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 non-linear 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 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.

References in corpus (1)

• A free boundary problem in thermal insulation with a prescribed heat source

Cited by in corpus (1)

• An isoperimetric result for an energy related to the $p$-capacity