Integration by parts for nonsymmetric fractional-order operators on a halfspace

Abstract

For a strongly elliptic pseudodifferential operator L of order 2a (0<a<1) with real kernel, we show an integration-by-parts formula for solutions of the homogeneous Dirichlet problem, in the model case where the operator is x-independent with homogeneous symbol, considered on the halfspace R+n. The new aspect compared to (−Δ)a is that L is nonsymmetric, having both an even and an odd part. Hence it satisfies a μ-transmission condition where generally μ≠a.We present a complex method, relying on a factorization in factors holomorphic in ξn in the lower or upper complex halfplane, using order-reducing operators combined with a decomposition principle originating from Wiener and Hopf. This is in contrast to a real, computational method presented very recently by Dipierro, Ros-Oton, Serra and Valdinoci. Our method allows μ in a larger range than they consider.Another new contribution is the (model) study of “large” solutions of nonhomogeneous Dirichlet problems when μ>0. Here we deduce a “halfways Green's formula” for L:∫R+nLuv¯dx−∫R+nuL⁎v‾dx=c∫Rn−1γ0(u/xnμ−1)γ0(v¯/xnμ⁎)dx′, when u solves a nonhomogeneous Dirichlet problem for L, and v solves a homogeneous Dirichlet problem for L⁎; μ⁎=2a−μ. Finally, we show a full Green's formula, when both u and v solve nonhomogeneous Dirichlet problems; here both Dirichlet and Neumann traces of u and v enter, as well as a first-order pseudodifferential operator over the boundary.