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An existence result for a nonconvex variational problem via regularity

Published online by Cambridge University Press:  15 September 2002

Irene Fonseca ,
Nicola Fusco  and
Paolo Marcellini
Irene Fonseca
Affiliation:
Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213, U.S.A.; fonseca@andrew.cmu.edu.
Nicola Fusco
Affiliation:
Dipartimento di Matematica “R. Caccioppoli”, Università di Napoli, Via Cintia, 80126 Napoli, Italy; fusco@matna1.dma.unina.it.
Paolo Marcellini
Affiliation:
Dipartimento di Matematica “U. Dini”, Università di Firenze, Viale Morgagni 67 A, 50134 Firenze, Italy; marcell@udini.math.unifi.it.
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Abstract

Local Lipschitz continuity of minimizers of certain integrals of the Calculus of Variations is obtained when the integrands are convex with respect to the gradient variable, but are not necessarily uniformly convex. In turn, these regularity results entail existence of minimizers of variational problems with non-homogeneous integrands nonconvex with respect to the gradient variable. The x-dependence, explicitly appearing in the integrands, adds significant technical difficulties in the proof.

Keywords

Nonconvex variational problemsuniform convexityregularityimplicit differential equations.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 7 , 2002 , pp. 69 - 95
Copyright
© EDP Sciences, SMAI, 2002

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