Nonexistence of Minimizers for a Nonlocal Perimeter Functional with a Riesz and a Background Potential

TL;DR

This paper proves that for large mass, the energy functional combining a nonlocal perimeter, Riesz potential, and background potential has no minimizers, highlighting limitations in variational problems with such nonlocal interactions.

Contribution

It establishes the nonexistence of minimizers for a broad class of nonlocal perimeter functionals with Riesz and background potentials at large mass.

Findings

No minimizers exist for sufficiently large mass.

The proof uses partitioning and energy comparison techniques.

Results extend previous work on nonlocal variational problems.

Abstract

We consider the nonexistence of minimizers for the energy containing a nonlocal perimeter with a general kernel $K$, a Riesz potential, and a background potential in $\mathbb{R}^N$ with $N\geq2$ under the volume constraint. We show that the energy has no minimizer for a sufficiently large mass under suitable assumptions on $K$. The proof is based on the partition of a minimizer and the comparison of the sum of the energy for each part with the energy for the original configuration. This strategy is shown in [R.A. Frank, R. Killip, P.T. Nam, 2016] and [D.A. La Manna, 2018]