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Some remarks concerning quasiconvexity and strong convergence. (English) Zbl 0628.49011

Let I be an integral of the calculus of variations of the form \(I[v]=\int_{\Omega}F(Dv)dx\), where \(\Omega\) is an open, bounded and smooth set of \({\mathbb{R}}^ n\), \(v\in W^{1,q}(\Omega;{\mathbb{R}}^ N)\) \((q>1)\), and Dv is the \(n\times N\) matrix of the gradient of v. The function \(F=F(P)\) satisfies the growth condition \(0\leq F(P)\leq c(1+| P|^ q)\) for some positive constant c and for all \(P\in {\mathbb{R}}^{nN}\). Moreover, F is uniformly strictly quasiconvex, in the sense that \[ \int_{\Omega}(F(A)+\gamma | D\phi |^ q)dx\leq \int_{\Omega}F(A+D\phi)dx, \] for some positive constant \(\gamma\) and for every \(\phi \in W_ 0^{1,q}(\Omega;{\mathbb{R}}^ N)\), and \(A\in {\mathbb{R}}^{nN}.\)
The authors prove that, if \(u_ K\) converges to u in the weak topology of \(W^{1,q}(\Omega;{\mathbb{R}}^ N)\) and if \(I[u_ k]\) converges to I[u], then \(u_ k\) converges, as \(k\to +\infty\), to u in the strong topology of \(W^{1,q}_{loc}(\Omega;{\mathbb{R}}^ N)\).
Reviewer: P.Marcellini

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
26B25 Convexity of real functions of several variables, generalizations
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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References:

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