On the boundary values of harmonic functions.

Garabedian, Paul R.. "On the boundary values of harmonic functions.." Revista Matemática Iberoamericana 1.2 (1985): 33-37. <http://eudml.org/doc/39319>.

@article{Garabedian1985,
abstract = {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.},
author = {Garabedian, Paul R.},
journal = {Revista Matemática Iberoamericana},
keywords = {Ecuación de Laplace; Función de Green; Función armónica; smooth boundary; finite Dirichlet integral; Dirichlet solution; Hilbert space; harmonic functions; orthogonal projection},
language = {eng},
number = {2},
pages = {33-37},
title = {On the boundary values of harmonic functions.},
url = {http://eudml.org/doc/39319},
volume = {1},
year = {1985},
}

TY - JOUR
AU - Garabedian, Paul R.
TI - On the boundary values of harmonic functions.
JO - Revista Matemática Iberoamericana
PY - 1985
VL - 1
IS - 2
SP - 33
EP - 37
AB - 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.
LA - eng
KW - Ecuación de Laplace; Función de Green; Función armónica; smooth boundary; finite Dirichlet integral; Dirichlet solution; Hilbert space; harmonic functions; orthogonal projection
UR - http://eudml.org/doc/39319
ER -