The space $SBV(\Omega)$ of special functions of bounded variation, and $GSBV(\Omega)$ of generalised special functions of bounded variation, have been introduced to study the so called free discontinuity problems. In these spaces it is possible to define a trace operator, whose definition coincides with the usual one, when is a Sobolev function. Unfortunately, due to the fact that a sequence in $(G)SBV(\Omega)$ may have jump sets getting infinitesimally close to the boundary of $\Omega$, the trace operator is not continuous. This lack of continuity leads for example to free discontinuity problems coupled with prescribed Dirichlet condition, having no solution. In this talk I present a possible way to overcome this problem, by restricting our attention to a smaller class of functions, i.e. $(G)SBV(\Omega;\Gamma)$ , which are the functions in $GSBV(\Omega)$ whose jump sets are obliged to lie in the $n1$dimensional set $\Gamma$. In these spaces it is possible to introduce a suitable weight function on the $H^{n1}$ measure of $\partial Omega$, to obtain some continuity results for the trace operator. Finally I will show some applications to the MumfordShah minimization problem with Dirichlet condition on the boundary, and to the existence of a solution to the wave equation in a domain with prescribed arbitrary growing cracks with Neumann condition on the boundary.
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Some continuity results for the trace operator in the contest of special functions of bounded variation
Research Group:
Emanuele Tasso
Institution:
SISSA
Location:
A133
Schedule:
Friday, June 1, 2018  14:00
Abstract:
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