Tangential contact between free and fixed boundaries for variational solutions to variable-coefficient Bernoulli-type free boundary problems

Abstract

In this paper, we show that, given appropriate boundary data, the free boundaries of minimizers of functionals of type $J(v;A,{\lambda }_{+},{\lambda }_{-},\Omega )={\int }_{\Omega }(⟨A(x)\nabla v,\nabla v⟩+\Lambda (v))dx$ and the fixed boundary touch each other in a tangential fashion. We extend the results of Karakhanyan, Kenig, and Shahgholian [Calc. Var. Partial Differential Equations 28 (2007), 15–31] to the case of variable coefficients. We prove this result via classification of the global profiles, as per Karakhanyan, Kenig, and Shahgholian [Calc. Var. Partial Differential Equations 28 (2007), 15–31].