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

Abstract

A classical pseudodifferential operator $P$ on ${ℝ}^n$ satisfies the $\mu$-transmission condition relative to a smooth open subset $\Omega$ when the symbol terms have a certain twisted parity on the normal to $\partial \Omega$. 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 $d^{\mu -k}$, $d\left(x\right)=dist\left(x,\partial \Omega \right)$. Examples include fractional Laplacians ${\left(-\Delta \right)}^a$ 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 ${ℝ}^n\setminus \Omega$ reduce to problems supported on $\overline{\Omega }$, and how the so-called “large” solutions arise. Moreover, the results are extended to general function spaces $F_{p,q}^s$ and $B_{p,q}^s$, including Hölder–Zygmund spaces $B_{\infty ,\infty }^s$. This leads to optimal Hölder estimates, e.g., for Dirichlet solutions of ${\left(-\Delta \right)}^{a}u=f\in L_{\infty }\left(\Omega \right)$, $u\in d^a{C}^a\left(\overline{\Omega }\right)$ when $0<a<1$, $a\ne \frac{1}{2}$.