W1,p-quasiconvexity and variational problems for multiple integrals

J. M. Ball; F. Murat

W^1,p-quasiconvexity and variational problems for multiple integrals

J. M. Ball, F. Murat

School of Mathematical & Computer Sciences

Research output: Contribution to journal › Article › peer-review

362 Citations (Scopus)

Abstract

Variational problems for the multiple integral I_O(u) = ?_O g(?u(x))dx, where O?R^m and u:O?Rⁿ 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_O in W^1,p(O;Rⁿ) and for the existence of minimizers for certain related integrals. Counterexamples are given concerning the weak continuity properties of Jacobians in W^1,p(O;Rⁿ), p = n = m. An existence theorem for nonlinear elastostatics is proved under optimal growth hypotheses. © 1984.

Original language	English
Pages (from-to)	225-253
Number of pages	29
Journal	Journal of Functional Analysis
Volume	58
Issue number	3
Publication status	Published - 1 Oct 1984

Cite this

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title = "W1,p-quasiconvexity and variational problems for multiple integrals",

abstract = "Variational problems for the multiple integral IO(u) = ?O g(?u(x))dx, where O?Rm and u:O?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 IO in W1,p(O;Rn) and for the existence of minimizers for certain related integrals. Counterexamples are given concerning the weak continuity properties of Jacobians in W1,p(O;Rn), p = n = m. An existence theorem for nonlinear elastostatics is proved under optimal growth hypotheses. {\textcopyright} 1984.",

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AU - Ball, J. M.

AU - Murat, F.

PY - 1984/10/1

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N2 - Variational problems for the multiple integral IO(u) = ?O g(?u(x))dx, where O?Rm and u:O?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 IO in W1,p(O;Rn) and for the existence of minimizers for certain related integrals. Counterexamples are given concerning the weak continuity properties of Jacobians in W1,p(O;Rn), p = n = m. An existence theorem for nonlinear elastostatics is proved under optimal growth hypotheses. © 1984.

AB - Variational problems for the multiple integral IO(u) = ?O g(?u(x))dx, where O?Rm and u:O?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 IO in W1,p(O;Rn) and for the existence of minimizers for certain related integrals. Counterexamples are given concerning the weak continuity properties of Jacobians in W1,p(O;Rn), p = n = m. An existence theorem for nonlinear elastostatics is proved under optimal growth hypotheses. © 1984.

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VL - 58

SP - 225

EP - 253

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

IS - 3

ER -

W^1,p-quasiconvexity and variational problems for multiple integrals

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