Homogenization of a Singular Perturbation Problem

Abstract

We discuss homogenization of the singular perturbation problem $$ {\Delta}_p{u}_{\delta}^{\varepsilon }={f}^{\varepsilon }{\beta}_{\delta}\left({u}_{\delta}^{\varepsilon}\right)\kern1em in\kern0.5em {\mathrm{\mathbb{R}}}^n\backslash \overline{B_1} $$ with a constant boundary value on the ball. Here, Δp is the usual p-Laplacian operator. It is generally understood that the two parameters δ and e are in competition and two different behaviors may be exhibited, depending on which parameter tends to zero faster. We consider one scenario where we assume that e, the homogenization parameter, tends to zero faster than δ, the singular perturbation parameter. We show that there is a universal speed for which the limit solves a standard Bernoulli free boundary problem.