Pohozaev-like identity for the regional fractional laplacian

TL;DR

This paper develops a new integration by parts formula for the regional fractional Laplacian in smooth bounded domains, leading to a Pohozaev-like identity useful for eigenvalue problems and boundary unique continuation.

Contribution

It introduces a novel integration by parts formula for the regional fractional Laplacian and derives a Pohozaev-like identity with explicit remainder, advancing analysis of fractional PDEs.

Findings

Established a new integration by parts formula for the regional fractional Laplacian.

Derived a Pohozaev-like identity with explicit remainder term.

Applied results to eigenvalue problems in the unit ball.

Abstract

We establish a new integration by parts formula for the regional fractional laplacian $(-\Delta)^s_\Omega$ in bounded open sets of class $C^2$. As a direct application, we prove that weak solutions to the corresponding Dirichlet problem satisfy a Pohozaev-like identity with an explicit remainder term. We apply the later to eigenvalue problems in the unit ball and discuss its potential use in establishing boundary-type unique continuation properties.