Published Paper

Inserted: 15 dec 2015
Last Updated: 16 mar 2018

Journal: J. Math. Pures Appl.
Volume: 108
Pages: 952-970
Year: 2017
Doi: https://doi.org/10.1016/j.matpur.2017.05.018

Abstract:

We study the lower semicontinuity in GSBV p (Ω; Rm ) of a free discontinuity functional F (u) that can be written as the sum of a crack term, depending only on the jump set Su , and of a boundary term, depending on the trace of u on ∂Ω . 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 Su and ∂Ω , respectively.

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