$L^\infty$ jenergies on discontinuous functions

Roberto Alicandro; Andrea Braides; Marco Cicalese

doi:10.3934/dcds.2005.12.905

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2005, Volume 12, Issue 5: 905-928. Doi: 10.3934/dcds.2005.12.905

This issue Previous Article Versal Unfoldings for rank--2 singularities of positive quadratic differential forms: The remaining case Next Article Blowing-up coordinates for a similarity boundary layer equation

$L^\infty$ jenergies on discontinuous functions

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

Received: December 31, 2003

Revised: September 30, 2004

Published: September 2005

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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 $SBV$.

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Mathematics Subject Classification: 49J45, 49Q20.

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