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2014 Local and nonlocal boundary conditions for $\mu$-transmission and fractional elliptic pseudodifferential operators
Gerd Grubb
Anal. PDE 7(7): 1649-1682 (2014). DOI: 10.2140/apde.2014.7.1649


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 nΩ 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=fL(Ω), udaCa(Ω¯) when 0<a<1, a12.


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Gerd Grubb. "Local and nonlocal boundary conditions for $\mu$-transmission and fractional elliptic pseudodifferential operators." Anal. PDE 7 (7) 1649 - 1682, 2014.


Received: 8 April 2014; Revised: 25 August 2014; Accepted: 23 September 2014; Published: 2014
First available in Project Euclid: 20 December 2017

zbMATH: 1317.35310
MathSciNet: MR3293447
Digital Object Identifier: 10.2140/apde.2014.7.1649

Primary: 35S15
Secondary: 45E99 , 46E35 , 58J40

Keywords: Besov–Triebel–Lizorkin spaces , boundary regularity , Dirichlet and Neumann conditions , elliptic pseudodifferential operators , fractional Laplacian , Hölder–Zygmund spaces , large solutions , singular integral operators , transmission properties

Rights: Copyright © 2014 Mathematical Sciences Publishers


Vol.7 • No. 7 • 2014
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