Published Paper

Inserted: 6 jun 2024
Last Updated: 10 nov 2024

Journal: Nonlinear Analysis
Volume: 251
Number: 113685
Year: 2025
Doi: https://doi.org/10.1016/j.na.2024.113685

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 $g(x)=1/\lvert x\rvert^\beta$ 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.