A free boundary problem for the p-Laplacian with nonlinear boundary conditions

Abstract

We study a nonlinear generalization of a free boundary problem that arises in the context of thermal insulation. We consider two open sets Omega subseteq A, and we search for an optimal A in order to minimize a nonlinear energy functional, whose minimizers u satisfy the following conditions: Delta _p u=0 inside A{setminus }Omega , u=1 in Omega , and a nonlinear Robin-like boundary (p, q)-condition on the free boundary partial A. We study the variational formulation of the problem in {{,textrm{SBV},}}, and we prove that, under suitable conditions on the exponents p and q, a minimizer exists and its jump set satisfies uniform density estimates.