A detailed analysis is presented of all pseudo-differential operators of orders up to 2 encountered in classical potential theory in two dimensions. Each of the operators under investigation turns out to be a sum of one or more of standard operators (second derivative, derivative of the Hilbert transform, etc.), and an integral operator with smooth kernel. This classification leads to an extremely simple analysis of spectra of such operators, and simplifies the design of procedures for their numerical evaluation. In a sequel to this paper, the obtained apparatus will be used to construct stable discretizations of arbitrarily high order for a variety of boundary value problems for elliptic partial differential equations.
All Science Journal Classification (ASJC) codes
- Applied Mathematics
- Hypersingular integral equations
- Laplace equation
- Potential theory
- Pseudo-differential operators