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.
Asymptotic analysis of the Mumford-Shah functional in periodically perforated domains / M. Focardi; M.S. Gelli. - In: INTERFACES AND FREE BOUNDARIES. - ISSN 1463-9971. - STAMPA. - 9:(2007), pp. 107-132. [10.4171/IFB/158]
Asymptotic analysis of the Mumford-Shah functional in periodically perforated domains.
FOCARDI, MATTEO;
2007
Abstract
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.File | Dimensione | Formato | |
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