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

Grubb, Gerd

doi:10.2140/apde.2014.7.1649

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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

DOI: 10.2140/apde.2014.7.1649

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 $\partial Ω$ . 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 $d^{μ - k}$ , $d (x) = dist (x, \partial Ω)$ . 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 $ℝ^{n} ∖ Ω$ reduce to problems supported on $\bar{Ω}$ , 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 ${(- Δ)}^{a} u = f \in L_{\infty} (Ω)$ , $u \in d^{a} C^{a} (\bar{Ω})$ when $0 < a < 1$ , $a \neq \frac{1}{2}$ .

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

Milestones

Received: 8 April 2014

Revised: 25 August 2014

Accepted: 23 September 2014

Published: 12 December 2014

Authors

Gerd Grubb
Department of Mathematical Sciences Copenhagen University Universitetsparken 5 DK-2100 Copenhagen Denmark

Vol. 7, No. 7, 2014