Vol. 7, No. 7, 2014

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
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 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, this condition assures solvability of Dirichlet-type boundary problems for P in full scales of Sobolev spaces with a singularity dμk, d(x) = dist(x,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 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 Fp,qs and Bp,qs, including Hölder–Zygmund spaces B,s. This leads to optimal Hölder estimates, e.g., for Dirichlet solutions of (Δ)au = f L(Ω), u daCa(Ω ¯) when 0 < a < 1, a1 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