Existence of minimizers for a generalized liquid drop model with fractional perimeter

TL;DR

This paper investigates the existence and asymptotic behavior of minimizers in a generalized liquid drop model involving fractional perimeter and Riesz potential, extending classical models and analyzing large-volume limits.

Contribution

It establishes existence results for minimizers under various decay conditions of the Riesz potential kernel and analyzes their asymptotic convergence to Euclidean balls.

Findings

Existence of minimizers for all volumes with fast-decaying kernels.

Existence of generalized minimizers when the kernel vanishes at infinity.

Minimizers converge to Euclidean balls as volume tends to infinity.

Abstract

We consider the minimization problem of the functional given by the sum of the fractional perimeter and a general Riesz potential, which is one generalization of Gamow's liquid drop model. We first show the existence of minimizers for any volumes if the kernel of the Riesz potential decays faster than that of the fractional perimeter. Secondly, we show the existence of generalized minimizers for any volumes if the kernel of the Riesz potential just vanishes at infinity. Finally, we study the asymptotic behavior of minimizers when the volume goes to infinity and we prove that a sequence of minimizers converges to the Euclidean ball up to translations if the kernel of the Riesz potential decays sufficiently fast.