Calculus of Variations and Geometric Measure Theory

S. Almi - G. Dal Maso - R. Toader

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

created by almi on 15 Dec 2015
modified by dalmaso on 16 Mar 2018


Published Paper

Inserted: 15 dec 2015
Last Updated: 16 mar 2018

Journal: J. Math. Pures Appl.
Volume: 108
Pages: 952-970
Year: 2017


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.