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A lower semicontinuity result for a free discontinuity functional with a boundary term

TitleA lower semicontinuity result for a free discontinuity functional with a boundary term
Publication TypePreprint
2015
AuthorsAlmi, S, Dal Maso, G, Toader, R

We study the lower semicontinuity in $GSBV^{p}(\Om;\R^{m})$ of a free discontinuity functional~$\F(u)$ that can be written as the sum of a crack term, depending only on the jump set~$S_{u}$, and of a boundary term, depending on the trace of~$u$ on~$\partial\Om$. We give sufficient conditions on the integrands for the lower semicontinuity of~$\F$. Moreover, we prove a relaxation result, which shows that, if these conditions are not satisfied, the lower semicontinuous envelope of~$\F$ can be represented by the sum of two integrals on~$S_{u}$ and~$\partial\Om$, respectively.

http://urania.sissa.it/xmlui/handle/1963/35146

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