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

TL;DR

This paper investigates a nonlinear free boundary problem related to thermal insulation, involving the p-Laplacian and nonlinear boundary conditions, establishing existence and regularity of minimizers within a variational framework.

Contribution

It introduces a novel variational formulation for a nonlinear free boundary problem with p-Laplacian and nonlinear boundary conditions, proving existence and density estimates of minimizers.

Findings

Existence of minimizers under certain conditions on p and q.

Uniform density estimates for the jump set of minimizers.

Variational framework in SBV for the problem.

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.