Lower semicontinuity and relaxation


I. Fonseca
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213


and

G. Leon
Dipartimento di Scienze e Tecnologie Avanzate
Universita del Piemonte Orientale
Alessandria, Italy 15100

ABSTRACT

Lower semicontinuity and relaxation results in BV are obtained for multiple integrals

$$ F(u,\Omega):=\int_\Omega f(x,u(x),\nabla u(x))\,dx, \quad u \in W^{1,1}(\Omega;\Bbb R^d), $$ where the energy density f satisfies lower semicontinuity conditions with respect to (x,u) and is not subjected to coercivity hypotheses. These results call for methods recently developed in the Calculus of Variations.

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