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

We study the lower semicontinuity in GSBV(P)(Omega; 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 derivative Omega We give necessary and sufficient conditions on the integrands for the lower semicontinuity of F. Moreover, we prove a relaxation result, which shows that the lower semicontinuous envelope of can still be represented as the sum of two integrals on S-u and partial derivative Omega, respectively. (C) 2017 Elsevier Masson SAS. All rights reserved.