A free boundary problem driven by boundary distance in the coincidence set

Abstract

We study a free boundary problem of minimising a functional containing a non-local term rewarding depth into the zero phase: for $u⩾0$ on a bounded, open set $\Omega \subset {\mathbb{R}}^d$, we minimise $$J(u)={\int }_{\Omega }\left(\frac{1}{2}|\nabla u{|}^2-fu\right)-{\int }_{\{u=0\}}F(\mathrm{dist}(x,\partial \{u=0\}))\mathrm{d}x.$$ This kind of functional arises, for example, from a two-membranes problem with an adhesive contact energy. We first address a well-definedness issue caused by the non-local, boundary-sensitive nature of the functional and prove existence of minimisers, establishing along the way a weak lower semicontinuity result for the non-local term. We then derive stationarity conditions for minimisers, including a variational (Euler--Lagrange type) inequality, a PDE on the positivity set, and, under a mild non-degeneracy assumption, a free boundary condition obtained via inner variations and a Danskin-type differentiation of the distance function.