Regularity results for an optimal design problem with a volume constraint

¹ Dipartimento di Ingegneria – Università del Sannio, Corso Garibaldi 82100 Benevento, Italy
carozza@unisannio.it
² Department of Mathematical Sciences, Carnegie Mellon University, PA 15213-3890 Pittsburgh, USA
fonseca@andrew.cmu.edu
³ Università di Napoli “Federico II” Dipartimento di Mat. e Appl. ‘R. Caccioppoli’, Via Cintia, 80126 Napoli, Italy
antpassa@unina.it

Received: 29 May 2013

Abstract

Regularity results for minimal configurations of variational problems involving both bulk and surface energies and subject to a volume constraint are established. The bulk energies are convex functions with p-power growth, but are otherwise not subjected to any further structure conditions. For a minimal configuration (u,E), Hölder continuity of the function u is proved as well as partial regularity of the boundary of the minimal set E. Moreover, full regularity of the boundary of the minimal set is obtained under suitable closeness assumptions on the eigenvalues of the bulk energies.