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Shaolin Chen

Publications and source records attributed to Shaolin Chen.

At least 37 records · Page 2Linked to original sources
arXiv2020

On certain quasiconformal and elliptic mappings

Let $\overline{\mathbb{D}}$ be the closure of the unit disk $\mathbb{D}$ in the complex plane $\mathbb{C}$ and $g$ be a continuous function in $\overline{\mathbb{D}}$. In this paper, we discuss some characterizations of elliptic mappings $f$ satisfying the Poisson's equation $Δf=g$ in $\mathbb{D}$, and then establish some sharp distortion theorems on elliptic mappings with the finite perimeter and the finite radial length, respectively. The obtained results are the extension of the corresponding classical results.

math.CV↗
arXiv2020

Modulus of continuity and Heinz-Schwarz type inequalities of solutions to biharmonic equations

For positive integers $n\geq2$ and $m\geq1$, suppose that function $f\in\mathcal{C}^{4}(\mathbb{B}^{n},\mathbb{R}^{m})$ satisfying the following: $(1)$ the inhomogeneous biharmonic equation $Δ(Δf)=g$ ($g\in \mathcal{C}(\overline{\mathbb{B}^{n}},\mathbb{R}^{m})$) in $\mathbb{B}^{n}$, (2) the boundary conditions $f=φ_{1}$ $(φ_{1}\in \mathcal{C}(\mathbb{S}^{n-1},\mathbb{R}^{m}))$ on $\mathbb{S}^{n-1}$ and $\partial f/\partial\mathbf{n}=φ_{2}$ ( $φ_{2}\in \mathcal{C}(\mathbb{S}^{n-1},\mathbb{R}^{m})$) on $\mathbb{S}^{n-1}$, where $\partial /\partial\mathbf{n}$ stands for the inward normal derivative, $\mathbb{B}^{n}$ is the unit ball in $\mathbb{R}^{n}$ and $\mathbb{S}^{n-1}$ is the unit sphere of $\mathbb{B}^{n}$. First, we establish the representation formula of solutions to the above inhomogeneous biharmonic Dirichlet problem, and then discuss the Heinz-Schwarz type inequalities and the modulus of continuity of the solutions.

math.CV↗
arXiv2019

The boundary behaviour of $K$-quasiconformal harmonic mappings

In this article, we first discuss the Lipschitz characteristic and the linear measure distortion of $K$-quasiconformal harmonic mappings. Then we give some characterizations of the radial John disks with the help of Pre-Schwarzian of harmonic mappings.

math.CV↗
arXiv2018

Bi-Lipschitz characteristic of quasiconformal self-mappings of the unit disk satisfying bi-harmonic equation

Suppose that $f$ is a $K$-quasiconformal self-mapping of the unit disk $\mathbb{D}$, which satisfies the following: $(1)$ the biharmonic equation $Δ(Δf)=g$ $(g\in \mathcal{C}(\overline{\mathbb{D}}))$, (2) the boundary condition $Δf=φ$ ($φ\in\mathcal{C}(\mathbb{T})$ and $\mathbb{T}$ denotes the unit circle), and $(3)$ $f(0)=0$. The purpose of this paper is to prove that $f$ is Lipschitz continuos, and, further, it is bi-Lipschitz continuous when $\|g\|_{\infty}$ and $\|φ\|_{\infty}$ are small enough. Moreover, the estimates are asymptotically sharp as $K\to 1$, $\|g\|_{\infty}\to 0$ and $\|φ\|_{\infty}\to 0$, and thus, such a mapping $f$ behaves almost like a rotation for sufficiently small $K$, $\|g\|_{\infty}$ and $\|φ\|_{\infty}$.

math.CV↗
arXiv2018

Schwarz type lemma, Landau type theorem and Lipschitz type space of solutions to biharmonic equations

The purpose of this paper is to study the properties of the solutions to the biharmonic equations: $Δ(Δf)=g$, where $g:$ $\overline{\mathbb{D}}\rightarrow\mathbb{C}$ is a continuous function and $\overline{\mathbb{D}}$ denotes the closure of the unit disk $\mathbb{D}$ in the complex plane $\mathbb{C}$. In fact, we establish the following properties for those solutions: Firstly, we establish the Schwarz type lemma. Secondly, by using the obtained results, we get a Landau type theorem. Thirdly, we discuss their Lipschitz type property.

math.CV↗
arXiv2017

Schwarz's Lemmas for mappings satisfying Poisson's equation

For $n\geq3$, $m\geq1$ and a given continuous function $g:~Ω\rightarrow\mathbb{R}^{m}$, we establish some Schwarz type lemmas for mappings $f$ of $Ω$ into $\mathbb{R}^{m}$ satisfying the PDE: $Δf=g$, where $Ω$ is a subset of $\mathbb{R}^{n}$. Then we apply these results to obtain a Landau type theorem.

math.CV↗
arXiv2017

Total energy of radial mappings

The main aim of this paper is to extend one of the main results of Iwaniec and Onninen (Arch. Ration. Mech. Anal., 194: 927-986, 2009). We prove that, the so called total energy functional defined on the class of radial streachings between annuli attains its minimum on a total energy diffeomorphism between annuli. This involves a subtle analysis of some special ODE.

math.AP↗
arXiv2017

Radial length, radial John disks and $K$-quasiconformal harmonic mappings

In this article, we continue our investigations of the boundary behavior of harmonic mappings. We first discuss the classical problem on the growth of radial length and obtain a sharp growth theorem of the radial length of $K$-quasiconformal harmonic mappings. Then we present an alternate characterization of radial John disks. In addition, we investigate the linear measure distortion and the Lipschitz continuity on $K$-quasiconformal harmonic mappings of the unit disk onto a radial John disk. Finally, using Pommerenke interior domains, we characterize certain differential properties of $K$-quasiconformal harmonic mappings.

math.CV↗
arXiv2016

Lipschitz type, radial growth and Dirichlet type spaces on functions induced by certain elliptic operators

In this paper, we investigate properties of classes of functions related to certain elliptic operators. Firstly, we prove that a main result of Dyakonov (Acta Math. 178(1997), 143--167) on analytic functions can be extended to this more general setting. Secondly, we study the radial growth on these functions and the obtained results are generalizations of the corresponding results of Makarov (Proc. London Math. Soc. 51(1985), 369--384) and Korenblum (Bull. Amer. Math. Soc. 12(1985), 99--102). Finally, we discuss the Dirichlet type energy integrals on such classes of functions and their applications.

math.CV↗
arXiv2015

John disk and $K$-quasiconformal harmonic mappings

The main aim of this article is to establish certain relationships between $K$-quasiconformal harmonic mappings and John disks. The results of this article are the generalizations of the corresponding results of Ch.~Pommerenke \cite{Po}.

math.CV↗
arXiv2014

Schwarz-Pick type estimates of pluriharmonic mappings in the unit polydisk

In this paper, we will give Schwarz-Pick type estimates of arbitrary order partial derivatives for bounded pluriharmonic mappings defined in the unit polydisk. Our main results are generalizations of results of Colonna for planar harmonic mappings in [Indiana Univ. Math. J. 38: 829--840, 1989].

math.CV↗