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

TY - JOUR

T1 - Local and nonlocal boundary conditions for μ-transmission and fractional elliptic pseudodifferential operators

AU - Grubb, Gerd

PY - 2014

Y1 - 2014

N2 - 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, this condition assures solvability of Dirichlet-type boundary problems for P in full scales of Sobolev spaces with a singularity dμ-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 generally, nonlocal type. 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 Fsp, q and Bs p, q, including Hölder-Zygmund spaces Bs∞,∞. This leads to optimal Hölder estimates, e.g., for Dirichlet solutions of (-Δ)au = f ∈ L∞(Ω), u ∈ daCa(Ω~)when 0

AB - 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, this condition assures solvability of Dirichlet-type boundary problems for P in full scales of Sobolev spaces with a singularity dμ-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 generally, nonlocal type. 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 Fsp, q and Bs p, q, including Hölder-Zygmund spaces Bs∞,∞. This leads to optimal Hölder estimates, e.g., for Dirichlet solutions of (-Δ)au = f ∈ L∞(Ω), u ∈ daCa(Ω~)when 0

U2 - 10.2140/apde.2014.7.1649

DO - 10.2140/apde.2014.7.1649

M3 - Journal article

SN - 2157-5045

VL - 7

SP - 1649

EP - 1682

JO - Analysis and PDE

JF - Analysis and PDE

IS - 7

ER -