# Hopf's lemma for viscosity solutions to a class of non-local equations with applications

## TL;DR

This paper extends Hopf's lemma and maximum principles to a broad class of non-local integro-differential equations using symbol analysis, with applications to eigenvalue problems and symmetry of solutions.

## Contribution

It introduces a non-local Hopf's lemma expressed via the operator's symbol, advancing boundary analysis for viscosity solutions of integro-differential equations.

## Key findings

- Established a non-local Hopf's lemma for viscosity solutions.
- Derived maximum principles for a wide class of non-local equations.
- Applied results to eigenvalue problems and symmetry of solutions.

## Abstract

We consider a large family of integro-differential equations and establish a non-local counterpart of Hopf's lemma, directly expressed in terms of the symbol of the operator. As closely related problems, we also obtain a variety of maximum principles for viscosity solutions. In our approach we combine direct analysis with functional integration, allowing a robust control around the boundary of the domain, and make use of the related ascending ladder height-processes. We then apply these results to a study of principal eigenvalue problems, the radial symmetry of the positive solutions, and the overdetermined non-local torsion equation.