\(W^{1,p}\)-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.


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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