Existence of minimizers for non-quasiconvex integrals · scalar case (for which N = 1) always falls...

Arch. Rational Mech. Anal. 131 (1995) 359-399. �9 Springer-Verlag 1995

Existence of Minimizers for Non-Quasiconvex Integrals

BERNARD D A C O R O G N A • P A O L O M A R C E L L I N I

Communicated by H. BREZIS

Table of contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 2. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 3. A general existence theorem in the hyperbolic case . . . . . . . . . . . . . . . . . . . . . . . . 364 4. 2 x 2 systems of linear Hamilton-Jacobi equations . . . . . . . . . . . . . . . . . . . . . . . . . 370 5. Necessary conditions for existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 6. Application to optimal design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 7. The case of the determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 8. The Saint Venant-Kirchhoff materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397

1. Introduction

Let ~ c IR" be a bounded open set. Let u:f~ ~ I R N, u = u(xl . . . . . xn) = (u~)l _< ~ _< ~; thus Du = (Ou~/~xi)l < ~ <_ ,, 1 <_ ~ <_ N, i.e:, Du ~ IR "N. Finally let 4o e IR ~N and f : IR ~N --+ IR be a lower semicontinuous function. The problem we consider is

(p) i n f { F ( u ) = S f ( D u ( x ) ) d x ' u ~ u ~ } a

where Uo(X) = ~oX, X el2. The quasiconvexity of f (cf. (1.1)) in t roduced by MORREY [52] guarantees, by

definition, that u0 is a minimizer for (P). The aim of this article is to give condit ions for existence and non-existence of

solutions of (P) for non-quasiconvex functions f. Before describing these conditions, we should stress that surprisingly there is more hope to solve the minimizat ion problem involving vector-valued functions than to solve problems for scalar- valued functions (for which N = 1). In fact, we show that the larger N is, the more freedom one has in the analysis.

360 B. DACOROGNA 8,~ P. MARCELLINI

To be more precise, we need to introduce some terminology and notations. Following MORREY [52], we say that f : IR "N --+ IR is quasiconvex if

~ f ( 4 + Vcp(x))dx >f(4)measf~ (1.1) a

for every bounded open set f~ ~ IR", ~ ~ IR "N and ~0 ~ Wo 1" oo(f~; IRN). Given a general f : l R "N ~ IR, the quasiconvex envelope Q f o r f is defined to be

the largest quasiconvex function less than or equal to f In the minimization problem the set which plays the important role is

K = {4 e IR"N: Q f (4) < f ( 4 ) ) . (1.2)

The nontrivial case is when Duo = 4o e K. Important observations in our analysis are that

(i) Q f should be quasiaffine around 4o (i.e., Q f and - Q f are quasiconvex); (ii) Q f need not be quasiaffine on the whole of K, but only on Kc~L (with

~0 e K c~ L) where L is a quasiaffine manifold of dimension nN - (N - 1) (i.e., we are free to impose N - 1 quasiaffine conditions; in particular, for the scalar case N = 1, this freedom does not exist).

Conditions (i) and (ii) are sufficiently general so as to include several well- known examples:

f (4) = g(det 4), (when n = N) and its generalizations, f ( r = g([ 41), where 14[ stands for the Euclidean norm of r e IR "N, the problem of KOHN ~; STRANG [-39] (cf. Example 3 in Section 2), the Saint Venant-Kirchhoff energy function (cf. Example 4 in Section 2), the scalar case (n > 1 and N = 1), the one-dimensional case (n = 1 and N > 1), that we do not consider explicitly here. These examples have been individually studied by several authors: AUBERT

& TAHRAOUI [2-51, BUTTAZZO, FERONE & KAWOHL [111, CELLINA [12, 131, CELLINA & ZAGATTI [15, 16], CESARI [17, 18], CUTRt ['-221, DACOROGNA [23,24], EKELAND [25], FRIESECKE [31], KLOTZLER [-37], MARCELLINI [43--461, MASCOLO [48], MAS- COLO & SCHIANCHI [-49 51], OLECH [541, RAYMOND [-56--59], TAHRAOUI [62, 63]. In particular, CELLINA [12, 131 and FRIESECKE [31] proposed, in the scalar case N = 1, some sharp conditions that turn out to be necessary and sufficient for existence of minimizers of some non-convex integrals with prescribed linear boundary data.

The existence of minimizers of (P) will be obtained by solving a system of first-order partial differential equations of Hamilton-Jacobi type (for more details see Sections 3 and 4). Although the case of a single Hamilton-Jacobi equation has a long history (cf. P.-L. LioNs [42]), systems of such equations have been less studied. In this paper we propose an approach to the study of some such systems related to variational problems. Our systems are composed of N equations, N - 1 of them being linear (i.e., we have chosen L, in (ii) above, to be a linear manifold).

In some cases, which are studied in Section 3, we obtain (cf. Lemma 3.5) that the gradient Du(x) of the solution belongs for almost every x to a manifold Lo c L of dimension lower than L. More precisely, dim Lo = n, while dim L = nN -- (N -- 1).

Existence of Minimizers 361

In Section 4 we describe the n -- N -- 2 case, by assuming that both first-order equations are linear. To find a solution of the system, it is then natural to separate the study into the h y p e r b o l i c (and parabo l i c ) case and the el l ip t ic case. The first will be reduced, through a linear transformation, to the case of a single equation which will be solved by the method of P.-L. LioNs [42], MASCOLO & SCHIANCHI [49] and TARTAR. The elliptic case will be reduced, again after a linear transformation, to Cauchy-Riemann equations, and will be solved explicitly by using the method of confocal ellipses introduced by MURAT & TARTAR (cf. TARTAR [64]).

In Section 7 a special application of the hyperbolic method leads to an existence result for a function of type

f(~) = g(~t . . . . . ~N-1) + h(det 4); (1.3)

we recall that the case g = 0 has been studied by MASCOLO & SCHIANCHI [49] and by CELLINA • ZAGATTI [15].

As mentioned earlier, in Section 6, these techniques will be applied in particular to the example of KOHN & STRANG [39] for n = 2 and N => 2. First we use the hyperbolic method to reduce the problem to n = N = 2 and then the elliptic method to find a solution. KOHN & STRANG had noticed that the elliptic method could be applied to the case n = N = 2.

Other important variational problems involving a non-convex integrand have been considered in the study of m i c r o s t r u c t u r e observed in certain materials, where the deformation gradients lie in some (symmetry related) potential wells. We refer to the papers by BALL 8~ JAMES [6,7] and also, fo r example, to [8, 27, 29, 36, 38, 53, 55, 611. We do not study these cases in this paper; however we observe that the dimensional analysis proposed here seems to agree with the condition in the two-well problem with rank-one connections (one condition on the determinant of the gradient, in the 2 x 2 case) to obtain a quasiaffine (in fact constant) envelope of the integrand.

In Section 5 we give some necessary conditions for existence. They also turn out to be sufficient in the scalar case, for functions depending on the norm only (i.e., f(~) = 9(l~l)), for functions of the type (1.3). For more general problems they are far from being sufficient. However, from a dimensional point of view, our necessary and our sufficient conditions match since, for the former, we require that f * * be affine on a manifold of dimension at least n N - ( N - 1) while for the latter, we require that Q f be quasiaffine on a manifold of the same dimension n N - ( N - 1).

Finally, in Section 8, we apply the necessary conditions of Section 5 to the study of the Saint Venant-Kirchhoff energy function. In particular we discover that the related minimization problem (P) may lack a solution for boundary slope 40 arbitrarily close to the identity.

2. Examples

We now list some non-convex problems that already appear in the literature (some of them have been intensively studied) and that can be solved by our methods.

362 B. DACOROGNA & P. MARCELLINI

Example 1 (The case of the determinant). Let n = N and

I :i) 4 = ( 4 ~ ) 1 _ _ < ~ . , 1 ~ N =

then let

f(~) = 0(41 . . . . . 4 N- ~) + h(det 4),

with g convex (and h non-convex). We emphasize that g is independent of one of the 4~; for notational convenience we choose e = N. This problem with g = 0 has been studied by DACOROCNA [-23] and MASCOLO & SCmANCHI [49]. Existence of minimizers when g = 0 has been obtained by MASCOLO & SCHIANCm [49] (see also [-15]) under more general boundary conditions Uo. We shall prove that if

lim h(3) ~__,oo ] - ~ - = +oO,

then (P) always has a solution, independently of the convexity of h and with no condition on g other than its convexity. Some more general cases will also be considered in Section 7.

Example 2 (The case of the norm). Let n > 1, N >= 1 and

fro = g(l l) where Z Z ( r)2 i = i ~ = 1 ,

with g non-convex. A special, but important, case is that of KOHN & STRANG given in Example 3 below. We shall prove in this case that ifg**([~o]) ~= g([~o[) and if g** is strictly increasing in ]40], then (P) has no solution if rank 4o = 1. Note that the scalar case (for which N = 1) always falls in this category (cf. MARCELLINI [43, 44]). Clearly, this condition is equivalent to saying that a necessary condition for existence of solutions in this context is that rank 4o _-> 2 (cf. the next example).

Example 3 (The case of KOHN & STRAN~ [39]). Let n = 2, N > 2 and

f ( 0 = g(l~l) = {10+ 1412 ifif ~=~0,4=0.

Our theorem will show that a necessary and sufficient condition for (P) to have a solution is that one of the following conditions hold:

(j) rank 4o = 2, (jj) ~o = 0,

( j j j ) 14ol 2 + 21adj24o[ > 1

(recall that adjz~ = det~ in the case N = 2, while for N >= 2 adjz~ is the N(N - 1)/2 vector composed of all the 2 x 2 minors of the matrix ~ ~ IR2N).


Note that the two last assertions (jj) and (jjj) correspond to the trivial case Qf(~o) =f(~0)- Therefore, if the boundary slope 3o is such that Qf(~o) <f(~o), then a necessary and sufficient condition for problem (P) to have a solution is rank 4o = 2.

Example 4 (The Saint Venant-Kirchhoff energy function). Let n = N and

E Ev f ( ~ ) - 8 ( 1 + v) 1~ '~ - I lz + 8 ( 1 + v ) (1 - 2v) (1412 - n)2

for ~ ~IR "• where ~ denotes the transpose of ~ and I the identity matrixl The constants E and v (0 < v < �89 are respectively the Young modulus and the Poisson ratio (cf. C~ARLET [20], DACOROONA [24], LE DR~T & RAOULT [40, 41], TRUESOELL & NOLL [65]). It is well known that f is not quasiconvex. Recently LE DRET & RAOULT [40, 41] have computed the quasiconvex envelope of f when n = 2 and n = 3 .

When n = 2 and r ~IR 2• let 0 < vl(~) ~< v2(~) denote the principal stretches (called also the singular values of 4, i.e., the eigenvalues of (r Our result shows that (P) has a solution with positive Jacobian if and only if

( 1 - - v)v2(~o) + vv~(~o) > 1.

According to the computations of LE DRET & RAOULT these conditions correspond to the case Qf(~o) =f(~o). Note that, in particular, problem (P) lacks a solution

ositive acobia if o = (: 0) w th0 = and ,

As a byproduct of our general theorem we obtain in the next examples some well-known results for non-genuinely vectorial problems.

Example 5 (The scalar case). Let n > 1 and N = 1. Our general theorems of Sections 3 and 5 apply to this example and give a necessary and sufficient condition for the existence of minimizers (see Corollary 5.2 and the remark that follows Corollary 5.2). After the existence results in [44, 49, 50], several authors proposed sufficient conditions with general boundary data, while necessary and sufficient conditions with linear boundary data were established by CELUNA [12, 13] and FRIESECKE [313.

Example 6 (The one-dimensional case). Let n = 1 and N > 1. If

l ira f ( ~ ) - + co, j,t-,+~o 1r

then (P) always has a solution, as already established by many authors (see for example Theorem 2.6 of Chapter 5 of [24]).

364 B. DACOROGNA 8,: P. MARCELLINI

3. A general existence theorem in the hyperbolic case

As we a l ready ment ioned in the introduct ion, w e should say that the terminology "hyperbol ic" takes its origins f rom the linear 2 x 2 systems of part ial differential equat ions studied in Section 4.

Recall tha t gl c IR ~ is a b o u n d e d open set, f : IR ~N ~ IR is a lower semicont inuous funct ion and the var ia t ional p rob lem is to find

(P) inf { F(u) = S f ( D u ( x ) ) d x : u ~u~ + W~176176

where Uo(X) = ~ox, x cO. We define

K = {4 ~ IR"U: Q f ( r < f ( ~ ) } , (3.1)

where Qf(~) = sup{g(r g quas iconvex and g < f } ,

To avoid a trivial s i tuat ion we assume that 4o ~ K. I f K has more than one componen t , we replace K by its connected c o m p o n e n t which contains 40. We then let

Lo = {{ = ({"), <_,_<NelR"N:r ~ = ~ + #~({o u - IN), 0~ = 1 , 2 , . . . , N - - 1} (3.2)

for certain #= elR(1 _< ct _< N - 1) (if N = 1, we let Lo = IR'). No te that

d i m L o = nN - n (N - 1) = n.

In order to p rove an existence theorem we need a boundedness a ssumpt ion for the set K. Since for the appl icat ions it is not na tura l to assume that K itself is bounded (see the next remark) , we state the required boundedness in terms of the following set K , ; to this goal let us define

Ko==-Kc~Lo = 4 = ~ L o : ( ( b j ; ~ N ) ~ , ) l < s < z ~ K l c l R l ,

where l is an integer in [1, n], b l , . � 9 b~ are l inearly independent vectors of IR" and K1 is a subset of IR ~ tha t we will assume convex and bounded.

The orem 3.1. Under the assumptions (i) ~o ~ K ,

(ii) K1 is bounded and convex subset of IR l, (iii) Q f is quasiaffine on Ko = K ~ L o ,

(P) has a solution.

The p roof of T h e o r e m 3.1 will be given at the end of the section, after several lemmas. Remarks. (i) In cont ras t to the scalar case it is, in general, unna tura l in the vectorial case to assume tha t K is bounded. One expects only that K is bounded in


certain directions, as for example when f ({ ) = g(det {) with g non-convex. H o w - ever, if K is bounded, we then choose l = n and bj = e a = ( 0 , . . . , 0, 1, 0 , . . . , 0).

(ii) At first glance T h e o r e m 3.1 m a y look much s t ronger than our assert ion in the in t roduct ion abou t the dimension. Indeed, it seems that we can arbi t rar i ly fix n ( N - 1) affine condit ions (cf. the definition of Lo) and not, as asserted, only N -- 1 conditions. This is only an illusory gain as shown by the next proposi t ion.

Proposition 3.2. L e t

L = ~=(~a) l<a_<NelR"N" ai(~i + l ~ f ) = c ~ , c ~ = l, 2 . . . . , N - - 1 , i=1

(3.3)

where c a, t ~ e IR f o r ~ = 1, 2 , . . . , N - 1, and U = (a~)l < i <_, are non-zero vectors o f

IR " ( i f N = 1, let L = IR "). L e t f~ c IR " be a bounded open set; f inal ly let Uo(X) = {oX,

with ~o E L . Then , f o r every u eUo + Wol' ~ (~ ; IRN),

Ou(x) e L a.e. ~=> Ou(x) e Lo a.e.

R e m a r k s (i) Observe that d im L = n N - ( N - 1) and tha t if ~o e L, then Lo c L, because if {0 e L, then

~a = i ] a~((~o)~ + #a(~o)~). i = l

(ii) P ropos i t ion 3.2 shows that, by fixing N - 1 condit ions (i.e., Du e L ) and the bou nda ry condition, au tomat ica l ly n ( N - 1) condit ions are fixed (i.e., D u e Lo).

(iii) The condi t ion in the definition (3.3) of L, when N = 2, reads

a,(~ + ~ ) = i = 1

where a = (al . . . . . a,) e 1R" and # e IR. This, in terms of scalar products , is equivalent to

(a ; ~l}~a,, + # ( a ; ~z}~, = c or, equivalently, {A; ~}e.• = c,

where

A = @]Rnx2' ~ - ~2 e]Rnx2;

up to a rea r rangement of the indices ~ = 1, 2 (i.e., up to a pe rmuta t ion of the lines of the matr ices A and ~), A is a generic mat r ix of rank one.

Proof of Proposition 3.2. ( ~ ) Cf. the remark (i) above. ( ~ ) Since Du(x) e L a.e.,

L u c~ N ai (ux~ + #aUx,) = c a a.e. in f~, c~ = 1, 2 . . . . , N - 1. (3.4)

i=1

366 B. DACOROGNA 86 P. MARCELLINI

If we denote 2 ~ = (a~)l _<~_<n elR", for ~ = l, 2 . . . . , N - 1, then (3.4) can be rewri t - ten as

d d 2 ~ ( u ~ + / u N ) = c ~ a.e. in f~, c~ = 1, 2 . . . . . N - - 1

or, equivalent ly ,

(Du~(x) + / D u N ( x ) ; 2 ~) = c a a.e. in f~, ~ = 1, 2 , . . . , N - 1. (3.5)

Since, Duo = 4o e L, by assumpt ion , (3.5) is also val id for Uo, i.e.,

c~ N (Du~o+# D u o ; , V ) = c a, e = 1 , 2 . . . . . N - 1 . (3.6)

Ex tend ing any u suo + W~'~176 IR N) by le t t ing u = uo in IR n - fL we find tha t if Du(x) e L a.e., then in view of (3.6) we can rewri te (3.5) on the whole of IR" as

(Du~(x) + / D u N ( x ) ; 2 ~) = c a for a lmos t all x e IR n, e = 1, 2 , . . . , N - 1.

(3.7)

Since f~ is bounded , we can find a hype rp l a ne H a o r t hogona l to 2 ~ with H ~ c~ f~ = 0. W e can therefore wri te any x e f~ as x = y~ + t2 ~, wi th ya e H a and

t e IR. Hence (3.7) can be in t eg ra ted and we get

~a(F + t,~ ~) + W u N ( F + t,~ a) = u~ (F ) + WuN(F) + tc a

N [ O~x = u ~ ( / ) + # u o t y ) + t c ~, a = l , 2 , . . . , N - 1 ,

(3.8)

the las t ident i ty coming f rom the fact tha t ya r f~ and tha t u = Uo outs ide of fL Since the b o u n d a r y cond i t ion Uo is affine, we deduce tha t in fact the lef t -hand side of (3.8)

is affine and hence

Du~(x) + ~'DuN(x) = Du~o +/DuUo a.e. in f~, ~ = 1, 2 . . . . . N -- 1, (3.9)

i.e., since Duo = 40, we get

Dua(x) = 4~o + #~(4~ - DuN(x)) a.e.

which is the asser t ion, Du(x) eLo a.e. [ ]

In the p r o o f of T h e o r e m 3.1 we shall use the fol lowing three lemmas.

L e m m a 3.3. Let Uo = 4oX, 4o ~ IR"N. Assume that there exists a K c IR € such that

(i) ~o ~g:, (ii) Q f is quasiaffine on Ir

(iii) there exists ~t~Uo + Wol'~(f~;IR N) with DgeI~ and f ( D ~ ) = Qf(Dg) a.e. Then (P) has a solution.

Remark. In T h e o r e m 3.1, /r is the c losure of K ~ L o , while in the case of KOHN & STRANG, for n = N = 2 , / r is the c losure of the set {~ e l R 2 • = ~t, d e t r > 0, t r 4 e (0, 1)}, where 4 t denotes the t r anspose of the mat r ix ~ and tr r = trace(4).


P roof of L e m m a 3.3. By the classical relaxat ion theorem (cf. DACORO~NA [24]) we have

= i n f { ! Q f ( D u ( x ) ) d x : u e u o + W l ' ~

= ~ Qf(Duo(x))dx = Q f ( 4 o ) m e a s n . f l

The last two identities come f rom the fact that Duo = 4o and that Q f is quasiconvex.

Since Q f is quasiaffine o n / { , we have that

Qf(Du(x))dx = ~ Qf(Duo(x))dx

for every u e Uo + Wo l' ~(f l ; IR N) with Du e K.. a.e. Using (iii) we immedia te ly obtain that

The next l emma is a general izat ion for l inear b o u n d a r y condit ions of an existence result for scalar Hami l ton - Jacob i equat ions (cf. P.-L. LIoNs [42], MAS- COLO & SCHIANCm [49], TARTAR).

L e m m a 3.4. Let f~ c IR n be abounded open set and let Ko c IR" be given by

K0 = {~ e JR" : ((b;; 4))1 _< ; ~l e K1} where bz, . �9 bz, 1 <- l <_ n, are given linearly independent vectors oflR ~, and K1 is a bounded convex set oflR z. Let 4o eKo and Uo(X) = (40; x ) for every x ~ . Then the problem

Du(x)eSKo for a lmost every x e f ~ , UeUo + W J ' ~ ( f l ) (3.10)

has a solution.

Proof. We divide the p roo f into four steps. Step 1. We assume first that bj = ej = (0 . . . . . 0, 1, 0 . . . . . 0) so that

Ko = {4 en~n:(~l . . . . , ~) e K d = K~ x n~ "-~ (3.11)

where K~ is a bounded convex set. We also assume that Uo = 0 and that

n = {x = (y, z) e IR ~ x lR"-~:lyl + Izl < 1}. (3.12)

So let ~o be the unit ball of lW, i.e., ~o = {Y elR~:]Yl < 1}. By using the existence result for Hami l ton - Jacob i equat ion for bounded sets (cf.

T h e o r e m 1.1. in MASCOLO & SCHIANCHI 1-49] and P.-L. L~ONS [42]) we obta in

368 B. DACOROGNA 8r P. MARCELLINI

a f u n c t i o n v such that

Dv(y)~OK1 for a lmos t every y e ~ 0 , veWol'~ (3.13)

The desired solut ion of (3.10) is then g iven by

( 1 - [ z [ ) v ( l ~ Y [ z l ) if [ z l < l , u(x) = u(y, z) = (3.14)

0 if ]zI = 1.

This funct ion satisfies u = 0 on 0f~; indeed, if [z[ = 1, we have the result by

Y definit ion and, if [z[ < 1 and ]y] + [z[ = 1, then ] ~ e~f lo and thus v = 0 and

u = 0 too. Observ ing tha t

we deduce that Du(x)= (Dyu(x),Dzu(x))e~K1 x lR " - l = OKo. I t remains to be checked tha t u E W l' ~(f~); a direct c o m p u t a t i o n gives

and thus u ~ W 1, ~o (~). Step 2. We still assume that Uo = 0 and tha t Ko is as in (3.11), but this t ime we take

S ~ a general ft. We then cover ~) by a countable family of sets { k}k= 2, each being homothe t i c to (3.12), and a set of measure zero So, i.e.,

~ = ~ S k . k=0

Using Step 1 we can find uk E Wo 1" ~ (Sk) with Duk ~ OKo. We m a y then define

U(X)=~Uk(X) for X~Sk, k > l ,

l o for x ~So.

Clearly u has all the asserted properties. Step 3. We now remove the a s sumpt ion (3.11) on Ko. We start by choosing bj+~ . . . . . b, so that b l , . . . , b , are l inearly independent . We then change co- ordinates

t / j = ( b j ; ~ } , l < j < n .

We let B be the n • n mat r ix whose lines are b~ and we apply Step 2 to the set BKo and to the domain B - ~ fL i.e., we find

DveO(BKo) a.e. inB- l f2 , veWol'~(B-lf~).

Setting u(x) = v(B-ix) we get the result.


Step 4. We finally consider the general case with uo = 4ox in ft. Solving the problem

D v ~ ( - - ~ o + Ko), v ~ W t ~ ' ~ ( n )

by Step 3 and setting u = v + u0, we have indeed obtained the conclusion of Lemma 3.4. [ ]

We now prove a lemma on the existence of solutions for (vertorial) Hamil ton- Jacobi systems.

L e m m a 3.5. Let ~ ~ IR ~ be a bounded open set. Le t

Ko = 4 = ~ l R " N w i t h ( ( b j , 4N}r~,)l<_j<=z~Kt c l R t , (3.15)

where l is an integer in [1, n], bl . . . . . bl are linearly independent vectors of IR" and K1 is bounded and convex. Let Uo(X) = ~oX in (), with 4o e K o ; then the problem

Du(x) e OKo for almost every x ~f2, u eUo + Wol'~(U~; IR N) (3.16)

has a solution. Moreover, Du(x) ~ Lo a.e. (cf. (3.2)).

Proof. We let

K N = {~N e l R ' : ( ( b j ; ~N))I_<j__<leK1 ~ IRt}. (3.17)

Recall that K1 is bounded and convex. Moreover , ~oNeK N, since 4o = (4~)1 _ < ~ N e K o . By applying Lemma 3.4 to K N, with the boundary da tum u~ = (4oN; x}, we get the existence of u N such that

DuN(x) e O K N a.e. in f2, UN eUNo + WJ'~( f~) . (3.18)

F r o m u ~ and (3.9) we can define u = (u~)l <~_<N E WI'~176 IR N) by

Du~ = Du~ + #~(Duno -- DuN(x)) a.e. in f~, c~ = 1, 2 . . . . . N - 1. (3.19)

It is clear by (3.19) and by the fact that u N = u~ on c?f~ that u = Uo on 0f~. No te also that (3.9) and (3.19) imply that D u ~ Lo a.e. (cf. (3.2)). Finally by the definition of Ko and K N, by the fact that Du N ~ OK N and by (3.19), we have that Du(x) ~ ~Ko a.e. in f~. [ ]

Proof of Theorem 3.1. Since 4o ~ Ko by assumption, Lemma 3.5 implies that there exists a solution to the problem

Du(x) e OKo = OK ~ L o for almost every x e f2, u ~ Uo + Woa' ~(~; IRW). (3.20)


Since f is lower semicontinuous and Q f is continuous, it follows that OK ~ {Q f(#) =f($)}. Moreover, by assumption, Q f is quasiaffine on Kc~Lo and, by the definition (3.2), #o e Lo. By applying Lemma 3.3, with/~ equal to the closure of the set K c~ Lo, we obtain the conclusion. []

4. 2 x 2 systems of linear Hamilton-Jacobi equations

We now deal with the case N = n = 2. We change slightly the notations and we denote by (x, y) e IR2 the independent variables and by (u, v) e IR2 the unknown functions.

In some examples, such as Example 1 or 3 in Section 2, the sets K , / ( or Ko appearing in Theorem 3.1 or in Lemmas 3.3, 3.4, 3.5, are in fact represented by two linear equations. So we assume here that the set Ko of Lemma 3.5 is given by

Ko = {~ e]R2• (A; ~)~• e(rz, f l )and (X; ~>a~• =y} , (4.1)

where ~, 8, 7 E IR, A, A e IR a • 2 are given (the second equation, (A; ~) = 7, charac- terizes the manifold L of Proposition 3.2).

We let ~ be a bounded open set of IR2 and

(po(X, y) = \Vo(X, y)] ~o ,

where ~0 eKo ~IR2• Let us also use the notation

al

A = bl b2 '

Therefore solving (as in Lemma 3.5)

DO(x, y) eOKo a.e., ~p ~Oo + Wo~'~~ IR2), (4.2)

and is equivalent to finding s f ~ 2 ~ with ~'~10~2-----0, ~lk.)~2=~ ~0 = (u, v)ecP0 + Wo~' | IR2) such that

alux+a2uy+blvx+b2vy=yo~ a.e. in ~t , a.e. in ~2,

(4.3) d l u x + d 2 u y + b l v x + b 2 v y = ~ a.e. inf~

Hence solving (4.2), which is the main step in finding a solution of the variational problem, is equivalent to solving a system of two linear equations (of Hamilton-Jacobi type).

When dealing with such systems it is natural to distinguish three cases, the hyperbolic, parabolic and elliptic cases. Lemma 3.5 has essentially handled the first two. We show below how to deal with the third.

Before giving any theorem we recall some well-known results concerning systems of the type (4.3) (see HEI~I~W~G [35] for more details).


Definition. The system (4.3) is said to be elliptic, hyperbolic or parabolic (in f~) if d > 0, d < 0 or d = 0 respectively, where

d= 4 d e t ( al b'~det(a2 b2) - ~det(a_ 1 b2) (a 2 bl ) ~2 \a l b l / \a2 b2 ( \ a l b2 + d e t ~/2 b , / J "

Remarks. (i) We are now in a position to "justify" the terminology used in Section 3, and this at least for linear 2 x 2 systems. Recall that we assumed there (cf. Proposition 3.2) that L is given by

In order to avoid confusion with the notations used in (4.3) we rewrite L as

~] ' a Ig l + a2{ 1 + ~i 2 + , (4.4)

i.e. b l = #a l and b2 = #a2. Wi th these restrictions (bl = #a l , b2 = pa2) the system (4.3) is then hyperbolic (parabolic in some degenerate situations) independently of al, a2, bl, b2, al, 672. Indeed, an elementary computation gives

d = -- [dl(#a2 - - b2) - az(#al - - b l ) ] 2 ~ 0 .

(ii) As is well known (since the coeffidents are constant), by writing, if necessary,

u(x, y) = A~o(lx + my, 2x + #y) + BO(lx + my, 2x + #y), (4.5)

v(x, y) = CO(Ix + my, 2x + #y) + DO(Ix + my, 2x + #y)

(with AD - BC 4:0 and I# - 2m 4= 0) one can reduce (4.3) to its normal form. In the three cases the first equation can be reduced to

U x -}- O r = { ~

(e, f ie 1R being, a priori, different from the original ones), while the second equation is reduced to u r + v~ = y, vy = y or u r - v ~ = y according to the hyperbolicity, parabolicity or ellipticity of the system. In the last case one recognizes, of course, the Cauchy-Riemann equations.

(iii) The problem considered in Section 3 deals with a generic hyperbolic (or parabolic) 2 x 2 system, and aI1 such systems can be written in the form considered in Section 3. Indeed, by an invertible transformation, any such system can be transformed into

u x + v y = , u y + v x = 7 ( o r v y = ? ) .

This standard wave system (or heat system) can then be brought back by an invertible transformation to the problem studied in Section 3.

372 B. DACOROGNA & P. MARCELL1N1

We are now in a posi t ion to state an existence theorem which includes the three cases. Let f~ c IR ~ be a bounded open set. Let

(u(x,y)) , PcP = (Pt ~ ) = ( a i u ~ + a 2 u , + b i v ~ + b 2 v ~ q~(x, y) = \v(x, y)} P~ \a~u~ + a~u, + Dtv~ + ?~v,)'

where al, b~, ai, b~ e IR, i = 1, 2. Let

~Oo(X, y) = \vo(x, y)/= 4o

and assume that

where 40 = (~ ) i =< i,~ __< 2 e IR 2 • 2

P, cPo = a ,~ l ~ + a 2 ~ + b,~ 2 + b2~ 2 e(~,/~),

P~q'o = a ,~ l + , i~g + ; d ~ + b ~ =

where cr fl, y e 1R, ~ </~.

(4.6)

(4.7)

Theorem 4.1. ~"~1 ('1~"~2 = 0 and ~0 e~Oo + Wol'~(~; IR~), such that

p i q ) ( x , y ) = f ~ a.e. in ~ l , a.e. in f~2,

Under assumptions (4.7), there exist f~i, f~2 ~ f~, with (~i w ~ 2 = ~-~,

P2q~(x, y) = ~ a.e. in f~.

(4.8)

Remark. Note that the measures of ~ l and ~')2 a r e automatical ly given by the bounda ry conditions. Indeed, since cpo is linear and Pl~oo e(c~,fl), there exists a 2 e(0, 1) such that PicPo = 2e + (1 - 2)B: The compatibi l i ty condit ions imply then that meas s = 2 meas f~ and meas ~ = (1 - 2 ) meas ~.

Proof. The p roof depends strongly on the type of operators considered in (4.6). Hyperbolic and parabolic cases. The p roof is an immediate consequence of Lemma 3.5 and of the fact that any parabol ic or hyperbolic system can be brought , by an invertible t ransformat ion as in (4.5), to a system where b i = #Lii for a certain # e IR (cf. the remarks above and (4.4)). Elliptic case. The proof is divided into three steps. Step 1. We first perform a t ransformat ion as in (4.5) to reduce (4.7) to a Cauchy- Riemann system, i.e.,

p c p = ( u x + v r ) . \ b l y - - V x

The compatibi l i ty condi t ion that (po(x ,y )= \Vo(X, y)

~ elR z• should satisfy is then


By writ ing

2 x - 4~Y ,

g ( x , y ) = f i @ ~ [ v ( x , y ) + 41 - - 3

a y - 4 xj,

we can t rans form the p rob lem into the following: Find f~l, ~"~2 a s in the theorem and ~0 = (u, v) e W I ' ~ IR 2) such tha t

01 a.e. in ~1, u ~ + vy -- uy - Vx = 0 a.e. in f~,

a.e. in f~2,

u(x, y) = 2x, v(x, y) = 2y o n 3 f ~ (4.9)

where 2 - 4~ + 42 - c~ e(0,�89 -

Step 2. We then solve (4.9) when ~ is the unit disk. This is easily done as for example in KOHN • STRANG [-39] (see also HASHIN & SHTRIKMAN [34], TARTAR [64]):

~1 = {(x, Y)elR2; x 2 + y2 < 1 - 2 2 } ,

f~2 = f~ - f i l = {(x, y) elR2 : 1 - 22 < x 2 -~- 22 < 1},

( (0, 0) in f~l,

Step 3. To deal with a general set f~, we express f~ as a countable union of disks Dk and a set Do of measure zero, and we use Step 2 on each one of these disks (as in Step 2 of L e m m a 3.4.). [ ]

This const ruct ion in the elliptic case is not sufficient to deal with the minimizat ion p rob lem of KOHN & STRAN6; we need in fact a s t ronger version of Step 2, as done in the following theorem.

Theorem 4.2. Let f) c 1R 2 be a bounded open set. Then there exists f~l, ~'~2 C ~'), with ~'~1 ('7 ~'~2 ~ ~) and (~1 w ff~2 = ff~ and u, v E W a' oo (f~) such that

u~ + vy = 1 a.e. in f~2, uy - vx = 0 a.e. in ~'~2, ( 4 . 1 1 )

u = v - = 0 i n f ~ l , u(x, y) = 2x, v(x, y) = #y o n ~ f ~

where 2, # > 0 and 0 < 2 + # < 1. In addition,

det(Du, Dr) = uxvy -- uyvx > 0 a.e. in f~.


Remarks. (i) When 2 = #, (4.10) gives a solut ion of (4.11), as was observed by HASHIN 8r SHTRIKMAN [34], MURAT & TARTAR (see TARTAR [64], KOHN & STRANO [39]).

(ii) When 2 =t = #, KoI4N & STRANO noticed tha t the me thod of confocal ellipses of MURAT & TARTAR could be applied. We prove this observa t ion below.

Oii) N o t e tha t T h e o r e m 4.2 is strictly s t ronger than T h e o r e m 4.1 for Cauchy- R iemann systems since, in (4.11), we require tha t u - v - 0 in ~)l, while in The- o rem 4.1 we require only tha t u~ + vy = 0 in f~,. Howeve r T h e o r e m 4.2 requires tha t the b o u n d a r y d a t u m is in d iagonal form; this restriction in T h e o r e m 4.1 is unnecessary.

(iv) Observe that in the theorem bo th 2 and # have to be non-zero. Indeed, if 2 = 0 for example, then using the fact that u should be ha rmon ic in ~'~2 (cU. (4.11)) and 0 on 8~2 we could deduce tha t u = 0 in f~2. Therefore in ~2 we should have vx = 0, vy = 1, v = 0 on 8f~l and # y on Of~, which is impossible.

Proof. We follow the me thod developed by MURAT 8r TARTAR (see [64]) for ha rmon ic functions. We divide the p roo f into four steps. Observe first that it is enough to do the cons t ruc t ion in the case where ~ is an ellipse. I f this is not the case, we can express f~ as a countable union of ellipses Ek and a set Eo of measure zero and then const ruct u and v in each of these ellipses (as in Step 2 of L e m m a 3.4). We also assume th roughou t that 2 4 = #; otherwise, see (4.10). Similarly we can assume that 2 + # < 1, since otherwise f~a = ~b, u - 2x, v = # y trivially give the result. Step 1. Since 2, # > 0, 2 4= # and 2 + # < 1, we can define

1 - 2 1 - ( 2 + # ) 1 - # 1- (2+ff ) c~ - - - /~ - - - (4.12)

2 2 - - (2 + #) ' # 2 - (2 + #)"

Direct compu ta t ions lead to

F T Z ) = 1 - (,t + #), (4.13)

2 # (1-~-fl~1/2 -[- (1 _~_ @)1/2 = 0 , (4.14)

D (4.15)

Step 2. Assume tha t f~ is given by the ellipse

X2 ~ = (x, y) fflR2:1 + (x

y2 } + < 1 . ( 4 . 1 6 )


Then define

X 2 y2 } ~1 = (x, y) e ]R 2 : - - + < 1 ~'~2 = ~'~ -- ~1 f l '

u(x, y) = f ( p ) x in ~2, v(x, y) = g(P)Y in f12

where for p e [0, 1] we let

(0'1 f ( P ) = L t , ~ ) - ' g(P) = - L t ~ )

2 C=(l + fi)l/2

and p is defined implicitly by

(4.17)

(4.18)

_ (_ Vq

\tU J' (4.19)

Py y p + e

By combining (4.22) and (4.23) we also obtain that

xpx (p + ~)3/2(p + ~)~/~

ypy 2 q (p + ~)1/2(p + fl)3/2 = (p + ~)1/2(p + fl)~/2. (4.25)

(4.24)

which imply that

Px x p + f i

X2 y2 + - 1 (4.20)

p + ~ p + / ~

(i.e., ~2 corresponds to p ~ (0, 1)). It remains to be checked that ~1, ~z, u and v have all the asserted properties. Step 3. We start with some preliminary computat ions. We h a v e

c ~ - / ~ c ~ - / ~ f t ( P ) = 2 (p + ~)3/2(p + fl)l/2' at(P) = 2 (p ..~ fl)3/2(p _~_ ~)1/2" (4.21)

Op ~p We next compute Px = ~xx and py = ~y. Using (4.20) we have

xZ(p + fl) + yZ(p + ~) = (p + e)(p + fl). (4.22)

Differentiating with respect to x and then y we get

2x(p + fl) 2y(p + ~) Px = 2p + ~ + fl - x 2 -- y2, PY = 2p + ~ + fl -- x 2 - y2, (4.23)

376 B. DACOROGNA r P. MARCELLINI

Step 4. We are now in a posi t ion to conclude that: (i) If (x ,y)~0f2a , then p = 0 and f rom (4.19) we get f ( 0 ) = 9 ( 0 ) = 0, i.e.

u = v = 0 on 0 f~ . (ii) I f (x, y) e Of 2, then p = 1 and by definit ion of c we get f (1 ) = 2 and thus

u = 2x on ~fl. Similarly we find using (4.14) in (4.19) tha t

o ( 1 ) = - c L - =

and hence v = #y on 8f). (iii) We now prove tha t uy - v~ = 0 in f~2; indeed,

uy - vx = x f ' ( p ) p y - Yg'(P)Px = 0

since (4.21) and (4.24) hold. (iv) We then show tha t u~ + vy - 1 in ~2- We have

u~ + vy = / ( p ) + x f ' ( p ) p ~ + g(P) + Yg'(P)Py

Using (4.25) we get

=C{(p"l-fl~l/2 (p-'l-O~k~l/2

q-2(Cr ( p q _ ~ ) 3 / Z ( p + f l ) l / 2 q-(p+c@/2(pq_fl)3/2 '

+ v , = c i c r - 1

where we have used (4.15) and the definition of c (cf. (4.19)) in the last identity. (v) We finally establish tha t

We have

det(Du, Dr) = uxvy - uyv~ > 0 a.e. in f~a. (4.26)

uxv, -- uyvx = f (P)g(P) + YP, f (P)9'(P) + xpx f ' (P )g (P) . (4.27)

Observe first that (~ - ~)c > 0, by (4.19). Therefore, using (4.21), we deduce tha t f ' (P) , g'(P) >= 0 and hence by (4.19), tha t f ( p ) , 9(P) >= 0 for every p ~ [0, 1]. There- fore if we can show that

xp~, ypy >= 0 for every p s [0, 1], (4.28)

we obta in f rom (4.27) the claimed result. In fact, (4.28) is an immedia te consequence of (4.22) and (4.23). Indeed f rom

(4.22) we have tha t

P q- flX2 P + ~ y 2 = p + ~ _ x 2, = p + / ~ _ y 2 , p + / ~ p + ~


which implies that

2p "{- O~ -[- f i - - X 2 _ y2 ~ 0. (4.29)

Combining (4.23) and (4.29) we obtain (4.28) and thus the proof is complete. []

5. Necessary conditions for existence

We now turn our attention to necessary conditions. In Theorem 5.1 we give a general condition, which is far from the sufficient one. It has however several advantages: (i) From the dimensional point of view the two conditions match, since both are expressed in terms of linearity, one of f * * the other of Q f on a manifold of dimension n N - (N - 1). (ii) In the scalar case N = 1, they turn out to be the same (cf. Corollary 5.2). (iii) In the case of the norm f(4) = g(14l), the necessary condition states that 40 is a matrix of rank at least two (cf. Corollary 5.3). In the particular case where g is the function of KOHN & STRAN6 (cf. Section 6) it turns out that this condition is also sufficient. (iv) In Section 7, we will show that for functions of the determinant and its generalizations the necessary and sufficient conditions are also the same. (v) In Section 8, we apply the necessary conditions to the study o f the Saint Venant-Kirchhoff energy function.

Recall that the problem under consideration is

(P) inf{F(u) = n~f (Du(x) )dx:u~u~ Wo~'~(f~; IRN)}, (5.1)

where f~ c IR ", Uo(X) = 4ox with 4o elR "N and f : l R "N ~ IR is lower semicontinuous. Furthermore f * * and Q f denote respectively the convex and the quasiconvex envelope of f, and

K = {4 elR"N:Qf(4) <f(4)}-

Before expressing our main result we need the following definition.

Definition. A convex function h:IRnN~ IR is said to be strictly convex at 4o = (4;)1 _<~___N EIRnN in at least N directions if there exists 2 = (2~)1 _<~_~N ~ IR'N such that

2 ~=#0 and (2~;4 ~ - 4 ; ) ~ . = 0 Vc~=l, 2 . . . . . N, (5.2)

whenever 4 = (~)1 _< ~ _< N satisfy the condition


Remarks. (i) Obviously if h is (globally) strictly convex, then it is strictly convex in at least N. directions at any 4o, since (5.3) holds then only if 4 = 4o. (ii) This definition means that there exist at least N independent directions, (2 ~, 0 , . . . , 0), (0, 22, 0 . . . . . 0) . . . . . ( 0 , . . . , 0, 2 N) orthogonal (cf. (5.2)) to the (pos- sibly existing) directions 4 - 4o where h is affine. In other words, h is not affine in at least N independent directions. (iii) In the scalar case (N = 1), the definition simply means that h is not affine in a neighbourhood of 40, i:e. there is at least one direction of strict convexity.

Theorem 5.1. Let 4o ~ IR"N be such that

(i) f**(~o) = Qf(4o) <f(4o), (ii) f ** is strictly convex at 4o in at least N directions; then (P) has no solution.

Remark. We already pointed out the interesting parts of this theorem; its weakness is that it is expressed in terms of f** and not of Q f

Before proving the theorem we immediately give two applications; a third one will be given in Section 7.

Corollary 5.2 (The scalar case). Let N = 1, 4o ~ 1R" be such that

(i) f**(40) <f(Co), (ii) f ** is strictly convex at 4o in at least one direction; then (P) has no solution.

Remark. Observe that Corollary 5.2, coupled with Theorem 3.1, implies that if f**(4o) <f(4o), then a necessary and sufficient condition for the existence of minimizers is that f * * be affine in all directions in a neighbourhood of 4o (by assuming, for the existence part, that K is bounded in some directions, as in Theorem 3.1). This can be compared with CELLINA 1-12, 13] and FRIESECKE [31].

We now turn our attention to the case n > 2, N > 1 and

f(4) = g(14]). (5.4)

We extend g to ( - o e , 0) as an even function and we define g** as the convex envelope of g in IR. We have the following result.

Corollary 5.3. Let 4o e ]R ~N be such thar

(i) g**(14ol) < g(14o[), (ii) g** is strictly increasing at 14ol. I f n > 2 and rank {40} < 1, then (P) has no solution.

Remarks. (i) In the next section we show that if

f l + t 2 if t + 0 , g(t)

0 if t = 0,


then the condit ion rank 4o =t = 1 turns out to (ii) If either n = 1 or N -- 1, one always has

g**([~l) = f * * ( r

while, in the genuinely vectorial case n, N >

g**(141) = f * * ( ~ )

(iii) In the one-dimensional case, i.e., when

be also sufficient for existence.

= Qf(~),

2, we only have in general that

< Qf(~). (5.5)

n = 1, then problem (P) always has a solution (see, for example, Theorem 2.6 of Chapter 5 of [24]). (iv) In the scalar case, i.e., when N = 1, if n > 2, problem (P), under the assumptions (i), (ii), of the previous corollary, never has a solution, as was pointed out by MARCELLINI [43, 44].

P roo f of Corollary 5.3. We divide the p roof into three steps. Step 1. First observe that the case 4o = 0 is excluded by our assumptions. Indeed if 4o = 0, then either g**(0) = g(0) or g**(0) < g(0). This second case implies that g** is constant on a ne ighbourhood of 4o and this is excluded by (ii). Step 2. We now show that if rank 4o = 1, then

g**(lr = f * * ( l ~ o l ) = Qf(~o) (5.6)

(while, for 4o with rank 40 > 2, one has in general Qf(4o) >f**(~o)) . We then denote by R f the rank-one convex envelope of f (cf. DACORO~NA [24] for more details concerning rank-one convex envelopes). As is well known one has

f * * ( 4 ) < Qf(~) < R f(4), (5.7)

R f(4) < inf{2f(~l ) + (1 - 2)f(~2): (5.8)

2 ~ [0, 1], rank{~: - 42} =< 1, 241 + (1 - 2)42 = 4}.

N o w let e > 0 be fixed; we then know, by Carath6odory 's theorem, that there exist 2 ~ (0, 1), tl, t2 > 0 such that

2 e ( t l ) + (1 -- 2)g(t2) ~ g**(1401) + Z, 2 t l + (1 -- /~)t 2 = 1401. (5.9)

t l t2 We then set 41 = 1~o1 40, 42 = ~ 40, which imply that

241 + (1 - 2)42 = 40, rank(~_l - 42) = rank4o = 1. (5.10)

Combining (5.8) and (5.9) we then get that

Rf(~o) < 2f(~1) + (1 -- 2)f(ff2) = 2g(1411) + (1 -- 2)g(1421) _-< g**(]~ol) + e.

Since ~ is arbitrary, we deduce from (5.7) that

g**(14o[) = f * * ( ~ o ) = Qf(~o) = Rf(~o). (5.11)


Step 3. We are now in a position to conclude the proof. Note that assumption (i) of the corollary and (5.11) imply hypothesis (i) of Theorem 5.1. To get the result we therefore only need to show that (ii) of the corollary implies assumption (ii) of Theorem 5.1. Since g** is strictly increasing at I~oL, we deduce that if

f**(~)+f**(~~176 , (5.12)

then Ir +[~ol = l f f + ~ o l , and hence r is parallel to 40. So choosing = (2~)1 _< ~ <_ N ~ IR"N, where U is orthogonal to ~ (this is possible, of course, only if

n = 2), we obtain immediately that f is strictly convex (at ~,o) in at least N directions. This concludes the proof of Corollary 5.3. []

Proof of Theorem 5.1. We divide the proof into two steps. Step 1. Using the relaxation theorem (cf. DACOROGNA [243) we have, as in Lemma 3.3, that

= [f~lQf(r -- If~lf**(~o) = m,

where we have used the hypothesis that Qf(r =f**(~o). We now proceed by contradiction and assume that (P) has a minimizer u ~Uo + W0~'~~ IRN). We therefore have

f(Du(x))dx = If~lf**(~o) = m. (5.13) a

We then follow the method of MARCELLINI [44]. By applying the Jensen inequality to f**, we obtain

m = Sf(Du(x))dx > Sf**(Du(x))dx > If~lf**(~o) = m. (5.14) f~ f~

Combining (5.13) and (5.14) and again using the convexity of f**, we have

l Il~[f **(~o) + f f**(Du(x))dx; m=-~ f~

We therefore obtain (recalling that Duo = 40)

f**(Du(x)) + f * * ( ' o ) = f * * ( D u ( x ) + ~o~ a.e. in fL (5.15) 2 /

Existence of Minimizers

Step 2. We now use assumpt ion (ii) of the theorem to deduce that

(2=; Du=(x) - {~) = 0 a.e. in f~

381

(5.16)

for every 1 _< e _< N. F r o m now on we reason componen t by componen t and so we fix e. Let x e f ~ be fixed. Let XoeSf~ be a point of intersection of the set {x + s2 ~ : s e IR} with Qf~, so that

x0 = x + t2 ~ (5.17)

for a certain t E IR. Therefore, since u e W 1' co (g2; IR N) (u -- u0 outside f~), by using (5.16) we get

U=(X)=H~(Xo)-- fdu~(x+sl~)ds o

t

0

t

= W(Xo) - f ( ~ ; 2~)ds 0

= u~o(Xo) - t C ~ ; ,1 ~5 = (Co; X o ) - t C ~ ; ;.~5

= C ; ; x0 - t,Z ~5 = C ~ ; x ) = uS(x),

where we have used the fact that uo(x) = {ox, u = uo on BY* and (5.17). Since x e ~ is arbitrary, we have obtained that u = uo in ~ and thus

[. f (Du)dx = ~. f (Duo)dx = f ( 4 o ) lf~l, f~ f~

which, combined with (5.13), contradicts the assumption that f**(~o) <f ({o) . This concludes the p roof of Theorem 5.1. [ ]

6. Application to optimal design

Recall that n = 2 and N > 2. For 4 ~]R2N, let

f(~x) = g([~[) = {1 01412

It was shown by KOHN & STRANG [39] that

~1 + 147 Q f (4 ) = [2(14[2 + 2[adj2~[),/2 _ 2[adj24[

if 4 4 0 , if 4 = 0 .

if 1412 + 21adj24[ _-> 1, if 14[ 2 + 21adj24] =< 1.


where adj2 ~ s tands for the N ( N - 1)/2 vector composed of all the 2 x 2 minors of the mat r ix 4 eIR2N; so that, in part icular , if N = 2, then adj2~ = det~.

The p rob l em under considera t ion is

+ t where ~ is a bounded open set o f l R 2 and Uo(x ,y)= ~o(X) , where 40~lR 2N.

kY/

Theorem 6.1. A necessary and sufficient condition for (P) to have a solution is that at least one of the following conditions hold:

(i) 4o = 0, (ii) [40[ 2 + 2[adjz 4o[ > 1, (iii) r ank 4o = 2.

Remarks. (i) No te that, if either (i) or (ii) holds, then Qf(4o) = f(4o); thus these two cases co r respond to the trivial situation. T h a t r ank 4o = 2 is a necessary condi t ion follows f rom the general results of Section 5 concerning functions f depending on the n o r m (see in par t icular Corol la ry 5.3). So the only thing which remains to be p roved is tha t r ank 4o = 2 is a sufficient condi t ion for the existence of a minimizer of (P).

(ii) W h e n N = 2, KOHN ~r STRANG E39, L e m m a 8.2 and Example 8.31 suggested the p roo f of the existence of minimizers and complete ly p roved it when 40 = 2I.

Proof. We divide the p roo f into three steps. Step 1. We start with some algebraic preliminaries. Recall first some notat ions. F o r

a = i , b = a N

we let

N N (a ; b ) = Y' aib ~, lal 2 = ~ (#)2, ladje(a, b)l a =

i = 1 /=1 l<=j<k<=N

Let us first p rove that

( ( a ; b ) ) 2 + ]adj2(a, b)] 2 --- ]a] 2 Ib[ 2. (6.1)

T o establish (6.1) we proceed by induction. I f N = 2, we have

( (a ; b)) 2 + ladja(a, b)l 2 = (alb 1 + a2b2) 2 + (a2b 1 - alb2) 2 = ]a[ 2 ]bl 2.

Assume now tha t (6.1) has been establ ished for N -- 1, i.e.,

aJb; + ~, ( a J b k - a k b J ) z = 2 (aJ) 2 Z (bY) 2" (6.2) \ j = l 1 <j<k<=N--1 j = l j = l

(aJb k _ akbJ) 2.


We now wish to show that (6.1) holds. We have

((a; b)) 2 + ladja(a, b)l 2 = aJb j + Y', j = l 1 < j < k < N

(aJb k _ a k b J ) 2

N~I . . )2 = aJb J

\ j = l

N - 1 +(aNbN) 2 + 2 a N b N ~ aJb j

j = l

§ N-1

~' (aJb k - akbJ) 2 + ~ (aJb N - aNbJ) 2 1 < j < k < N - - 1 j = l

N-1 N-1 N-1 N-1 = 2 (aJ) 2 Y', (b J) 2 +(aNbN) 2 +(bN) 2 ~ (a J) 2 :t-(aN) 2 ~ (b J) 2,

j = l j = l j = l j = l

where we have used (6.2) in the last identity. Then we immediately get (6.1). Then let a, b ~IR N, (p, 1//e C1(IR2) and let us define u e C1(IR2; IR N) in the

following way

Du = aD(p + bDO =

u~ \aNgx + bNOx aN(p, + bSOr/

We assert that

IOul 2 -4- 2[adj2 Du[ = la] 2 IO(pl 2 + Ibl 2 IDOI 2 + 2(a ; b)aN(D(p; Oi//)i.2

+ 2ladj2(a, b)[ Idet (D(p, D0)I. (6.3)

Furthermore, if det (Do, D0) = (px0y - (Py0~ > 0, then

[Dul2+21adj2Dul = [la[(p~ + ~ 0~ + ladj2(a, b)l la[ 1//'] 2

+[]al(p, ladj2(a,b)l[al 0 ~ + ~ 0 Y ] 2"

(6.4)

The factorisation into a sum of two squares will be important to reduce our problem to Theorem 4.2. One should note that (6.4) is not the only possible decomposition into a sum of two squares; we more generally have

]Dul 2 + 21adj2Du[ = [la](pxcosc~ + laJq)ysinc~ + JbJ0xcosB + [bJ0ysinfl] 2

+ [lal(pxsin c~ - l a l (pycos e + Ibl0xsinfl - I b l 0 r c o s f l ] 2

(6.5)

384

where

B. DACOROGNA & P. MARCELLIN1

[a] Ib[ cos(fl - c0 = (a ; b) or la] Ib[ sin(fl - ~) = ladj2(a, b)l

(the second identity is permitted because of (6.1)). In (6.4) we have chosen ~ = 0. We now establish (6.3) and (6.4).

IDul 2 = [aDq~ + bDO] a = lal z IDq~J 2 + 2(aDq);bDO)~N + [b] 2 IO012

= lal 2 [O~ol 2 + Ibl e IO012 + 2(a; b)~N(Ocp; OO)r~2. (6.6)

Observe then that

j k i k (aib k a k b J ) ( p ~ y q),,O~), U x l A y - - 1 A y U x ~ - -

so that

]adj2Du] 2 = ]adjz(a, b)12(det (Dcp; D0)) z. (6.7)

Combining (6.6) and (6.7) we obtain (6.3). To establish (6.4) we develop its right- hand side:

V( (a ; b)~ 2 ( ]ad j2 (a ,b ) l ) z ] (Oz+02) [alZ(q~ + ~~ + I \ lal J + \ lal

+ 2(a; b)(q)~0x + (Py0r) + 2[adj2(a, b)l(cP~0r - ~0r0x).

By using (6.1) and (6.3) we get (6.4). This concludes Step 1. Before proceeding further one should note that if

then (6.4) is read as

IDul 2 + 2ldetDul = [~0~ + 0y] 2 -1- [-q)y - - 0 x ] 2, (6.9)

(a; b) [adj2(a, b)l lal(p~ + ~ h T - 0~ + [ a ~ 0y = 1 a.e. in f~2,

(a ; b) ladjz(a, b)l O~ + Oy = 0 a.e. i n ~2, ]alcp" ]a[ ~ I -

since Ial = Ibl = 1, (a ; b) = 0 and adj2(a, b) = 1. Step 2. We now solve a Hamil ton-Jacobi system associated with (6.4). So we let a, b e lR N - {0}, and a not parallel to b, i.e., [adj2(a, b)[ + 0. We let f2 c IR z be a bounded open set and 2, # > 0 such that 2 + / z < 1. We then assert that we can find f21, ~'~2 C ~ with f~l c~ f~2 = O, f i l u f i2 = f i , and cp, 0 e W I ' 0o (~-~) such that


~ o = 0 - - 0 in f~l,

2 (a ; b ) ~o(x, y) = ~ x - # [al [adjz(a, b)l y on 8n, (6.10)

lal 0(x ' Y) = #]adj2(a, b)l y on 8f~,

det(D~o, DO) = ~ox0y -- (PyOx > 0 a.e. in f~.

As before, observe that, if N = 2 and a, b are given as in (6.8), then (6.10) is exactly the system that we have solved in Theo rem 4.2. No te also that the equat ions in (6.10) are elliptic, since 6 = 41adjz(a, b)l 2 > 0. By setting

~1 r(x, y), 0(x, y) z(x, y), (a; b ) lal

(p(x, y) = or(x, y) -- lal ladj2(a, b)[ - ladj2(a, b)] (6.11)

we have t ransformed (6.10) into

o - ~ + ~ r = l a . e , i n ~ 2 , % - ~ = 0 a . e . i n ~ 2 , a = ~ = 0 i n ~ l ,

a(x, y) = 2x, z(x, y) = #y on 8~ (6.12)

det(D~, Dz) = a~zy - % ~ __> 0 a.e. in ~ ,

which was explicitly solved in Theo rem 4.2. Step 3. We are now in a posi t ion to conclude the p roo f of the theorem. Recall tha t the only thing to be p roved is that, if r ank 40 = 2 and 14ol 2 + 21adj2 4ol _-< 1, then (P) admits a solution. Let

4o = (41 , 42) = " 4!2~ / e ~N x IR N = IR 2N. (6.13)

Since rank 4o = 2 and the in tegrand f is invar iant under o r thogona l t ransforma- tions, there is no loss of generali ty if we assume tha t the two vectors 41, 42 ~ IR N are non-zero and or thogonal . So we m a y assume tha t

(41; ~2) = 0. (6.14)

Observe that in this case

14o[ e + 2[adj2~ol = I~112 + 14212 + 214111421 = (1411 + 1421) 2,

so we also assume tha t 1411 + 14z[ < 1. We then define u = (u=)l _<=_<N ~ Wl'~176 IR N) by

u = 41q) + 420, (6.15)


where (0 and O~Wt '~~ are solutions of (6.10) with a = ( 1 , b = ~ 2 , 2 = I(1], # = 1~21; i.e., in view of (6.14) we have

I~11~o~ + 1{2l~by = 1 a.e. in f12, ]{l lcPr - 1r = 0 a.e. in fZ2,

q9 = ~b -= 0 in f21,

~(x, y) = y on aO, ~Pxffr - cPy~bx ---- 0 a.e. in fl.

i.e., u(x, y) = ~o(~) on a~ .

~0 (x, y) = x,

Observe that o n 0~,

u ~ = ~ i x + ~ y ,

Using (6.3), (6.4) and (6.16) we obtain that

[DulZ+gladj2Du,={lo a.e. in f~2,

in ffl~;

(6.16)

therefore

(6.17)

Qf(Du) =f(Du) a.e. in f~. (6.18)

Fo r 4o -- ({~)1 __< ~ __< N.1 _< i ~ 2 we define

rF1 = r + r

~ = ~ A 2 + ~ M :

IglIA, + 1~2]M2 @[0, 13 .

[{11A2 -1~21M1 -- 0

A1M2 - A2M1 > 0

~]R2N : Ai, M i ~]R, . = fl =

Observe that, by (6.3) and (6.4), Qf is quasiaffine o n / ( , since

QfOl) = [~llA1 + I~2IM2 + 2ladj2~l(a~M2 - A2M1).

Since 4o ~ K (in this case A1 = M2 = 1, A2 = M~ = 0), we can apply L e m m a 3.3 to our problem and thus conclude the p roof of Theorem 6.1. [ ]

7. T h e c a s e o f t h e d e t e r m i n a n t

We now return to the first example of Section 2. We first recall some notat ions:

= �9 . ~ l R n • ~ IR "~.

We denote by ~ the vector in IR ~ - 1 composed of all the elements of ~ but one; for nota t ional convenience we assume that the missing one is ~,". We start with a representat ion of a quasiconvex envelope.


Proposition 7.1. Let g : IR "2-1 __. IR be convex, h : IR ~ IR be lower semi-continuous and bounded from below. I f

f (4 ) = 9(~') + h(det ~), (7.1)

then

P f (4) = Qf(~) = R f (4 ) = g(~) + h**(det 4). (7.2)

Remarks. (i) Recall (ef. DACOROGNA [24]) that P f and R f respectively denote the polyconvex and the rank-one convex envelope of f. We always have

P f (4) < Q f (4) < R f (4) .

The only proper ty of R f (already used in Section 5) that we need in the p roof is that

R f(4) < i n f { 2 f ( ~ ) + (1 - )~)f(42) : J( ~ [0, 1], 24~ + (1 -- ~)42 = 4,

rank(41 -- 42) < 1}. (7.3)

Similarly, the only fact about polyconvex functions that we use is that if

c?(~) = ~(~) + z(det4) V4 e lR "•

with ~ : IR" • ~ ]R, )~ : IR --. IR convex, then ~o is polyconvex. (ii) Propos i t ion 7.1 remains valid under weaker conditions. Fo r example, if we let 4o e IR "• be such that

h**(det 4o) < h(det 40)

and such that the component containing det4o where h # h** is given by (tl, t2) c IR, we know that in this interval we must have

h**(t) = m t + q for every t e It1, t2]

and for certain m, q e JR. We can assume that g : IR"2 ~ IR is convex and that there exists an t / e IR "2 with rank ~/= 1 such that

t ~ g(~o + tq) is affine for t ~ [tl , t2].

This happens, for example, if

g(~) = m[4[ + q ~'~ elR"• e [ t l , t2]

for some m, qe lR , where 4o~lR "• is such that r a n k 4 o = l . Then t ~ g(4o + t~o) --= m]4o](1 + t) + q is affine, in the interval of nonnegat ive real numbers [tl, t2], in the rank-one direction q = 40.

Of course this assumption also contains the hypothesis of the proposit ion. (iii) One could also prove the proposi t ion in a more general context, with the determinant replaced by any quasiaffine function, the p roof being identical (cf. also DACOROGNA [24]).

388 B. DACOROGNA & 1 ). MARCELLINI

Proofi Let F({) -~ g(~') + h**(det 4). Since g and h** are convex, F is po lyconvex and therefore

F(4) <= P f ( 4 ) <= Q f ( ~ ) <= R f ( 4 ) .

Hence, if we can show that

R f ( ~ ) < F(~), (7.4)

we immedia te ly have the result. We need to consider two cases. Case 1. Assume first tha t

/ 41 ... eL, t d e t [ : ' ~=i =0.

\4,'-' " - " (7.5)

Let e > 0; using the proper t ies of h** we find tha t there exist 2 e [0, 1], x, y e lR so that

h**(det 4) < 2h(x ) + (1 - 2)h(y) < h**(det 4) + e,

We then let

X =

~ . . 1 \ �9 ~, ,-~ ~

/ 4~-* .-1 U - l /

�9 �9 �9 4n - -1

2x + (1 -- 2)y = de t4 .

(7.6)

r IR,*X.

and let Y be defined similarly with a replaced by fi, ~ and fl being chosen so that

2e + (1 - 2)fl = ~,~,, d e t X = x, det Y = y, (7.7)

This is always possible, but before showing this, we do it when n = 2. If

with a =~ 0 (cf. (7.5)), we let ' a

X = x + bc , Y = y + bc C

, a ~ a

We obta in immedia te ly (7.7) since, by (7.6), 2x + (1 - 2)y = det ~ = ad - bc. We now repeat it for n > 2. We choose e such that

. . 4 71 t / d e t X = d e t " �9 a + - - ' + ( - 1 ) " 4 ] d e t | "

t n-- �9 n--i 41 1 "" r 4~ -1

~ X

. . . 4 ~ - 1 /


and similarly for Y, with e replaced by fl and x by y. Since 2e + (1 - 2)fl = ~ and 2x + (1 - 2)y = det 3, we have indeed obta ined (7.7).

Therefore, summar is ing (7.7) and the second equat ion of (7.6), we have found X, Y E I R "• so that

2 X + ( 1 - 2 ) Y = ~ , r a n k { X - Y } _ < _ l , J ? = P = ~ , d e t X = x , d e t Y = y ,

(7.8)

where we recall that ~ is the vector in IR ~2-1 composed of all the elements of ~ IR, x ~ but the last one ~ .

By (7.6), (7.8) we obta in

F(~) + e = g(~) + h**(det 4) + e > g(~) + 2h(x) + (1 - 2)h(y)

= 2[g(X) + h (de tX) ] + (1 - 2)[g(Y) + h(det Y)]

= 2 f ( X ) + (1 - 2 ) f (Y) .

Using the fact that r a n k { X - Y} < 1, by (7.3) we deduce tha t

F(~) + e > R f(~).

Since e is arbi t rary, we have indeed obta ined (7.4) and the propos i t ion is established, p rovided (7.5) holds. Case 2. We now assume that (7.5) does not hold. Then, by the cont inui ty of h**, for every ~ > 0 we find ~ e lR ~• satisfying (7.5) and

I t / - - ~] < e, h**(dett/) =< e + h**(det~).

Applying Case 1 to t /we get

e + 901) + h**(det ~) > g(t~) + h**(det t/) = Rf(rt).

Since R f is also cont inuous (cf. DACOROGNA [24]), we deduce the result by letting e tend to zero. This concludes the p roof of Propos i t ion 7.1. [ ]

We are now in a posi t ion to state the main theorem of this section. Let

4 = : EIR,•

f (4 ) = 9(~ ~ . . . . . 4 "-1) + h(det ~) (7.9)

where g : IR "(n- ~) ~ IR is convex and h : IR ~ IR is lower semicont inuous, with

lira ~ - = + oo. (7.10) b ---~ oO


Let us consider the variational problem

(P) inf {! f(Du(x))dx: u ~uo + Wo*'~176 lR")}

where uo(x) = 4ox, x e~.

Theorem 7.2. Under assumptions (7.9) and (7.10) problem (P) has a solution.

Remarks. (i) The case where g -= 0 was solved by MASCOLO • SCHIANCHI [-49] under more general boundary conditions. (ii) It is interesting to note that the theorem hides in fact a necessary and sufficient condition similar to that of Section 5. In view of the remarks following Proposit ion 7.1 one can prove Theorem 7.2 under weaker conditions on 9. Instead of (7.9), we can assume that, given 40, there exists a e {1, 2 , . . . , N} such that the function g : IR "~ ~ IR is convex and

4 ~ --, g(4o ~ . . . . , ~ - 1 , 4 ~, ~+~ . . . . ,40 N) (7.11)

is affine. The necessary condition then reads: If g is strictly convex at 4o in at least N = n directions, then (P) has no solution (we shall prove this necessary part of the remark after the proof of the theorem). In Theorem 7.2 the function 9 cannot be strictly convex in the 4 ~ direction for every e = 1, 2 , . . . , n. (iii) A similar result to that of Theorem 7.2 can be proved if h is a function of a quasiaffine function, as in CELLINA & ZAGATTI [15], instead of a function depending only on the determinant.

Proof of Theorem 7.2. We adopt the notations of Section 3. Since (7.10) holds, there exist tl, t2 ~]R, so that the connected component of K containing 4o satisfies

K={4zN'•162 (7.12)

We then choose

L o = 4 = EIR . . . . 4 ~=(4o) ~ , e = l , 2 . . . . . n - - 1 . (7.13) n

We now need to express the fact that the set where the gradient of the solution takes its values is bounded in certain directions. For this we choose (cf. Theorem 3.1) I = 1, b = (adj , - 1 40)" ~IR", so that

(b; 4~}~n = ((adj._ 1 40)"; 4~} = det 4o

and, more generally,

(b; ~"}~, = ((adj,_ 140)"; 4"} = det ~, (7.14)


for every ~ = (~)1 _<~_<n such that ~ = (4o) ~ for every e = 1, 2 . . . . . n - 1; i.e., for every 4 e Lo.

Let us define Ko = K c~ L0 (cf. Section 3) by

K o = 4 = n

Lo c IR" x,: 4~ = (4o)~, c~ = 1, 2 . . . . . n -- 1 and det 4 ~ (tl, t2) I"

d

(7.15)

Since (2f is (quasi)affine on Kc~Lo (because n - 1 components of = (~)1 _< ~ _<, are fixed), we deduce from Theorem 3.1 that (P) has a solution, and

this concludes the p roof of Theorem 7.2. [ ]

P roo f of (ii) of the Remark. We discuss here the necessity par t of the condition. We have

f (~) = g(~) + h(det 4) (7.16)

with g : IR "2 ~ IR convex and g strictly convex at 4o in at least n = N directions (cf. the Definition in Section 5). We wish to show that in this case (P) has no solution. By (7.11), there exist m, q e lR so that

By defining

h * * ( t ) = m t + q f o r t e [ h , t2].

f l (4) = f ( ~ ) - (mdet 4 + q), (7.17)

we consider the variat ional problem

(P~) inf {F~(u) = ! fl(DU(x))dx:u ~uo + W~'~ ;

we find that any minimizer of (P) is a minimizer of (Pz) and conversely. So we show that i fg is strictly convex in at least n directions, then (P~) has no solution. Fo r this purpose we define

f2(4) = g(4) + h2(det 4) (7.18)

where

~h(t) - (mr + q) if t ~ [tl , t2] ,

h2(t) = {h**(t) - (mr + q) if t~) It1, t2].

Correspondingly we consider the variat ional problem

(P2) inf{F2(u) = a 5 f 2 ( D u ( x ) ) d x ' u e u ~ W~ IR")}"


Since fl >fe we have F1 > Fe. Moreover , by Propos i t ion 7.1,

inf{F, (u)} = inf{F2(u) } = Qf2(4o) lg~l = g(4o)[Ol; (7.19)

therefore (P1) lacks a solut ion if (P2) lacks a solution. Since g is convex and f2 > 9, we deduce tha t f * * > 9. Recalling that, by

Propos i t ion 7.1 Qf2({o) = g({o), we obta in

f**( r = Qfz(r = 9(~0), (7.20)

and in fact f * * ( r = g(r for every ~ such that det 4 ~ [q , t2]. F u r t h e r m o r e 9 is strictly convex at 40 in at least n directions, so we can apply

T h e o r e m 5.1 to deduce tha t (P2) has no solut ion and therefore (Pl) and (P) have no solut ion either. Hence the result. [ ]

8. The Saint Venant-Kirchhoff materials

We now return to Example 4 of Section 2. Here n = N. The Saint Venant- Ki rchhof f s tored-energy funct ion is given by

E Ev f ( 4 ) - - - 1 ~ 4 - - 112 + (1412 -- n) 2

8(1 + v) 8(1 + v)(1 - 2v)

where ~ ~ IR "• i f denotes the t ranspose of 4, I the identi ty matr ix , E the Young modu lus and v e (0, �89 the Poisson rat io (for m o r e details cf. CIARLET [20], LE DRET & RAOULT 1-40, 41], TRUESDELL & NOLL [65]).

The funct ion f is not quasiconvex. Recently LE DRET & RAOULT [40, 41] have c o m p u t e d the quas iconvex envelope Qf of f when n = 2 and n = 3. I t turns out tha t

Qf(~) = f * * ( ~ ) .

We deal here with the case n = 2. Since f rom the point of view of minimizat ion, E

the cons tan t - - is irrelevant, we work with the following function 8(1 + v)

f ( r = 1~r - 112 + ~ ( 1 4 l 2 - 2) 2 (8.1)

where this t ime 4 ff]R2X2- In terms of the singular values vl and /)2 (ordered such tha t vl </)2) of the mat r ix ~ (i.e., the eigenvalues of ( ~ ) l / z ) , f takes the form

V 2 f (4 ) = (V~ -- 1) 2 + ( v2 -- 1) 2 + ~ ( v * -- 1 + v22 -- 1) 2 . (8.2)


where

LE DRET • RAOULT have shown that

1 = [ ~ - 112+ Q f(4) =f**(4) ]~- v

1 +

(1 - - v ) ( 1 - - 2v) [ ( 1 - - v ) ( v ~ - - 1) + v ( v ~ - - 1)]~ ( 8 . 3 )

0 2 if X >= 0, Ix] 2 = if x < 0.

Remarks. (i) In our framework the expression (8.3) is not very convenient, since it is expressed in terms of the singular values. However, one should keep in mind that for 4 ~l Rz• we have

2 / ) 1 ( 4 ) = ( [ 4 1 2 + 2]det 4]) 1/2 - - ( ] 4 ] 2 - - 2]det 41) 1/2, (8.4)

2V2(4) = (14l 2 + 2]det 41) 1/2 + (]41 z - 2ldet 41)1/2.

In terms of the notations introduced by ALIBERT & DACOROGNA [1], we can rewrite

~ / ~ ( 4 ) = 114+1- lu l l , 4 5 ~ 2 ( 0 = 14+1 + 14-1 (8.5)

where given 4 = (4~)1 _<i,~__<2, 4 + and 4- are defined by

24+ = - (42 ' -42) ~:I+ 2{2 ' 24 -= \{2'+42 - ( {~ -42 ) , ] "

(ii) In the case n = 3, a simple expression for the singular values in terms of [41, ladj2 4[ and ]det 4[, like (8.4) or (8.5), is not known. Only this obstacle does not allow us to reproduce the following analysis for the case n = 3. (iii) For 4 e]R2X2 and 0 < v1(4) < vz(4) letting

D~ = {4 elRZ• 1 and (1 - v ) v 2 + vv 2 < 1}, (8.6)

D2 = {4 EIR2• > 1 and (1 - v)v~ + vvzz < 1},

we can rewrite (8.3) as

I f(41) if 4r

(v~-- 1) 2 if ~eD2,

10 ]-Z~- v if 4eD, .

(8.7) Q f(4) =f**(4) =

The minimization problem under consideration is

where f~elR 2, Uo(X, y ) = 4o( xx] with 4o e 1R2• We then have kY /

394 B. DACOROGNA 8z P. MARCELLINI

Theorem 8.1. Let det 4o > 0. Problem (P) has a solution u, with det Du > 0 a.e., if and only if

(1 -- v)v~(4o) + vv~({o) > 1. (8.8)

Remarks. (i) The Saint Venant-Kirchhoff energy function (8.1) is a nonlinear approximation in a neighbourhood of the identity of more general stored-energy functions (see CIARLET [20]). Thus in the applications to nonlinear elasticity (see [20]), this energy function is expected to be useful in a range of"small strains" (e.g., with Du close to the identity). Nevertheless, our theorem shows that there are

b ~ 1 7 6 1 7 6 1 7 6 1 7 6 1 7 6 l - e 0 ~ ) w i t h

small e > 0) so that (P) has no solution. (ii) CIARLZT & G~VMONAT [21] (see also Theorem 4.10-2 in CIARLET [20]) have proved that it is possible to construct a coercive polyconvex energy function which approximates the Saint Venant-Kirchhoff function in a neighbourhood of the identity. Their approximated energy has therefore a minimizer for any boundary slope 4o with det 40 > 0. Our Theorem 8.1 (and the previous remark (i)) shows that their existence result is not preserved when passing to the limit. (iii) The condition that detDu > 0 a.e. in Theorem 8.1 is motivated by nonlinear elasticity, since the analytic expression of f neither prevents det Du from approach- ing 0, nor even becoming negative. The constraint det Du > 0 is used only in Step 2 of the proof, but not in the other step. (iv) The proof of the theorem useg a recent characterization by LE DRET &; RAOULT [40, 41] of the quasiconvex envelope Q f In fact, (8.8) corresponds to the case where

Qf({o) =f({o). Proof of Theorem 8.1. ( ~ ) Note that assumption (8.8) implies that v2 > 1 and that, combined with (8.3), gives

Qf(~o) =f**(~o)

1 1 = - (v~ - 1) 2 + 1 -- v (1 -- v)(1 -- 2v)

[ (1 - v)(v~ - 1 ) + v(v~ - 1)] 2

I v2 1 1 --v 2 1 (V~ - - 1) 2 1 + + 1) 2 = 1 ~ - 7 ~ ~ ~ ( V l -

2~

1 - v 2v vZ 1)(v2 2 1) - i _ - 2 ~ [ M - 1)2 + (v~- 1)23 + T c G ( 1 - -

V 2 = ( ~ - 1)~ + ( '~ - 1)~ + t - - Y 5 7 v [ M - 1) + (v~ - 1)] ~ =f(~o).

Therefore the Problem (P) has u = Uo as a minimizer.


( ~ ) We now assume that (8.8) does not hold. Observe that, as in the previous computation, this implies that Qf(4o) <f(4o). We have to show that (P) has no solution u with det Du > 0 a.e. Because of the invariance under rotations of the function f, there is no loss of generality if we assume that

with d > a > 0. Since (8.8) does not hold, 40 is either in D1 or D2 (cf. (8.6), (8.7)). We analyze

these two cases separately. Step 1. Assume that 4o e D2. The non-existence of solutions would then follow from our general Theorem 5.1. So we need to show that f * * is strictly convex at 4o in at least two directions. For this we consider 4 E D2 (recall that D2 is open) such that

Consider the function g : IR ---> IR defined by

1 g(x) = ~ (x 2 - 1) 2

Observe that g is strictly convex when x > 1. In D2 we can write

f ** (4) = g (v2 (4)). (8.11)

Since v2(4) > 1 in Dz, we deduce from (8.10), (8.11) and the strict convexity of g that

( 4 + 4o~ _ v~(4) + v2(4o) v2t~il 2

Using (8;5) we obtain that

/ 4+ _ + 4 ~ + 4- _+ 4~ = ILT+I + IU I +,4o,+l~ol (8.12) 2 2 2

By the convexity of the norm we deduce that

14 + +4o1=14+1+1431 , IU + 4 o 1 = 1 4 - 1 + 1 4 7 1 . (8.13)

From (8.13) we obtain that there exist t +, t - > 0 so that

= t 40, ~- = t ~o. (8.14)


(Note that by hypothesis nei ther ~ ~- = 0 nor ~ o = 0.) F r o m the definitions of ~ + and 4 - in (8.6) we get

4 - ~o = (t + - 1)4o + ( t - - 1)~o.

Since 4o satisfies (8.9), we deduce that

Therefore, choosing

f rom (8.15) and (8.16) we obta in tha t

< U ; ~ - ~ ) = 0 f o r c ~ = l , 2. (8.17)

Summar i s ing our computa t ions , we have just p roved that if ~ satisfies (8.10), then necessarily (8.17) holds. This is exactly saying, in our terminology, that f*-* is strictly convex at 4o in at least two directions. Applying T h e o r e m 5.1 we get that, if ~o ~ 1)z, then (P) has no solution. Step 2. We now assume tha t ~o e1)1; in this case,

Qf(~o) = f * * ( ~ o ) = 0.

I f (P) has a solut ion u E Uo + W01' 0o (fa; •z), then we necessarily should have

f(Du) = 0 a.e. in f~.

This therefore implies that

vl(1)u) = v z ( O u ) = 1 a.e. in f)

or, in other words, u should satisfy

[1)ul 2 = 2]detDul a.e. in f), ]Du[ 2 + 2]detDu[ = 4 a.e. in f),

u(x,y)= ~ O ( y ) on 0f~.

Since we assumed that det Du > 0 a.e., we deduce that

de tDu = 1 a.e.; (8.18)

therefore, using b o u n d a r y data, (8.18) and the fact tha t u 6 W 1' oo, we get

meas f~ = j J det Du(x, y)dx dy = det ~o' meas f~ = a d - m e a s f~, g~

i.e., ad = 1. H o w e v e r this is impossible since ~o ~D1 (i.e., vl = a < v2 = d < 1 and (1--v)v 2 + vv 2 < 1 ) . []


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D6partement de Math6matiques Ecole Polytechnique F6d6rale de Lausanne

1015 Lausanne

and

Dipartimento di Matematica Viale Morgagni 67A

50134 Firenze

(Accepted November 14, 1994)

Existence of minimizers for non-quasiconvex integrals · scalar case (for which N = 1) always falls...

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