Local and nonlocal boundary conditions for μ-transmission and fractional elliptic pseudodifferential operators

Abstract

A classical pseudodifferential operator P on Rn satisfies the μ-transmission condition relative to a smooth open subset Ω when the symbol terms have a certain twisted parity on the normal to ∂Ω. As shown recently by the author, this condition assures solvability of Dirichlet-type boundary problems for P in full scales of Sobolev spaces with a singularity dμ-k, d(x)=dist(x, ∂Ω). Examples include fractional Laplacians (-Δ)a and complex powers of strongly elliptic PDE. We now introduce new boundary conditions, of Neumann type, or, more generally, nonlocal type. It is also shown how problems with data on Rn\Ω reduce to problems supported on Ω~, and how the so-called "large" solutions arise. Moreover, the results are extended to general function spaces Fsp, q and Bsp, q, including Hölder-Zygmund spaces Bs∞,∞. This leads to optimal Hölder estimates, e.g., for Dirichlet solutions of (-Δ)au = f ∈ L∞(Ω), u ∈ daCa(Ω~)when 0