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 p-growth in periodically perforated domains via the Gamma-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 g(n)/((n-1)), epsilon being its periodicity. We give two different formulations of the obstacle problem to include also perforations with Lebesgue measure zero.