W1,p-quasiconvexity and variational problems for multiple integrals

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Abstract

Variational problems for the multiple integral IΩ(u) = ∝Ω g(▽u(x))dx, where Ω⊂Rm and u:Ω→Rn are studied. A new condition on g, called W1,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 IΩ in W1,p(Ω;Rn) and for the existence of minimizers for certain related integrals. Counterexamples are given concerning the weak continuity properties of Jacobians in W1,p(Ω;Rn), pn = m. An existence theorem for nonlinear elastostatics is proved under optimal growth hypotheses.

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