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 R with N ≥ 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. Mathematics Subject Classification (2010). 49J40, 49Q10, 49Q20, 28A75.