Existence and nonexistence of minimizers for classical capillarity problems in presence of nonlocal repulsion and gravity

TL;DR

This paper studies the existence and nonexistence of minimizers for a capillarity problem with nonlocal interactions and gravity, revealing conditions under which solutions exist or fail to exist.

Contribution

It establishes new existence results for small masses and a generalized existence for all masses, including nonlocal kernels, and identifies nonexistence in large mass regimes for specific kernels.

Findings

Existence of minimizers in small mass regime.

Nonexistence of minimizers for large mass with certain kernels.

Generalized existence of minimizers as disjoint unions at infinite distances.

Abstract

We investigate, under a volume constraint and among sets contained in a Euclidean half-space, the minimization problem of an energy functional given by the sum of a capillarity perimeter, a nonlocal interaction term and a gravitational potential energy. The capillarity perimeter assigns a constant weight to the portion of the boundary touching the boundary of the half-space. The nonlocal term is represented by a double integral of a positive kernel $g$, while the gravitational term is represented by the integral of a positive potential $G$. We first establish existence of volume-constrained minimizers in the small mass regime, together with several qualitative properties of minimizers. The existence result holds for rather general choices of kernels in the nonlocal interaction term, including attractive-repulsive ones. When the nonlocal kernel $g(x)=1/|x|^\beta$ with $\beta \in (0,2]$, we also obtain nonexistence of volume constrained minimizers in the large mass regime. Finally, we prove a generalized existence result of minimizers holding for all masses and general nonlocal interaction terms, meaning that the infimum of the problem is realized by a finite disjoint union of sets thought located at ``infinite distance'' one from the other. These results stem from an application of quantitative isoperimetric inequalities for the capillarity problem in a half-space.