On the Boundary Values of Harmonic Functions.
- Autores: P.R. Garabedian
- Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 1, Nº 2, 1985, págs. 33-37
- Idioma: inglés
- Títulos paralelos:
- Sobre los valores en la frontera de las funciones armónicas.
- Enlaces
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Resumen
Over the years many methods have been discovered to prove the existence of a solution of the Dirichlet problem for Laplace's equation. A fairly recent collection of proofs is based on representations of the Green's functions in terms of the Bergman kernel function or some equivalent linear operator [3]. Perhaps the most fundamental of these approaches involves the projection of an arbitrary function onto the class of harmonic functions in a Hilbert space whose norm is defined by the Dirichlet integral [5]. Here a problem has remained open concerning continuity at the boundary of the solution that is constructed by orthogonal projection. Past discussion of this question turned out to be successful in spaces of two or three dimensions, but failed for larger numbers of independent variables [2]. It is the purpose of the present note to remove any such restriction and simultaneously to give a concise treatment of the boundary condition that is applicable to other existence proofs.
