On the relaxation of some classes of pointwise gradient constrained energies

摘要

The integral representation problem on BV(Omega) for the L-1(Omega)-lower semicontinuous envelope (F) over bar of the functional F: u is an element of W-1,W-infinity(Omega) (bar right arrow) integral(Omega) f(del u) dx is approached when f is a Borel function, not necessarily convex, with values in [0, + infinity]. The presence of the value +infinity in the image of f involves a pointwise gradient constraint on the admissible configurations, since those generating the relaxation process must satisfy the condition del u(x) is an element of dom f for a.e. x is an element of Omega. The main novelty relies in the absence of any convexity assumption on the domain of f. For every convex bounded open set Omega, (F) over bar is represented on the whole BV(Omega) as an integral of the calculus of variations by means of the convex lower semicontinuous envelope of f. Due to the lack of the convexity properties of dom f, the classical integral representation techniques, based on measure theoretic arguments, seem not to work properly, thus an alternative approach is proposed. Applications are given to the relaxation of Dirichlet variational problems and to first order differential inclusions. (C) 2006 Elsevier Masson SAS. All rights reserved.