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

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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.

Keywords: 35J05; 65N99; AMS Subject Classifications: 35J40; Elliptic boundary value problem; Numerical solution

Document Type: Research Article


Affiliations: Communicated by V. Maz'ya

Publication date: August 1, 2005

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