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 & PDE
JF - Analysis & PDE
IS - 7
ER -