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

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 $\partialΩ$. As shown recently by the author, the condition assures solvability of Dirichlet-type boundary problems for elliptic $P$ in full scales of Sobolev spaces with a singularity $d^{μ-k}$, $d(x)=\operatorname{dist}(x,\partialΩ)$. Examples include fractional Laplacians $(-Δ)^a$ 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 $R^n\setminus Ω$ reduce to problems supported on $\barΩ$, 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 $B^s_{\infty ,\infty}$. This leads to optimal Hölder estimates, e.g. for Dirichlet solutions of $(-Δ)^au=f\in L_\infty (Ω)$, $u\in d^aC^a(\barΩ)$ when $0<a<1$, $a\ne 1/2$ (in $d^aC^{a-ε}(\barΩ)$ when $a=1/2$).