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

Abstract

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 ${\epsilon }^{\frac{n}{n-1}}$, being $\epsilon$ its periodicity. We give two different formulations of the obstacle problem to include also perforations with Lebesgue measure zero.