# On overdetermind problems for a general class of nonlocal operators

## TL;DR

This paper investigates overdetermined boundary value problems for a broad class of nonlocal operators, demonstrating symmetry of solutions and domains using combined analytic and probabilistic methods.

## Contribution

It extends symmetry results to various nonlocal operators including fractional Laplacians and relativistic stable operators, in multiple domain types.

## Key findings

- Solutions and domains are radially symmetric under certain conditions.
- The methods combine analytic and probabilistic techniques.
- Results apply to bounded, exterior, and annular domains.

## Abstract

We study the overdetermined problem for a large family of non-local operators given by generators of subordinate Brownian motions. In particular, this family includes the fractional Laplacian, relativistic stable operators etc. We consider these problems in bounded domains, exterior domains, and in annular domains and we show that under suitable conditions, the domains and solutions are both radially symmetric. Our method uses both analytic and probabilistic tools.