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

Abstract

Consider a classical elliptic pseudodifferential operator $P$ on ${\mathbb{R}}^n$ 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 $\Omega \subset {\mathbb{R}}^n$, we show an integration-by-parts formula with a boundary integral involving $(d^{-a}u){|}_{\partial \Omega }$, where $d(x)=\mathrm{dist}(x,\partial \Omega )$. 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.

Final version to appear in J. Differential Equations, 42 pages. References added

References in corpus (4)

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Cited by in corpus (6)

• On higher dimensional singularities for the fractional Yamabe problem: a non-local Mazzeo-Pacard program
• Exact Green's formula for the fractional Laplacian and perturbations
• Integration by parts for nonsymmetric fractional-order operators on a halfspace
• Semiclassical analysis of a nonlocal boundary value problem related to magnitude
• The principal transmission condition
• On Landis Conjecture for the Fractional Schrödinger Equation