Flux in axiomatic potential theory. II. Duality

Bertram Walsh

Displaying similar documents to “Flux in axiomatic potential theory. II. Duality”

Perturbation of harmonic structures and an index-zero theorem

Bertram Walsh (1970)

Annales de l'institut Fourier

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In the framework of an axiomatic theory of sheaves of “harmonic” functions, a notion of perturbation of these sheaves is introduced which corresponds to the replacement of the operator $Δ$ by an operator $Δ + f$ , in the classical situation. The “harmonic” functions with which the paper is concerned are assumed to satisfy certain hypotheses (weaker than the axioms of Bauer); it is shown that the perturbed harmonic functions also satisfy these hypotheses. Moreover, the results obtained imply that...

The Martin boundaries of equivalent sheaves

John C. Taylor (1970)

Annales de l'institut Fourier

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The Martin compactification of $X$ defined by a Brelot sheaf $H_{1}$ satisfying proportionality is shown to be the same as for $H_{2}$ if the sheaves agree outside a compact set. Minimal points coincide and hence $S_{1}^{+}$ and $S_{2}^{+}$ are isomorphic topological cones. Nakai’s result on the extension to $X$ of a function harmonic outside a compact set is extended to Bauer’s theory. The connected components of the Martin boundary $Δ$ correspond to the ends of $X$ which are related to direct decomposition of the cone $H^{+}$ . ...

Biharmonic morphisms

Mustapha Chadli, Mohamed El Kadiri, Sabah Haddad (2005)

Commentationes Mathematicae Universitatis Carolinae

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Let $(X, ℋ)$ and $(X^{'}, ℋ^{'})$ be two strong biharmonic spaces in the sense of Smyrnelis whose associated harmonic spaces are Brelot spaces. A biharmonic morphism from $(X, ℋ)$ to $(X^{'}, ℋ^{'})$ is a continuous map from $X$ to $X^{'}$ which preserves the biharmonic structures of $X$ and $X^{'}$ . In the present work we study this notion and characterize in some cases the biharmonic morphisms between $X$ and $X^{'}$ in terms of harmonic morphisms between the harmonic spaces associated with $(X, ℋ)$ and $(X^{'}, ℋ^{'})$ and the coupling kernels of them.

Axiomatic theory of harmonic functions. Balayage

Nicu Boboc, Corneliu Constantinescu, A. Cornea (1965)

Annales de l'institut Fourier

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Dans une axiomatique des fonctions harmoniques un peu plus générale que celle de H. Bauer, on démontre les relations suivantes : $R_{s + t}^{A} = R_{s}^{A} + R_{t}^{A},$ $R_{s}^{A \cup B} + R_{s}^{A \cap B} \leq R_{s}^{A} + R_{s}^{B},$ $A_{n} ↑ A, S_{n} ↑ s \Rightarrow R_{s_{n}}^{A_{n}} ↑ R_{s}^{A},$ où $A$ , $B$ , $A_{n}$ , (resp. $s$ , $t$ , $s_{n}$ ) sont des ensembles (resp. fonctions hyperharmoniques non-négatives) arbitraires. Les mêmes relations sont valables pour $\hat{R}$ . On démontre aussi que la relation $\int^{*} s d μ^{A} = \int^{*} {\hat{R}}_{s}^{A} d μ$ a lieu si l’espace de base a une base dénombrable ou si l’axiome $D$ de M. Brelot est satisfait,...

A maximal regular boundary for solutions of elliptic differential equations

Peter Loeb, Bertram Walsh (1968)

Annales de l'institut Fourier

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Soit $𝒜$ une classe harmonique de Brelot, définie sur $W$ . Il est donné un critère de régularité en termes de barrières, pour les points d’une frontière idéale. Soit $ℌ$ un sous-treillis banachique de ${ℬ𝒜}_{W}$ . Si $𝒜$ est hyperbolique, la frontière idéale compactifiante déterminée par $ℌ$ contient une “frontière harmonique” $Γ_{ℌ}$ qui satisfait le critère de régularité et $ℌ ≅ 𝒞_{R} (Γ_{ℌ})$ . Entre autres applications, on a la théorie des frontières de Wiener et Royden et des comparaisons de classes harmoniques.