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

摘要

Consider a classical elliptic pseudodifferential operator P on 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 of R-n, we show an integration-by-parts formula with a boundary integral involving (d(-a)u)vertical bar(partial derivative Omega), where d(x)= dist (x, Omega partial derivative). 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 similar to P-P+, where we set up a calculus for the generalized pseudodifferential operators P-+/- that come out of the construction. (C) 2016 Elsevier Inc. All rights reserved.