Calculus of Variations and Geometric Measure Theory

M. Focardi - M. S. Gelli

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

created by gelli on 27 Jan 2006
modified by focardi on 02 May 2008


Published Paper

Inserted: 27 jan 2006
Last Updated: 2 may 2008

Journal: Inter. Free Boundaries
Volume: 9
Pages: 107-132
Year: 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 $\epsilon^{n/(n-1)}$, being $\eps$ its periodicity. We give two different formulations of the obstacle problem to include also perforations with Lebesgue measure zero.