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A necessary and sufficient condition for the weak lower semicontinuity of one-dimensional non-local variational integrals

Published online by Cambridge University Press:  12 July 2007

Jonathan Bevan  and
Pablo Pedregal
Jonathan Bevan
Affiliation:
Mathematical Institute, University of Oxford, 24–29 St Giles', Oxford OX1 3LB, UK
Pablo Pedregal
Affiliation:
Escuela Técnica Superior de Ingenieros Industriales, Universidad de Castilla-La Mancha, Avenida Camilo José Cela, s/n, 13071 Ciudad Real, Spain
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Abstract

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In this short note we prove that the functional I : W1,p(J;R) → R defined by is sequentially weakly lower semicontinuous in W1,p(J,R) if and only if the symmetric part W+ of W is separately convex. We assume that W is real valued, continuous and bounded below by a constant, and that J is an open subinterval of R. We also show that the lower semicontinuous envelope of I cannot in general be obtained by replacing W by its separately convex hull Wsc.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 136 , Issue 4 , August 2006 , pp. 701 - 708
Copyright
Copyright © Royal Society of Edinburgh 2006