964), pp. 35 – 92.

4 J. M. Ball, Convexity conditions and existence theorems in nonlinear

elasticity, Arch. Ration. Mech. Analysis, Vol. 63 (1978), pp. 337 – 403.

5 B. Dacorogna, Direct Methods in the Calculus of Variations, Springer-Verlag,

Berlin, 1989.

6 A. Friedman, Foundations of Modern Analysis, Dover Publications, Inc., New

York, 1982.

7 I. M. Gelfand and S. V. Fomin, Calculus of Variations, Prentice-Hall,

Englewood Cliffs, New Jersey, 1963.

8 E. Hewitt and K. Stromberg, Real and Abstract Analysis, a Modern Treatment

of the Theory of Functions of a Real Variable, Springer-Verlag, Berlin, 1965.

9 C. B. Morrey, Multiple Integrals in the Calculus of Variations, Springer-

Verlag, Berlin, 1966.

10 H. Simpson, by private communication, 2003.

28 11 V. Sverak, Quasiconvex functions with subquadratic growth, Proc. R. Soc.

Lond. A, Vol. 433 (1991), pp. 723-725.

12 F. Y. M. Wan, Introduction to the Calculus of Variation and Its Applications,

Chapman & Hall, New York, 1995.

29 Vita

Catherine S. Remus was born in Great Lakes Naval Hospital, Illinois on October

29, 1952. She lived with her family in Illinois and attended Parochial school through tenth grade. She graduated from Willowbrook High School in June of 1970. She attended various colleges from 1970 until 1976 because of family relocations. She worked in Civil

Engineering from 1972 until 1988 in increasingly responsible positions. She was married in 1984 and gave birth to her son in 1988. She focused on family responsibilities until

1997 when she divorced and went back to school. She earned a Bachelor of Science in

Mathematics in May, 1999 from the University of West Florida and began her graduate studies at the University of Tennessee in August, 1999 where she earned a Master of

Science in Mathematics in December, 2003.