# American Institute of Mathematical Sciences

October  2005, 12(5): 905-928. doi: 10.3934/dcds.2005.12.905

## $L^\infty$ jenergies on discontinuous functions

 1 DAEIMI, Università di Cassino, via Di Biasio, 03043 Cassino (FR), Italy 2 Dipartimento di Matematica, Università di Roma 'Tor Vergata', via della Ricerca Scientifica, 00133 Roma, Italy 3 SISSA, via Beirut 2-4, 34100 Trieste, Italy

Received  January 2004 Revised  October 2004 Published  February 2005

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 $SBV$.
Citation: Roberto Alicandro, Andrea Braides, Marco Cicalese. $L^\infty$ jenergies on discontinuous functions. Discrete & Continuous Dynamical Systems, 2005, 12 (5) : 905-928. doi: 10.3934/dcds.2005.12.905
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