Uniform approximation of harmonic functions

G. F. Vincent-Smith

Displaying similar documents to “Uniform approximation of harmonic functions”

The Martin boundaries of equivalent sheaves

John C. Taylor (1970)

Annales de l'institut Fourier

Similarity:

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 $H^{+}$ . ...

Complex Ginzburg-Landau equations in high dimensions and codimension two area minimizing currents

Fanghua Lin, Tristan Rivière (1999)

Journal of the European Mathematical Society

Similarity:

There is an obvious topological obstruction for a finite energy unimodular harmonic extension of a $S^{1}$ -valued function defined on the boundary of a bounded regular domain of $R^{n}$ . When such extensions do not exist, we use the Ginzburg-Landau relaxation procedure. We prove that, up to a subsequence, a sequence of Ginzburg-Landau minimizers, as the coupling parameter tends to infinity, converges to a unimodular harmonic map away from a codimension-2 minimal current minimizing the area within...

Harmonic measures for symmetric stable processes

Jang-Mei Wu (2002)

Studia Mathematica

Similarity:

Let D be an open set in ℝⁿ (n ≥ 2) and ω(·,D) be the harmonic measure on $D^{c}$ with respect to the symmetric α-stable process (0 < α < 2) killed upon leaving D. We study inequalities on volumes or capacities which imply that a set S on ∂D has zero harmonic measure and others which imply that S has positive harmonic measure. In general, it is the relative sizes of the sets S and $D^{c} ∖ S$ that determine whether ω(S,D) is zero or positive.

On separately subharmonic functions (Lelong’s problem)

A. Sadullaev (2011)

Annales de la faculté des sciences de Toulouse Mathématiques

Similarity:

The main result of the present paper is : every separately-subharmonic function $u (x, y)$ , which is harmonic in $y$ , can be represented locally as a sum two functions, $u = u^{*} + U$ , where $U$ is subharmonic and $u^{*}$ is harmonic in $y$ , subharmonic in $x$ and harmonic in $(x, y)$ outside of some nowhere dense set $S$ .

Axiomatic theory of harmonic functions. Balayage

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

Annales de l'institut Fourier

Similarity:

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

Similarity:

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.

Relative Fatou theorem for harmonic functions of rotation invariant stable processes in smooth domains

Krzysztof Bogdan, Bartłomiej Dyda (2003)

Studia Mathematica

Similarity:

For $C^{1, 1}$ domains we give exact asymptotics near the domain’s boundary for the Green function and Martin kernel of the rotation invariant α-stable Lévy process. We also obtain a relative Fatou theorem for harmonic functions of the stable process.

Duality on vector-valued weighted harmonic Bergman spaces

Salvador Pérez-Esteva (1996)

Studia Mathematica

Similarity:

We study the duals of the spaces $A^{p α} (X)$ of harmonic functions in the unit ball of $ℝ^{n}$ with values in a Banach space X, belonging to the Bochner $L^{p}$ space with weight ${(1 - | x |)}^{α}$ , denoted by $L^{p α} (X)$ . For 0 < α < p-1 we construct continuous projections onto $A^{p α} (X)$ providing a decomposition $L^{p α} (X) = A^{p α} (X) + M^{p α} (X)$ . We discuss the conditions on p, α and X for which $A^{p α} (X) * = A^{q α} (X *)$ and $M^{p α} (X) * = M^{q α} (X *)$ , 1/p+1/q = 1. The last equality is equivalent to the Radon-Nikodým property of X*.