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

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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.

Article information

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

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35S15: Boundary value problems for pseudodifferential operators
Secondary: 45E99: None of the above, but in this section 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems 58J40: Pseudodifferential and Fourier integral operators on manifolds [See also 35Sxx]

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


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.

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