Lower semicontinuity of multiple µ-quasiconvex integrals

Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy; fragala@mate.polimi.it.

Received: 25 February 2002

Abstract

Lower semicontinuity results are obtained for multiple integrals of the kind , where μ is a given positive measure on , and the vector-valued function u belongs to the Sobolev space associated with μ. The proofs are essentially based on blow-up techniques, and a significant role is played therein by the concepts of tangent space and of tangent measures to μ. More precisely, for fully general μ, a notion of quasiconvexity for f along the tangent bundle to μ, turns out to be necessary for lower semicontinuity; the sufficiency of such condition is also shown, when μ belongs to a suitable class of rectifiable measures.