Asymptotic analysis of the Mumford-Shah functional in periodically perforated domains.

We study the asymptotic limit of obstacle problems for Mumford-Shah type functionals with p-growth 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 the power n/(n-1) of ε, being ε the size of the unit cell of the underlying lattice. We give two different formulations of the obstacle problem to include also perforations with Lebesgue measure zero.