Calculus of Variations and Geometric Measure Theory
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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 on 29 Jan 2016


Submitted Paper

Inserted: 15 dec 2015
Last Updated: 29 jan 2016

Year: 2015


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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