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

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