Integration by parts and Pohozaev identities for space-dependent fractional-order operators

TL;DR

This paper extends integration-by-parts and Pohozaev identities to space-dependent fractional-order operators, enabling analysis of boundary behavior and solution properties for a broader class of elliptic pseudodifferential operators.

Contribution

It generalizes recent boundary identities to operators with variable coefficients, nonsymmetry, and lower-order terms, using a novel factorization approach.

Findings

Derived boundary integration-by-parts formula for space-dependent fractional operators.

Extended Pohozaev identities to nonsymmetric and variable coefficient operators.

Provided applications to unique continuation and nonexistence results.

Abstract

Consider a classical elliptic pseudodifferential operator $P$ on ${\Bbb R}^n$ of order $2a$ ($0<a<1)$ with even symbol. For example, $P=A(x,D)^a$ where $A(x,D)$ is a second-order strongly elliptic differential operator; the fractional Laplacian $(-\Delta )^a$ is a particular case. For solutions $u$ of the Dirichlet problem on a bounded smooth subset $\Omega \subset{\Bbb R}^n$, we show an integration-by-parts formula with a boundary integral involving $(d^{-a}u)|_{\partial\Omega }$, where $d(x)=\operatorname{dist}(x,\partial\Omega )$. This extends recent results of Ros-Oton, Serra and Valdinoci, to operators that are $x$-dependent, nonsymmetric, and have lower-order parts. We also generalize their formula of Pohozaev-type, that can be used to prove unique continuation properties, and nonexistence of nontrivial solutions of semilinear problems. An illustration is given with $P=(-\Delta +m^2)^a$. The basic step in our analysis is a factorization of $P$, $P\sim P^-P^+$, where we set up a calculus for the generalized pseudodifferential operators $P^\pm$ that come out of the construction.