0 then r(0) = 0. The results and comments in this section apply with appropriate modifications to incompressible elasticity, in which the deformation u is required to satisfy the pointwise constraint (6.5 j
a.e. .YE n.
det Vu(x) = 1
The constraint (6.5) is conveniently incorporated into our framework by requiring that the stored-energy function g: M3 x3 --t R satisfy g(A ) = 00 if and only if det A # 1; note that such g are Bore1 measurable but not continuous. If, for example, if
g(A) =fl(vy; + cy + v;j =+a
detA =P,L’?P~=
1,
otherwise,
where p > 0, a > 1, then g is WI+-quasiconvex if and only if max(a,p) > 3. In fact, if max(a,p) > ? then g is W’+ quasiconvex by Theorem 4.5(i). If max(u.p) < 3 and i, > 1 let U(X) = (r(R)/R)x be the radial mapping of B= (1x1 < 11 defined by r(R)=
(R” +A’-
1)’ ‘.
Then u E W’%“(B; IF;‘) and U(X) la* = Ax but g(1l) = 03. jH g(Vu(x)) d.u < co. so that g is not W’.P-quasiconvex. 7. COUNTEREXAMPLES CONCERNING WEAK CONTINUITY OF JACOBIANS
Let II > 1 and let R c R” be open. In the preceding sections we used the fact that the mapping u I-+ det Vu is sequentially continuous from WL3”(Q; R”) endowed with the weak topology to L’(a) endowed with the topology of measures. By means of two counterexamples of a different nature we now show that u ++ det Vu is not sequentially weakly continuous from W’*“(Q; R”) to L’(B), that is, uj--\ u in W’*“(R; R”) does not imply that det Vuj - det Vu in L’(Q). This should be contrasted with the fact that if h is convex and lower semicontinuous then J‘* h(det Vu(x)) dx is swlsc on W’*“(Q; F?“) (Th eorem 4.l(ii)), and the fact that, for example, the functional u H jc, a(x) ]det Vu(x)\ dx is swlsc on I+“-“(R; P”) for all a E L”(Q). a > 0 (see Acerbi and Fusco [3, Theorem 2.41). COUNTEREXAMPLE 7.1. Let B = (xE R”: 1x1 < l}, II > 1. Consider for j = 1, 2,..., the radial mappings rW Uj(X) = *
x,
R = 1x1,
246
BALL AND MURAT
where if
0 < R < l/j,
=2-jR
if
l/j < R < 2/j,
=o
if
2/j 1. Consider the space Rad’.“(B) of radial mappings u(x) = (r(R)/R)x belonging to W’*“(B; IR”), with norm N(r) given by (7.3). Supposing that 0 < R < R < q and using Holder’s inequality with exponents II and n/(n - 1) we have It-“(R) - r”(E)1 1. For x = (xl,..., x”) E Rn lxkl and Q = {x E R”: 11x/I< l} = (-1, 1)“. We let =maxlGkGn
COUNTEREXAMPLE
M
w=+
for all
denote
x E Q.
This function is continuous except at zero and vanishes on 3Q. We extend u as a Q-periodic function to the whole of R”. In Q\{O} u = (u’,..., u”) has distributional derivatives
g
(x)’ St,--
II4
a llxll F=
sign xk
=o
xi a llxll _ 6’ k’ ll-#~
(7.5)
if Il.4 = lxkI, if
11x/I= Ixi/, i# k.
We note that
and show that (7.5) defines the derivatives of u in the sense of distributions in Q (and not just in Q\ (0)). In fact, if 4 E 9(Q) and E > 0,
where ra denotes terms tending to zero as E + 0. Finally, since by (7.6)
it follows that u E W’*p(Q; R”) for all p, 1
1, is not easy to find proved completely in the literature. LEMMA
fE L&,(ip”)
A.1. Let QC IF?“’ be an m-cube. Let 1 < p < co and be Q-periodic. Define&(x) =f( jx). Then as j + co
let
in Lp(ll) for any bounded open subset fJ c Rm. Proof. We suppose without loss of generality that Q = (0, 1)“; in this case a function f is Q-periodic provided f(x + ei) =f(x) for all x and i = l,..., m, where (ei} denotes the standard basis of pm.
250
BALLANDMURAT
Suppose first that fE Lm(IRm). Then 4 is bounded in Lm(G) contains a subsequence f, -* 0 in L”O(J2). We show that B(x) = j f(y)
dy = const.
and so
a.e. x E a.
(A.1)
Q
In fact, let Q, c R be an m-cube with edges parallel to the axes and of side a. If pus > 1 then PQ, is a union of ([pa - 11)” unit cubes of the fundamental lattice and a set E, with meas E, < ([pa + 21)” - ([pa - l])m, where [ ] denotes “integer part.” Thus I!,,
(.twjQf(r)dv)d~~
= lpi,,,
(f(4-jQf(YPY)
dzi
meas E,,
< Cp-”
which tends to zero as p --t co. Thus JQ, (0(x) - J&(Y) dy) dx = 0 for all Q, and (A.l) follows by a density argument. Since 0 is uniquely determined byf it follows that the whole sequence fj - * JQf(y) dy in L”O(R). Next suppose that fE Lo,, with 1
O, lRmp(x)dx = 1. For E > 0 let P,(x)=~~’ E-“‘p(x/&), and let f” = pE*J Then f ‘+ f in L”(Q) and clearly f E is Q-periodic. We claim that fi-+ fj in L”(R) as E + 0, uniformly inj. We have that
Jo If;(x) -fj(x)l”
dx =jPm ho If”(y) -f (y)l’ dy
k
1
QkVYY) -f(y)l”
dy,
where the sum JJk is over cubes Qk = ak + Q of the fundamental lattice, the number of these cubes being less than Cjm. But
jQ, lf"(~> -f (YIP dy = jQ In”
-f (y)l’ dy
since f”, f are Q-periodic. Therefore ]Ifj” -fj]lLe(q) < C ]]f” - f ]lLPcQ,,which proves the claim. Since 7 is smooth we know that f;-* 1 f’(y) dy in L”O(s2) as j + co. Writing 4 = fj +fj - fj” we deduce that 4 - 7Qf(y) dy in Lp(12) as required. 1 COROLLARY A.2. Let Q c I?“’ be an m-cube. Let 1 < p < cq let v E W:,$(Rm; R”) be such that Vu is Q-periodic, and define u/(x) = j- ‘v(jx). Then v.---& (jQ Vv(y)dy) x (-* if-P= a) 3
in W’qp(12;R”) for any bounded open subset l2 c R”.
QUASICONVEXITYANDVARIATIONALPROBLEMS
251
ProoJ Let Q = (0, 1)” and consider the function zi(x) = v(x + ei) - P(X). Since Vz,(x) = Vtl(x + ei) - VU(X) = 0, zi(x) is actually independent of X: let A E I’W’~“’ be defined by Aei = zi(x) and w by w(x) = v(x) -Ax. Then w(x + ei) - w(x) = t’(x + ei) - c(x) - Aei = 0. Hence u’ is Q-periodic. But c’~(x) = Ax + + w(jx), Vcj(x) = A + VW($). in ,!,P(R; IR”), and by the lemma VC __-\ Therefore cj+Ax ,fv (A + VW(Y)) dq’ = IQ VV(JJ) 4~ in LP(R; MnYfl) (-* if P = co), which completes the proof. 1 The final result is a lower semicontinuity
theorem.
PROPOSITION A.3. Let R c iR”’ be bounded open, and let H: 7” + E be convex, lower semicontinuous and bounded below. Let 6,, 8 E L ‘(0; F, ‘) with ei -* 0 in the sense of measures (i.e., ./‘()Bjq4dx + .(‘Ij $4 dx for all continuous functions I$: f2 + IR with compact support in Q). Then
\ H(B(x)) dx < lim inf (_ H(B,(x)) dx. i-a .n . I>
(A.21
Remark A.4. This proposition was proved by Reshetnyak [ 17, p. 805 1 for the case when H depends also on x, but takes only finite values. In our case we are able to give a slightly simpler proof. but still based on Reshetnyak’s idea. (Note added in proof: see also Marcellini 121 1.) Proof of Proposition A.3. We suppose first that H(z) = max, 41s,,, (Ui + @iy Z)) is P’iecewise aftine, convex and nonnegative. For each z E I!‘\ we choose an element A(z) E aH(z). There exists a sequence ck --f 0 a.e. in R and strongly in L’(l2; IR”) with the property that z’~ is constant on each open s-cube Qk,,, = k-‘[m + (0, l)‘], m E Z’. Also, there is a sequence $k E Ci(O1 such that 0 < 4k < 1, #k = 0 in a neighborhood of each aQk,, and tik + I a.e. in R. By the convexity of H,
H(ej(x))> H(u~(x))+ (A(v/ct-x)), e,(X)- u/c(X)) a.e. x E R.
252
BALL ANDMURAT
We multiply
this inequality
by $,Jx) and integrate over ~2 to obtain
!, H(ej(X)) dx> I, #/c(X) ff(ej(X))dx 2 I, $/c(X) H(u/c(X)) dx + !, #L(X)@ (u/c(X)>, ej(X)- u/c(X)) dx. Fix k and let j+ co. Since #,&x) A(n,Jx)) support in Q we obtain
is continuous
with compact
lim inf J H(ej(X)) dx > I, #/c(X) H(u/AX)) dx j-100 0
+ j, !h(-w(%@))~ e(X)- Q(X))dx. Let k-+co. By Fatou’s lemma lim inf,,, j”c #,Jx) H(D~(x)) dx > J”, H(d(x)) dx. S’mce A(.) is uniformly bounded a simple estimate shows that
;;% j, h(X)(wdX))y e(X)- Q(X))dx = 0. Hence we obtain (A.2). Now let H: IRS+ R be convex, lower semicontinuous and bounded below. Any such function is the supremum of a countable family of afftne functions. Assuming without loss of generality that H > 0 it follows that H can be written as the limit of an increasing sequence H, of piecewise affne, convex, nonnegative functions. For each I 1 H/(0(x)) dx < li,m_&f j H,(6Jj(x)) dx < liz&f R R
j
R
H(Bj(x)) dx.
Letting I+ co we deduce from the monotone convergence theorem that 1 H(O(x)) dx < lirn&f R as required.
1 H(Bj(x)) dx R
I ACKNOWLEDGMENTS
We thank P. Marcellini for interesting discussions. The research of JMB was supported by a U.K. Science and Engineering Research Council Senior Fellowship. Parts of the research were also done as JMB was visiting the Universitk Pierre et Marie Curie and then as FM visited Heriot-Watt University as an S.E.R.C. Visiting Fellow.
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253
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