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

TL;DR

This paper demonstrates that, under suitable boundary conditions, the free boundary and fixed boundary of minimizers in certain variational problems touch tangentially, using classification of global profiles and adapting existing methods.

Contribution

It establishes the tangential contact between free and fixed boundaries in variational solutions to Bernoulli type problems, extending previous classifications to this context.

Findings

Free and fixed boundaries contact tangentially under certain conditions

Classification of global profiles adapted for this problem

Extension of boundary contact results to variable coefficient Bernoulli problems

Abstract

In this paper, we show that given appropriate boundary data, the free boundary and the fixed boundary of minimizers of functionals of type \eqref{functional} contact each other in a tangential fashion. We prove this result via classification of the global profiles, adapting the ideas from \cite{KKS06}.