A class of overdetermined problem for fractional Capacity

TL;DR

This paper investigates a novel overdetermined problem related to fractional capacity, using concavity properties and Brunn-Minkowski inequalities to characterize geometric shapes like balls, extending previous p-capacity results.

Contribution

It introduces a new overdetermined problem framework for fractional capacity and generalizes p-capacity characterizations using concavity and geometric inequalities.

Findings

Characterization of balls via concavity and Brunn-Minkowski inequalities.

Extension of p-capacity concepts to fractional capacity.

New insights into geometric properties related to fractional capacities.

Abstract

In this paper, we consider an unconventional overdetermined problem through a property of concavity, which provides some characterizations of balls via Brunn-Minkowski inequalities. In this setting, our rsults can be viewed as the generalization of $p$-capacity in [14], which have its own interest.