Local and nonlocal boundary conditions for \(\mu\)-transmission and fractional elliptic pseudodifferential operators. (English) Zbl 1317.35310

Grubb, Gerd

Anal. PDE 7, No. 7, 1649-1682 (2014).

The interesting paper under review deals with \(\mu\)-transmission and fractional elliptic pseudodifferential operators. More precisely, a classical pseudodifferential operator \(P\) on \(\mathbb{R}^n\) satisfies the \(\mu\)-transmission condition relative to a smooth open subset \(\Omega\) when the symbol terms have a certain twisted parity on the normal to \(\partial\Omega.\) As shown recently by the author [Adv. Math. 268, 478–528 (2015; Zbl 1318.47064)], that condition ensures solvability for the Dirichlet boundary value problems for the operator \(P\) in full scales of Sobolev spaces with a singularity of the type \(d^{\mu-k}\) with \(d(x)= \text{dist} (x,\partial\Omega).\) Known examples include fractional Laplacians \((-\Delta)^a\) and complex powers of strongly elliptic PDE.
The author introduces here new types of boundary conditions, especially of Neumann type, or, more generally, of nonlocal type. It is shown how problems with data on \(\mathbb{R}^n\setminus\Omega\) reduce to problems supported on \(\overline{\Omega}\) and how the so-called “large” solutions arise. Moreover, the results are extended to Triebel-Lizorkin and Besov spaces \(F^s_{p,q}\) and \(B^s_{p,q},\) including Hölder-Zygmund spaces \(B^s_{\infty,\infty}.\) This leads to optimal Hölder estimates for Dirichlet solutions of \((-\Delta)^au=f\in L^\infty(\Omega),\) \(u\in d^{a}C^a(\overline{\Omega})\) with \(a\in(0,1),\) \(a\neq 1/2.\)

Reviewer: Dian K. Palagachev (Bari)

MSC:

35S15	Boundary value problems for PDEs with pseudodifferential operators
35R11	Fractional partial differential equations
46E35	Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
58J40	Pseudodifferential and Fourier integral operators on manifolds

Keywords:

fractional Laplacian; boundary regularity; Dirichlet condition; Neumann condition; Hölder-Zygmund spaces; Besov-Triebel-Lizorkin spaces; transmission properties; elliptic pseudodifferential operators; singular integral operators

Citations:

Zbl 1318.47064