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New relaxation theorems with applications to strong materials

Part of: Existence theories

Published online by Cambridge University Press:  22 April 2018

Jean-Philippe Mandallena  and
Mikhail Sychev
Jean-Philippe Mandallena
Affiliation:
Université de Nîmes, Laboratoire MIPA, Site des Carmes, Place Gabriel Péri, 30021 Nîmes, France ([email protected])
Mikhail Sychev
Affiliation:
Laboratory of Differential Equations and Related Problems in Analysis, Sobolev Institute of Mathematics, 4 Acad. Koptyug Avenue, Novosibirsk 630090, Russia and Novosibirsk State University, Pirogova str. 1, Novosibirsk 630090, Russia ([email protected])
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Abstract

Recently, Sychev showed that conditions both necessary and sufficient for lower semicontinuity of integral functionals with p-coercive extended-valued integrands are the W1,p-quasi-convexity and the validity of a so-called matching condition (M). Condition (M) is so general that we conjecture whether it always holds in the case of continuous integrands. In this paper we develop the relaxation theory under the validity of condition (M). It turns out that a better relaxation theory is available in this case. This motivates our research since it is an important old open problem to develop the relaxation theory in the case of extended-value integrands. Then we discuss applications of the general relaxation theory to some concrete cases, in particular to the theory of strong materials.

Keywords

lower semicontinuityrelaxationW1;p-quasi-convexityextended-valued integrandstrong materials

MSC classification

Primary: 49J45: Methods involving semicontinuity and convergence; relaxation
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 148 , Issue 5 , October 2018 , pp. 1029 - 1047
Copyright
Copyright © Royal Society of Edinburgh 2018 

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