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.

A lower semicontinuity result for a free discontinuity functional with a boundary term / Almi, Stefano; Dal Maso, Gianni; Toader, Rodica. - In: JOURNAL DE MATHÉMATIQUES PURES ET APPLIQUÉES. - ISSN 0021-7824. - 108:6(2017), pp. 952-990. [10.1016/j.matpur.2017.05.018]

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

#### Abstract

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.
##### Scheda breve Scheda completa Scheda completa (DC)
108
6
952
990
https://doi.org/10.1016/j.matpur.2017.05.018
http://preprints.sissa.it/xmlui/handle/1963/35146
Almi, Stefano; Dal Maso, Gianni; Toader, Rodica
File in questo prodotto:
File
Alm-DM-Toa-JMPA-2017.pdf

non disponibili

Tipologia: Versione Editoriale (PDF)
Licenza: Non specificato
Dimensione 688.37 kB
Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/15979