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Weak lower semicontinuity for polyconvex integrals in the limit case

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Abstract

We prove a lower semicontinuity result for polyconvex functionals of the Calculus of Variations along sequences of maps \(u:\Omega \subset \mathbb{R }^n\rightarrow \mathbb{R }^m\) in \(W^{1,m}\), \(2\le m\le n\), bounded in \(W^{1,m-1}\) and convergent in \(L^1\) under mild technical conditions but without any extra coercivity assumption on the integrand.

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Acknowledgments

The research of N. Fusco, C. Leone and A. Verde was supported by the 2008 ERC Advanced Grant N. 226234 “Analytic Techniques for Geometric and Functional Inequalities”. Part of this work was conceived when M. Focardi visited the University of Naples. He thanks N. Fusco, C. Leone, and A. Verde for providing a warm hospitality and for creating a stimulating scientific atmosphere. The authors wish to thank the referee for the careful reading of the manuscript.

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Authors and Affiliations

  1. Dipartimento di Matematica e Informatica “U. Dini”, Università degli Studi di Firenze, Viale Morgagni 67/A, 50134 , Firenze, Italy

    M. Focardi, P. Marcellini & E. Mascolo

  2. Dipartimento di Matematica “R. Caccioppoli”, Università degli Studi “Federico II” di Napoli, Via Cinthia-Monte S. Angelo, 80126 , Napoli, Italy

    N. Fusco, C. Leone & A. Verde

Authors
  1. M. Focardi
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  2. N. Fusco
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  3. C. Leone
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  4. P. Marcellini
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  5. E. Mascolo
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  6. A. Verde
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Corresponding author

Correspondence to N. Fusco.

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Communicated by L.Ambrosio.

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Focardi, M., Fusco, N., Leone, C. et al. Weak lower semicontinuity for polyconvex integrals in the limit case. Calc. Var. 51, 171–193 (2014). https://doi.org/10.1007/s00526-013-0670-0

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  • DOI: https://doi.org/10.1007/s00526-013-0670-0

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