Vol. 7, No. 7, 2014

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

Gerd Grubb

Vol. 7 (2014), No. 7, 1649–1682
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, $a\ne \frac{1}{2}$.

Keywords
fractional Laplacian, boundary regularity, Dirichlet and Neumann conditions, large solutions, Hölder–Zygmund spaces, Besov–Triebel–Lizorkin spaces, transmission properties, elliptic pseudodifferential operators, singular integral operators
Mathematical Subject Classification 2010
Primary: 35S15
Secondary: 45E99, 46E35, 58J40