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

We study a free boundary problem of minimising a functional containing a non-local term rewarding depth into the zero phase: for $u\ge0$ on a bounded, open set $\Omega\subset\mathbb R^d$, we minimise $$J(u) = \int_\Omega \left(\frac12|\nabla u|^2 - fu\right) \;-\; \int_{\{u=0\}} F\big(\mathrm{dist}(x,\partial \{u=0\})\big)\;\mathrm{d}x.$$ 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.