Calculus of Variations and Geometric Measure Theory

R. Alicandro - A. Braides - M. Cicalese

$L^\infty$ energies on discontinuous functions

created on 08 May 2003
modified by braidesa on 10 May 2005


Published Paper

Inserted: 8 may 2003
Last Updated: 10 may 2005

Journal: Discrete and Continuous Dynamical Systems
Volume: 12
Pages: 905-928
Year: 2005


\noindent We study necessary and sufficient conditions for the lower-semicontinuity of one-dimensional energies defined on ($BV$ and) $SBV$ of the model form $F(u)= \sup f(u')\vee\sup g([u])$, and prove a relaxation theorem. We apply these results to the study of problems with Dirichlet boundary conditions, highlighting a complex behaviour of solutions. We draw a comparison with the parallel theory for integral energies on $SBV$.

Keywords: Supremal functionals, semicontinuity, nonconvex energies