Singular solutions of elliptic boundary value problems in polyhedra. (English) Zbl 0567.35025

Grisvard, P.

Port. Math. 41, 367-382 (1982).

The behaviour of solutions of elliptic equations near the boundary \(\partial \Omega\) (which is supposed to be a polyhedron) is studied on a simple model example of a Dirichlet boundary value problem for the Laplace equation on an open bounded set \(\Omega \subset R^ 3\). The right hand side is supposed to belong to the Sobolev space \(H^ m(\Omega)\). In a very nice exposition the author describes sets of singular solutions near the edge (or near the vertex) of \(\Omega\) having the following property: the difference between the solution u and a suitable linear combination of singular solutions belong to \(H^{m+2}\) on a neighbourhood of the edge or the vertex.

Reviewer: J.Stará

MSC:

35J25	Boundary value problems for second-order elliptic equations
35B65	Smoothness and regularity of solutions to PDEs
35D10	Regularity of generalized solutions of PDE (MSC2000)
46E35	Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems

Keywords:

domain with corners; regularity; Dirichlet boundary value problem; Laplace equation; Sobolev space; singular solutions