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

2014/03/31 by Gerd Grubb · 1 citation
Mathematics · #math.AP #math.FA #msc:35J75 #msc:35S15 #msc:45E99 #msc:46E35 #msc:58J40

paper · pdf · doi:10.2140/apde.2014.7.1649

published as Anal. PDE 7 (2014) 1649-1682 · Title slightly changed, 34 pages

arxiv created 2014/12/20 · arxiv updated 2016/01/20

Abstract

A classical pseudodifferential operator P on Rⁿ 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, the condition assures solvability of Dirichlet-type boundary problems for elliptic P in full scales of Sobolev spaces with a singularity d^μ-k, 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 general nonlocal. It is also shown how problems with data on Rⁿ∖ Ω reduce to problems supported on Ω, and how the so-called "large" solutions arise. Moreover, the results are extended to general function spaces F^s_p,q and B^s_p,q, including Hölder-Zygmund spaces B^s_{∞ ,∞}. This leads to optimal Hölder estimates, e.g. for Dirichlet solutions of (-Δ)^au=f∈ L_∞ (Ω), u∈ d^aC^a(Ω) when 0<a<1, a≠ 1/2 (in d^aC^a-ε(Ω) when a=1/2).

Cited by

• Boundary Control and Calderón type Inverse Problems in Non-local heat equation