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

$L^\infty$ jenergies on discontinuous functions

Roberto Alicandro 1, , Andrea Braides 2, and  Marco Cicalese 3,

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 $SBV$.
Keywords: $L^\infty$ energies, functions of bounded variation, lower semicontinuity..
Mathematics Subject Classification: 49J45, 49Q2.
Citation: Roberto Alicandro, Andrea Braides, Marco Cicalese. $L^\infty$ jenergies on discontinuous functions. Discrete and Continuous Dynamical Systems, 2005, 12 (5) : 905-928. doi: 10.3934/dcds.2005.12.905
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