Displaying similar documents to “Infinite dimension of solutions of the Dirichlet problem”

On the boundary values of harmonic functions.

Paul R. Garabedian (1985)

Revista Matemática Iberoamericana

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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 problem with L p -boundary data in contractible domains of Carnot groups

Andrea Bonfiglioli, Ermanno Lanconelli (2006)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Let ℒ be a sub-laplacian on a stratified Lie group G . In this paper we study the Dirichlet problem for ℒ with L p -boundary data, on domains Ω which are contractible with respect to the natural dilations of G . One of the main difficulties we face is the presence of non-regular boundary points for the usual Dirichlet problem for ℒ . A potential theory approach is followed. The main results are applied to study a suitable notion of Hardy spaces.

Finite distortion functions and Douglas-Dirichlet functionals

Qingtian Shi (2019)

Czechoslovak Mathematical Journal

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In this paper, we estimate the Douglas-Dirichlet functionals of harmonic mappings, namely Euclidean harmonic mapping and flat harmonic mapping, by using the extremal dilatation of finite distortion functions with given boundary value on the unit circle. In addition, ∂ ¯ -Dirichlet functionals of harmonic mappings are also investigated.