## Analysis & PDE

• Anal. PDE
• Volume 7, Number 7 (2014), 1649-1682.

### Local and nonlocal boundary conditions for $\mu$-transmission and fractional elliptic pseudodifferential operators

Gerd Grubb

#### 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, 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 $Fp,qs$ 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, $a≠12$.

#### Article information

Source
Anal. PDE, Volume 7, Number 7 (2014), 1649-1682.

Dates
Revised: 25 August 2014
Accepted: 23 September 2014
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.apde/1513731607

Digital Object Identifier
doi:10.2140/apde.2014.7.1649

Mathematical Reviews number (MathSciNet)
MR3293447

Zentralblatt MATH identifier
1317.35310

#### Citation

Grubb, Gerd. Local and nonlocal boundary conditions for $\mu$-transmission and fractional elliptic pseudodifferential operators. Anal. PDE 7 (2014), no. 7, 1649--1682. doi:10.2140/apde.2014.7.1649. https://projecteuclid.org/euclid.apde/1513731607

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