An existence result for a nonconvex variational problem via regularity

¹ Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213, U.S.A.; fonseca@andrew.cmu.edu.
² Dipartimento di Matematica “R. Caccioppoli”, Università di Napoli, Via Cintia, 80126 Napoli, Italy; fusco@matna1.dma.unina.it.
³ Dipartimento di Matematica “U. Dini”, Università di Firenze, Viale Morgagni 67 A, 50134 Firenze, Italy; marcell@udini.math.unifi.it.

Received: 9 May 2001

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.