A Free Boundary Problem with Facets

Abstract

We study a free boundary problem on the lattice whose scaling limit is a harmonic free boundary problem with a discontinuous Hamiltonian. We find an explicit formula for the Hamiltonian, prove the solutions are unique, and prove that the limiting free boundary has a facets in every rational direction. Our choice of problem presents difficulties that require the development of a new uniqueness proof for certain free boundary problems. The problem is motivated by physical experiments involving liquid drops on patterned solid surfaces.

Limit shapes of local minimizers for the Alt-Caffarelli energy functional in inhomogeneous media

“…In a forthcoming companion article we show how discontinuities in Q * , Q * are responsible for formation of facets in the free boundary under a monotone quasi-static motion [16]. It was already discovered by Caffarelli and Lee [6], and explored further by the author and Smart [15], that, in a convex setting, discontinuities in Q * result in facets in the minimal supersolution of (1.4).…”

“…In a previous work with Smart [15] we considered the scaling limit of a free boundary problem on the lattice Z d analogous to (1.1). In that case we were able to find an explicit formula for I(p) = [Q * (p), Q * (p)].…”

“…Here the subsolution condition is upper semicontinuous and so needs to be interpreted carefully. The theory for this type of problem was developed in the previous paper of the author and Smart [15]. The augmented pinning problem can also be stated in the case when U is compact and convex.…”

“…Such localized construction is exactly the content of Theorem 1.3. In a previous paper the author and Smart [15] developed viscosity solution tools to prove the convergence of minimal supersolutions and maximal subsolutions. We will use those tools again here, with some necessary refinements.…”

“…One is led to wonder whether these shapes are a microscale phenomenon, or a macroscale phenomenon that remains in the homogenization limit. Starting in our previous work with Smart [15] we have been investigating this question. In that paper we derived an equation like (1.4) from a scaling limit for a discrete version of the Alt-Caffarelli functional.…”