Nonexistence of minimizers for a nonlocal perimeter with a Riesz and a background potential

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 $N\ge 2$ under the volume constraint. We show that the energy has no minimizer for a sufficiently large volume 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.

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