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

In this paper, we show that, given appropriate boundary data, the free boundaries of minimizers of functionals of type (Formula Presented) 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].