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

Alexey Karapetyants; Joel Esteban Restrepo

Displaying similar documents to “Generalized Hölder type spaces of harmonic functions in the unit ball and half space”

Landau's theorem for p-harmonic mappings in several variables

Sh. Chen, S. Ponnusamy, X. Wang (2012)

Annales Polonici Mathematici

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A 2p-times continuously differentiable complex-valued function f = u + iv in a domain D ⊆ ℂ is p-harmonic if f satisfies the p-harmonic equation $Δ^{p} f = 0$ , where p (≥ 1) is a positive integer and Δ represents the complex Laplacian operator. If Ω ⊂ ℂⁿ is a domain, then a function $f : Ω \to ℂ^{m}$ is said to be p-harmonic in Ω if each component function $f_{i}$ (i∈ 1,...,m) of $f = (f ₁, . . ., f_{m})$ is p-harmonic with respect to each variable separately. In this paper, we prove Landau and Bloch’s theorem for a class of p-harmonic mappings...

A class of functions containing polyharmonic functions in ℝⁿ

V. Anandam, M. Damlakhi (2003)

Annales Polonici Mathematici

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Some properties of the functions of the form $v (x) = \sum_{i = 0}^{m} {| x |}^{i} h_{i} (x)$ in ℝⁿ, n ≥ 2, where each $h_{i}$ is a harmonic function defined outside a compact set, are obtained using the harmonic measures.

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

Fanghua Lin, Tristan Rivière (1999)

Journal of the European Mathematical Society

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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...

A critical growth rate for harmonic and subharmonic functions in the open ball in $R^{n}$

Krzysztof Samotij (1987)

Colloquium Mathematicae

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On Polynomially Bounded Harmonic Functions on the $Z^{d}$ Lattice

Piotr Nayar (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

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We prove that if $f : Z^{d} \to R$ is harmonic and there exists a polynomial $W : Z^{d} \to R$ such that f + W is nonnegative, then f is a polynomial.

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 $S$ .

Uniform bounds for quotients of Green functions on $C^{1, 1}$ -domains

H. Hueber, M. Sieveking (1982)

Annales de l'institut Fourier

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Let $Δ u = Σ_{i} \frac{\partial^{2}}{\partial_{x_{i}}^{2}}$ , $L u = Σ_{i, j} a_{i j} \frac{\partial^{2}}{\partial x_{i} \partial x_{j}} u + Σ_{i} b_{i} \frac{\partial}{\partial x_{i}} u + c u$ be elliptic operators with Hölder continuous coefficients on a bounded domain $Ω \subset R^{n}$ of class $C^{1, 1}$ . There is a constant $c > 0$ depending only on the Hölder norms of the coefficients of $L$ and its constant of ellipticity such that $c^{- 1} G_{Δ}^{Ω} \leq G_{L}^{Ω} \leq c G_{Δ}^{Ω} on Ω \times Ω,$ where $γ_{Δ}^{Ω}$ (resp. $G_{L}^{Ω}$ ) are the Green functions of $Δ$ (resp. $L$ ) on $Ω$ .

Duality on vector-valued weighted harmonic Bergman spaces

Salvador Pérez-Esteva (1996)

Studia Mathematica

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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*.

On the order of starlikeness and convexity of complex harmonic functions with a two-parameter coefficient condition

Agnieszka Sibelska (2010)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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The article of J. Clunie and T. Sheil-Small [3], published in 1984, intensified the investigations of complex functions harmonic in the unit disc $Δ$ . In particular, many papers about some classes of complex mappings with the coefficient conditions have been published. Consideration of this type was undertaken in the period 1998–2004 by Y. Avci and E. Złotkiewicz [2], A. Ganczar [5], Z. J. Jakubowski, G. Adamczyk, A. Łazinska and A. Sibelska [1], [8], [7], H. Silverman [12] and J. M. Jahangiri...

On a question of T. Sheil-Small regarding valency of harmonic maps

Daoud Bshouty, Abdallah Lyzzaik (2012)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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The aim of this work is to answer positively a more general question than the following which is due to T. Sheil-Small: Does the harmonic extension in the open unit disc of a mapping f from the unit circle into itself of the form $f (e^{i t}) = e^{i φ (t)}$ , $0 \leq t \leq 2 π$ where $φ$ is a continuously non-decreasing function that satisfies $φ (2 π) - φ (0) = 2 N π$ , assume every value finitely many times in the disc?

Liouville theorems for self-similar solutions of heat flows

Jiayu Li, Meng Wang (2009)

Journal of the European Mathematical Society

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Let $N$ be a compact Riemannian manifold. A quasi-harmonic sphere on $N$ is a harmonic map from $(ℝ^{m}, e^{{| x |}^{2} / 2 (m - 2)} / d s_{0}^{2})$ to $N$ ( $m \geq 3$ ) with finite energy ([LnW]). Here $d s 2_{0}$ is the Euclidean metric in $ℝ^{m}$ . Such maps arise from the blow-up analysis of the heat flow at a singular point. In this paper, we prove some kinds of Liouville theorems for the quasi-harmonic spheres. It is clear that the Liouville theorems imply the existence of the heat flow to the target $N$ . We also derive gradient estimates and Liouville theorems...

On the fusion problem for degenerate elliptic equations II

Stephen M. Buckley, Pekka Koskela (1999)

Commentationes Mathematicae Universitatis Carolinae

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Let $F$ be a relatively closed subset of a Euclidean domain $Ω$ . We investigate when solutions $u$ to certain elliptic equations on $Ω ∖ F$ are restrictions of solutions on all of $Ω$ . Specifically, we show that if $\partial F$ is not too large, and $u$ has a suitable decay rate near $F$ , then $u$ can be so extended.

Harmonie reflections

Lieven Vanhecke, Maria-Elena Vazquez-Abal (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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We study local reflections $\phi_{\sigma}$ with respect to a curve $\sigma$ in a Riemannian manifold and prove that $\sigma$ is a geodesic if $\phi_{\sigma}$ is a harmonic map. Moreover, we prove that the Riemannian manifold has constant curvature if and only if $\phi_{\sigma}$ is harmonic for all geodesies $\sigma$ .

Harmonic functions in four variables with rational and algebraic $p_{4}$ -associates

R. P. Gilbert (1964)

Annales Polonici Mathematici

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