Local and nonlocal boundary conditions forμ-transmission and fractional elliptic pseudodifferential operators

Abstract

Abstract. A classical pseudodifferential operator P on R n 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, the condition assures solvability of Dirichlet-type boundary problems for P in full scales of Sobolev spaces with a singularity d µ−k , d(x) = dist(x, ∂Ω). Examples include fractional Laplacians (−∆) a and complex powers of strongly elliptic PDE.We now introduce new boundary conditions, of Neumann type or more general nonlocal. It is also shown how problems with data on R n \ Ω reduce to problems supported on Ω, and how the so-called "large" solutions arise. Moreover, the results are extended to general function spaces F s p,q and B s p,q , including Hölder-Zygmund spaces B s ∞,∞ . This leads to optimal Hölder estimates, e.g. for Dirichlet solutions of (−∆) a u = f ∈ L ∞ (Ω), u ∈ d a C a (Ω) when 0 < a < 1, a = 1 2 .

Regularity of spectral fractional Dirichlet and Neumann problems

“…The complex powers of can be defined by spectral theory in in the cases where is selfadjoint, but Seeley has shown in how the powers can be defined more generally in a consistent way, acting in ‐based Sobolev spaces (), by a Cauchy integral of the resolvent around the spectrum Here is the set of distributions u (functions if ) such that , and (denoted in , ), where stands for restriction to Ω. The general point of view is that the resolvent is constructed as an integral operator (found here by pseudodifferential methods) that can be applied to various function spaces, e.g.…”

“…4.3.3.) We here use a notation of , , where stands for the space of functions in with support in .…”

“…Escher and Seiler , can be or has been included for these boundary value problems. (It was possible to include in the regularity study for the restricted fractional Laplacian in using Johnsen .) Such an extension would allow removing the “−0” in some formulas in Corollaries and above.…”

“…More generally, one can replace by an elliptic pseudodifferential operator P of order 2 a having the so‐called a ‐transmission property at , cf. , .…”

“…For the restricted Dirichlet fractional Laplacian, detailed regularity properties of solutions of in Hölder spaces and Sobolev spaces have just recently been shown, in Ros‐Oton and Serra –, Grubb , .…”