An operational method to solve a Dirichlet problem for the Laplace operator in a plane sector. (English) Zbl 0735.35045
This important paper is devoted to the solution of a Dirichlet problem for the Laplace operator in a plane sector. Grisvard’s method and some results in operator theory concerning the closedness of the sum of two operators and the decomposition of a Banach space according to a decomposition of the spectrum of an operator are employed. Main result: The authors present a proof for the existence and uniqueness of solutions in suitably weighted \(W^{2,p}\)-spaces. Moreover a representation formula for the solution is given.
Reviewer: J.Lovíšek (Bratislava)
MSC:
35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
35J25 | Boundary value problems for second-order elliptic equations |
35C99 | Representations of solutions to partial differential equations |
47A60 | Functional calculus for linear operators |