On the Dirichlet problem for pseudodifferential operators generating Feller semigroups
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.
scientific article; zbMATH DE number 871728