Pohozaev identities for anisotropic integro-differential operators

TL;DR

This paper derives Pohozaev identities and integration by parts formulas for anisotropic fractional operators of order 2s, involving boundary terms with the quantity u/d^s, extending classical identities to nonlocal anisotropic contexts.

Contribution

It establishes new Pohozaev identities and integration by parts formulas for anisotropic integro-differential operators of order 2s, incorporating boundary terms with u/d^s.

Findings

Derived Pohozaev identities for anisotropic fractional operators

Established boundary term expressions involving u/d^s

Extended classical identities to nonlocal anisotropic operators

Abstract

We establish Pohozaev identities and integration by parts type formulas for anisotropic integro-differential operators of order $2s$, with $s\in(0,1)$. These identities involve local boundary terms, in which the quantity $u/d^s|_{\partial\Omega}$ plays the role that $\partial u/\partial\nu$ plays in the second order case. Here, $u$ is any solution to $Lu=f(x,u)$ in $\Omega$, with $u=0$ in $\mathbb R^n\setminus\Omega$, and $d$ is the distance to $\partial\Omega$.