On the Dirichlet problem for pseudodifferential operators generating Feller semigroups
type: Item
instance of: scholarly article
title: On the Dirichlet problem for pseudodifferential operators generating Feller semigroups
zbMATH Open document ID: 0901.47027
published in: Journal of Functional Analysis
publication date: 1998-11-16T00:00:00Z
review text: The authors prove that pseudodifferential operators \(p(x,D)\) with continuous symbols \(p:\mathbb{R}^n\times \mathbb{R}^n\rightarrow \mathbb{C}\), such that \(p(x,\cdot): \mathbb{R}^n\rightarrow \mathbb{C}\) is negative definite in the sense of A. Beurling and J. Deny, generate strong Feller semigroups under some natural additional assumptions. From this, it is shown that one can construct a balayage space starting with \(p(x,D)\). A first approach to the Dirichlet problem \[ p(x,D)u\bigl|_\Omega=0 \quad\text{and}\quad u\bigl|_{\Omega^c}= f\bigl|_{\Omega^c} \tag{1} \] is presented in the setting of this space. Here \(\Omega^c\) is the complement of \(\Omega\subset \mathbb{R}^n\). If \(p(x,D)\) is symmetric, a second approach is to work in the associated Dirichlet space. In particular, just using the Hilbert space structure in the Dirichlet space, one has a weak formulation of (1) and can apply results of \textit{N. Jacob} [Math. Nachr. 164, 63-89 (1993; Zbl 0817.35011)]. The authors also use the Dirichlet space structure via the spectral synthesis theorem and obtain the weak solution to (1) by an orthogonal projection of the data. They identify the solutions of (1) obtained by the first and second approaches, respectively. The key for this identification is a representation of the solution by using the Feller process associated with \(p(x,D)\). In the first approach, boundary regularity questions can be handled.
reviewed by: Rémi Vaillancourt
Mathematics Subject Classification ID: 47D07
Mathematics Subject Classification ID: 47G30
Mathematics Subject Classification ID: 35S15
Mathematics Subject Classification ID: 35B65
zbMATH DE Number: 871728
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: pseudodifferential operators
zbMATH Keywords: Feller semigroups
zbMATH Keywords: probabilistic potential theory
zbMATH Keywords: balayage space
zbMATH Keywords: spectral synthesis theorem
zbMATH Keywords: Feller process
author: Walter Hoh
author: Niels Jacob
MaRDI profile type: Publication
full work available at URL: jfan.1996.0039
OpenAlex ID: W2089077966
DOI: 10.1006/JFAN.1996.0039
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description: scientific article; zbMATH DE number 871728