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Article Contents

# $L^\infty$ jenergies on discontinuous functions

• 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 $SBV$.
Mathematics Subject Classification: 49J45, 49Q20.

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