2026/07/23 by Amal Alphonse, Enrico Valdinoci, Marcelo Bongarti
#math.AP #math.OC
We study a free boundary problem of minimising a functional containing a non-local term rewarding depth into the zero phase: for u\geqslant 0 on a bounded, open set Ω⊂\mathbb Rd, we minimise J(u) = ∫Ω(\frac12|∇ u|2 - fu) - ∫_\u=0\ F(dist(x,∂ \u=0\)) dx. 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.