A relaxation result for energies defined on pairs set-function and applications

Abstract

Abstract. We consider, in an open subset Ω of R N , energies depending on the perimeter of a subset E ⊂ Ω (or some equivalent surface integral) and on a function u which is defined only on Ω \ E. We compute the lower semicontinuous envelope of such energies. This relaxation has to take into account the fact that in the limit, the "holes" E may collapse into a discontinuity of u, whose surface will be counted twice in the relaxed energy. We discuss some situations where such energies appear, and give, as an application, a new proof of convergence for an extension of Ambrosio-Tortorelli's approximation to the Mumford-Shah functional.Mathematics Subject Classification. 49J45, 49Q20, 49Q10.

Epitaxially strained elastic films: the case of anisotropic surface energies

“…2.38 and 2.39) to treat the presence of anisotropy in the surface term (we refer also to the recent works [3,6] for related relaxation results in higher dimension).…”

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“…This is a formalization of an argument that is present for example in the classical approximation by elliptic functionals by Ambrosio and Tortorelli [2] or in the finite-element schemes by Chambolle and Dal Maso [14][15][16], that actually is immediately derived from [13].…”

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