Inserted: 15 dec 2015
Last Updated: 29 jan 2016
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.