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Asymptotic analysis of Mumford–Shah type energies in periodically perforated domains

We study the asymptotic limit of obstacle problems for Mumford–Shah type functionals with pgrowth in periodically perforated domains via the Γ -convergence of the associated free-discontinuity energies. In the limit a non-trivial penalization term related to the 1-capacity of the reference hole appears if and only if the size of the perforation scales like en/(n−1), e being its periodicity. We give two different formulations of the obstacle problem to include also perforations with Lebesgue measure zero.