Asymptotic analysis of Mumford–Shah type energies in periodically perforated domains
Matteo Focardi, Maria Stella Gelli · Interfaces and Free Boundaries Mathematical Analysis Computation and Applications · 2007
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 \varepsilon^{\frac n{n-1}} , being \varepsilon its periodicity. We give two different formulations of the obstacle problem to include also perforations with Lebesgue measure zero.