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\(W^{1,p}\)-quasiconvexity and variational problems for multiple integrals. (English) Zbl 0549.46019

Variational problems for the multiple integral \(I_{\Omega}(u)=\int_{\Omega}g(\nabla u(x))dx,\) where \(\Omega\subset {\mathbb{R}}^ m\) and \(u:\Omega\to {\mathbb{R}}^ n\) are studied. A new condition on g, called \(W^{1,p}\)-quasiconvexity is introduced which generalizes in a natural way the quasiconvexity condition of C. B. Morrey, it being shown in particular to be necessary for sequential weak lower semicontinuity of \(I_{\Omega}\) in \(W^{1,p}(\Omega;{\mathbb{R}}^ n)\) and for the existence of minimizers for certain related integrals. Counterexamples are given concerning the weak continuity properties of Jacobians in \(W^{1,p}(\Omega;{\mathbb{R}}^ n), p\leq n=m\). An existence theorem for nonlinear elastostatics is proved under optimal growth hypotheses.

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
49J27 Existence theories for problems in abstract spaces
74B20 Nonlinear elasticity
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