Inner estimate and quasiconformal harmonic maps between smooth domains

Kalaj, David; Mateljević, Miodrag

doi:10.1007/bf02916757

Cited by 68 publications

(66 citation statements)

References 17 publications

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“…and satisfies Pisson differential inequality, then the function χ(x) = −d(u(x)), where d(u) = dist(u, ∂Ω), satisfies as well Pisson differential inequality in some neighborhood of the boundary. By using this fact and Theorem B we prove Theorem C. This extends some results of the author, Mateljevic and Pavlovic ([23], [26], [20], [21] and [32]) from the plane to the space. It is important to notice that, the conformal mappings and decomposition of planar harmonic mappings as the sum of an analytic and an anti-analytic function played important role in establishing some regularity boundary behaviors of q.c.…”

Section: Introduction and Statement Of Main Resultssupporting

confidence: 83%

“…An application of Theorem B yields the following theorem which is a generalization of analogous theorems for plane domains due to the author and Mateljevic (see [23] and [20]). …”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 89%

“…One of the starting points of this paper is the following theorem which was one of the main tools in proving some recent results of the author and Mateljevic (see [21] and [20] [3] and [13]). Let s : U → R (s : U → B m ) be a continuous function from the closed unit disc U into the real line (closed unit ball) satisfying the conditions:…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

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A priori estimate of gradient of a solution of a certain differential inequality and quasiconformal mappings

Kalaj

2013

JAMA

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Abstract. We will prove a global estimate for the gradient of the solution to the Poisson differential inequality |∆u(x)| ≤ a|∇u(x)| 2 + b, x ∈ B n , where a, b < ∞ and u| S n−1 ∈ C 1,α (S n−1 , R m ). If m = 1 and a ≤ (n + 1)/(|u|∞4n √ n), then |∇u| is a priori bounded. This generalizes some similar results due to E. Heinz ([13]) and Bernstein ([3]) for the plane. An application of these results yields the theorem, which is the main result of the paper: A quasiconformal mapping of the unit ball onto a domain with C 2 smooth boundary, satisfying the Poisson differential inequality, is Lipschitz continuous. This extends some results of the author, Mateljević and Pavlović from the complex plane to the space.

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Section: Introduction and Statement Of Main Resultssupporting

confidence: 83%

“…An application of Theorem B yields the following theorem which is a generalization of analogous theorems for plane domains due to the author and Mateljevic (see [23] and [20]). …”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 89%

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

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A priori estimate of gradient of a solution of a certain differential inequality and quasiconformal mappings

Kalaj

2013

JAMA

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“…Hence, the image domains of K-qch mappings can not be always confined to a canonical domain such as the unit disk or the upper half plane. Kalaj [9][10][11][12][13] did a lot of work on studying the Lipschitz continuity for different image domains from D, for example he proved that every K-qch mapping between a Jordan domain with C 1,α (α < 1) and a Jordan domain with C 1,1 compact boundary is bi-Lipschitz [9]. The fact that every K-qch mapping of D or H onto itself is hyperbolically bi-Lipschitz has been showed by Knežević and Mateljević [14].…”

Section: P(z T)w(t) Dtmentioning

confidence: 99%

(K,K')-Quasiconformal harmonic mappings of the upper half plane onto itself

Chen¹,

Chen²

2012

Ann. Acad. Sci. Fenn. Math.

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Abstract. In this paper we show that (K, K )-quasiconformal mappings with unbounded image domains are not Hölder continuous, which is different from the case with bounded image domains given by Kalaj and Mateljević. For a (K, K )-quasiconformal harmonic mapping of the upper half plane onto itself, we prove that it is Lipschitz and hyperbolically Lipschitz continuous. Moreover, we get four equivalent conditions for a harmonic mapping of the upper half plane onto itself to be a (K, K )-quasiconformal mapping.

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“…Martio [1968] was the first to consider harmonic quasiconformal mappings on the complex plane. Recent papers [Kalaj 2004[Kalaj , 2008Kalaj and Mateljević 2006;Pavlović 2005, 2009;Manojlović 2009;Pavlović 2002] shed much light on this topic. Proposition 1.1 [Kalaj 2009].…”

Section: Introductionmentioning

confidence: 99%

Harmonic quasiconformal self-mappings and Möbius transformations of the unit ball

Kalaj¹,

Mateljević²

2010

Pacific J. Math.

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We prove in dimension n > 2 that a K -quasiconformal harmonic mapping u of the unit ball B n onto itself is Euclidean bi-Lipschitz if u(0) = 0 and K < 2 n−1 . This is an extension of a similar result of Tam and Wan for hyperbolic harmonic mappings with respect to a hyperbolic metric. The proof uses Möbius transformations on the related space and a recent result of the first author, which states that harmonic quasiconformal self-mappings of the unit ball are Lipschitz continuous.

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Inner estimate and quasiconformal harmonic maps between smooth domains

Cited by 68 publications

References 17 publications

A priori estimate of gradient of a solution of a certain differential inequality and quasiconformal mappings

A priori estimate of gradient of a solution of a certain differential inequality and quasiconformal mappings

(K,K')-Quasiconformal harmonic mappings of the upper half plane onto itself

Harmonic quasiconformal self-mappings and Möbius transformations of the unit ball

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