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We consider, in an open subset Ω of
energies depending on the perimeter of a subset
(or some equivalent surface integral) and on a function u which is
defined only on
. 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.
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