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

الأصل · EN

Consider a classical elliptic pseudodifferential operator P on Rⁿ 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 (-Δ)ᵃ is a particular case. For solutions u of the Dirichlet problem on a bounded smooth subset Ω⊂ Rⁿ, we show an integration-by-parts formula with a boundary integral involving (d⁻ᵃu)|∂Ω, where d(x)=dist(x,∂Ω). 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=(-Δ+m²)ᵃ. The basic step in our analysis is a factorization of P, P P-P+, where we set up a calculus for the generalized pseudodifferential operators P± that come out of the construction.