Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

$L^\infty$ jenergies on discontinuous functions

Pages: 905 - 928, Volume 12, Issue 5, May 2005      doi:10.3934/dcds.2005.12.905

       Abstract        Full Text (334.9K)       Related Articles

Roberto Alicandro - DAEIMI, Università di Cassino, via Di Biasio, 03043 Cassino (FR), Italy (email)
Andrea Braides - Dipartimento di Matematica, Università di Roma 'Tor Vergata', via della Ricerca Scientifica, 00133 Roma, Italy (email)
Marco Cicalese - SISSA, via Beirut 2-4, 34100 Trieste, Italy (email)

Abstract: 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:  $L^\infty$ energies, functions of bounded variation, lower semicontinuity.
Mathematics Subject Classification:  49J45, 49Q20.

Received: January 2004;      Revised: October 2004;      Available Online: February 2005.