On explicit solvability of an elliptic boundary value problem and its application

Abstract

A homogeneous boundary condition is constructed for the equation (I − Δ)u = f in an arbitrary bounded or exterior domain Ω ⊆ (I and Δ being the identity operator and the Laplacian), which generates a boundary value problem with an explicit formula of the solution u. The problem creates an isomorphism between the appropriate Sobolev spaces with an explicitly written inverse operator. In the article, all results are obtained not just for the operator I − Δ but also for an arbitrary elliptic differential operator in of an even order with constant coefficients. As an application, the usual Dirichlet boundary value problem for the homogeneous equation (I − Δ)u = 0 in a bounded or exterior domain is reduced to an integral equation in a thin boundary layer. An approximate solution of the integral equation generates a rather simple new numerical algorithm solving the 2D and 3D Dirichlet problem.