Displaying similar documents to “Constructing a Laplacian on the diamond fractal.”

Infinite dimension of solutions of the Dirichlet problem

Vladimir Ryazanov (2015)

Open Mathematics

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It is proved that the space of solutions of the Dirichlet problem for the harmonic functions in the unit disk with nontangential boundary limits 0 a.e. has the infinite dimension.

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.

Domains of Dirichlet forms and effective resistance estimates on p.c.f. fractals

Jiaxin Hu, Xingsheng Wang (2006)

Studia Mathematica

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We consider post-critically finite self-similar fractals with regular harmonic structures. We first obtain effective resistance estimates in terms of the Euclidean metric, which in particular imply the embedding theorem for the domains of the Dirichlet forms associated with the harmonic structures. We then characterize the domains of the Dirichlet forms.

Probability and a Dirichlet problem for multiply superharmonic functions

John B. Walsh (1968)

Annales de l'institut Fourier

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Soit ℋ un préfaisceau complet de fonctions “harmoniques” définies sur W . Un critère de régularité pour les points des frontières idéales de W est donné. Pour chaque sous-treillis banachique ℌ de ℬℋ W , il existe une frontière idéale qui compactifie W et qui contient une “frontière harmonique” Γ ℌ qui est l’ensemble des points réguliers ; ℌ est isométriquement isomorphe à 𝒞 ( Γ ℌ ) Parmi des applications se trouvent les théories frontières de Wiener et Royden et aussi les classes comparables harmoniques. ...