Finite distortion functions and Douglas-Dirichlet functionals

Qingtian Shi

Displaying similar documents to “Finite distortion functions and Douglas-Dirichlet functionals”

Dirichlet's principle, distortion and related problems for harmonic mappings.

Mateljević, Miodrag (2004)

Publications de l'Institut Mathématique. Nouvelle Série

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A Maximum Principle for Harmonic Mappings Which Solve A Dirichlet Problem.

Jürgen Jost (1982)

Manuscripta mathematica

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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.

A counterexample to the $L^{p}$ -Hodge decomposition

Piotr Hajłasz (1996)

Banach Center Publications

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We construct a bounded domain $Ω \subset ℝ^{2}$ with the cone property and a harmonic function on Ω which belongs to $W_{0}^{1, p} (Ω)$ for all 1 ≤ p < 4/3. As a corollary we deduce that there is no $L^{p}$ -Hodge decomposition in $L^{p} (Ω, ℝ^{2})$ for all p > 4 and that the Dirichlet problem for the Laplace equation cannot be in general solved with the boundary data in $W^{1, p} (Ω)$ for all p > 4.

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.

Harmonic majorization of $| x_{1} |$ in subsets of $ℝ^{n}$ , $n \geq 2$ .

Eiderman, Vladimir, Essén, Matts (1996)

Annales Academiae Scientiarum Fennicae. Series A I. Mathematica

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Convolutions of harmonic right half-plane mappings

YingChun Li, ZhiHong Liu (2016)

Open Mathematics

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We first prove that the convolution of a normalized right half-plane mapping with another subclass of normalized right half-plane mappings with the dilatation [...] −z(a+z)/(1+az) $- z (a + z) / (1 + a z)$ is CHD (convex in the horizontal direction) provided [...] a=1 $a = 1$ or [...] −1≤a≤0 $- 1 \leq a \leq 0$ . Secondly, we give a simply method to prove the convolution of two special subclasses of harmonic univalent mappings in the right half-plane is CHD which was proved by Kumar et al. [1, Theorem 2.2]. In addition, we derive...

Harmonic mappings in the exterior of the unit disk

Magdalena Gregorczyk, Jarosław Widomski (2010)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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In this paper we consider a class of univalent orientation-preserving harmonic functions defined on the exterior of the unit disk which satisfy the condition $\sum_{n = 1}^{\infty} n^{p} (| a_{n} | + | b_{n} |) \leq 1$ . We are interested in finding radius of univalence and convexity for such class and we find extremal functions. Convolution, convex combination, and explicit quasiconformal extension for this class are also determined.

On separately subharmonic functions (Lelong’s problem)

A. Sadullaev (2011)

Annales de la faculté des sciences de Toulouse Mathématiques

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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 $S$ .

The Dirichlet problem with non-compact boundary.

Stephen J. Gardiner (1993)

Mathematische Zeitschrift

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Generalized Hölder type spaces of harmonic functions in the unit ball and half space

Alexey Karapetyants, Joel Esteban Restrepo (2020)

Czechoslovak Mathematical Journal

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We study spaces of Hölder type functions harmonic in the unit ball and half space with some smoothness conditions up to the boundary. The first type is the Hölder type space of harmonic functions with prescribed modulus of continuity $ω = ω (h)$ and the second is the variable exponent harmonic Hölder space with the continuity modulus ${| h |}^{λ (\cdot)}$ . We give a characterization of functions in these spaces in terms of the behavior of their derivatives near the boundary.