摘要：
The purpose of this paper is to study the properties of the solutions to the inhomogeneous biharmonic equations:
$$\Delta (\Delta f)=g$$
, 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.
摘要：
In this paper, we are concerned with the following Schrodinger-Poisson-Slater problem with critical growth: -Delta u + (u(2) * 1/vertical bar 4 pi x vertical bar)u = mu k(x)vertical bar u vertical bar(p-2) u + vertical bar u vertical bar(4) u in R-3.( ) We use a measure representation concentration-compactness principle of Lions to prove that the (PS), condition holds locally. Via a truncation technique and Krasnoselskii genus theory, we further obtain infinitely many solutions for mu is an element of (0, mu*) with some mu* > 0.
摘要：
For $$b\in L_{\mathrm{loc}}({\mathbb {R}}^n)$$ and $$0<\alpha <1$$, we use fractional differentiation to define a new type of commutator of the Littlewood-Paley g-function operator, namely $$\begin{aligned} g_{\Omega ,\alpha ;b}(f )(x) =\bigg (\int _0^\infty \bigg |\frac{1}{t} \int _{|x-y|\le t}\frac{\Omega (x-y)}{|x-y|^{n+\alpha -1}}(b(x)-b(y))f(y)\,dy\bigg |^2\frac{dt}{t}\bigg )^{1/2}. \end{aligned}$$Here, we obtain the necessary and sufficient conditions for the function b to guarantee that $$g_{\Omega ,\alpha ;b}$$ is a bounded operator on $$L^2({\mathbb {R}}^n)$$. More precisely, if $$\Omega \in L(\log ^+ L)^{1/2}{(S^{n-1})}$$ and $$b\in I_{\alpha }(BMO)$$, then $$g_{\Omega ,\alpha ;b}$$ is bounded on $$L^2({\mathbb {R}}^n)$$. Conversely, if $$g_{\Omega ,\alpha ;b}$$ is bounded on $$L^2({\mathbb {R}}^n)$$, then $$b \in Lip_\alpha ({\mathbb {R}}^n)$$ for $$0<\alpha < 1$$.
期刊：
Complex Analysis and Operator Theory,2019年13(4):2049-2068 ISSN：1661-8254
通讯作者：
Chen, Shaolin
作者机构：
[Chen, Shaolin] Hengyang Normal Univ, Coll Math & Stat, Hengyang 421008, Hunan, Peoples R China.;[Kalaj, David] Univ Montenegro, Fac Nat Sci & Math, Cetinjski Put B B, Podgorica 81000, Montenegro.
通讯机构：
[Chen, Shaolin] H;Hengyang Normal Univ, Coll Math & Stat, Hengyang 421008, Hunan, Peoples R China.
关键词：
Landau type theorem;Poisson’s equation;Schwarz’s Lemma
摘要：
For a given continuous function
$$g:~\overline{\mathbb {D}}\rightarrow {\mathbb {C}}$$
and a given continuous function
$$\psi :~{\mathbb {T}}\rightarrow {\mathbb {C}}$$
, we establish some Schwarz type Lemmas for mappings f satisfying the PDE:
$$\Delta f=g$$
in
$${\mathbb {D}}$$
, and
$$f=\psi $$
in
$${\mathbb {T}}$$
, where
$${\mathbb {D}}$$
is the unit disk of the complex plane
$${\mathbb {C}}$$
and
$${\mathbb {T}}=\partial {\mathbb {D}}$$
is the unit circle. Then we apply these results to obtain a Landau type theorem, which is a partial answer to the open problem in Chen and Ponnusamy (Bull Aust Math Soc 97: 80–87, 2018).
作者机构：
[胡立军] College of Mathematics and Statistics, Hengyang Normal University, Hengyang, Hunan, 421002, China;[袁礼] Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China;[Zhai J.] Dawning Information Industry Co., Ltd., Beijing, 100193, China
作者机构：
[Chen, Shaolin] Hengyang Normal Univ, Coll Math & Stat, Hengyang 421008, Hunan, Peoples R China.;[Ponnusamy, Saminathan] Indian Inst Technol Madras, Dept Math, Chennai 600036, Tamil Nadu, India.
通讯机构：
[Ponnusamy, Saminathan] I;Indian Inst Technol Madras, Dept Math, Chennai 600036, Tamil Nadu, India.
关键词：
K-quasiconformal harmonic mapping;Radial John disk;Radial length;Pommerenke interior domain
摘要：
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.
期刊：
Computers & Mathematics with Applications,2019年78(8):2560-2574 ISSN：0898-1221
通讯作者：
Wang, Qisheng
作者机构：
[Wang, Keyan] Hengyang Normal Univ, Sch Math & Stat, Hengyang 421008, Hunan, Peoples R China;[Wang, Qisheng] Wuyi Univ, Sch Math & Computat Sci, Jiangmen 529020, Guangdong, Peoples R China
通讯机构：
[Wang, Qisheng] W;Wuyi Univ, Sch Math & Computat Sci, Jiangmen 529020, Guangdong, Peoples R China.
关键词：
Hyperbolic equations;Expanded mixed finite element method;Error estimates
摘要：
In this paper, an expanded mixed finite element method is developed to approximate the solution of second order hyperbolic equations. We discuss and theoretically prove the existence and uniqueness of solution for the semidiscrete formulation, and obtain the optimal order error estimates in the L-infinity(L-2) norm. In addition, we also construct the fully discrete approximations and provide the corresponding error analysis. Some numerical examples are presented to demonstrate the theoretical analysis. (C) 2019 Elsevier Ltd. All rights reserved.
作者机构：
[Chen, Shaolin] Hengyang Normal Univ, Coll Math & Stat, Hengyang 421008, Hunan, Peoples R China.;[Ponnusamy, Saminathan] Indian Inst Technol Madras, Dept Math, Chennai 600036, Tamil Nadu, India.
通讯机构：
[Chen, Shaolin] H;Hengyang Normal Univ, Coll Math & Stat, Hengyang 421008, Hunan, Peoples R China.
期刊：
Results in Mathematics,2019年74(3):1-16 ISSN：1422-6383
通讯作者：
Chen, Shaolin
作者机构：
[Chen, Shaolin] Hengyang Normal Univ, Coll Math & Stat, Hengyang 421008, Hunan, Peoples R China.
通讯机构：
[Chen, Shaolin] H;Hengyang Normal Univ, Coll Math & Stat, Hengyang 421008, Hunan, Peoples R China.
关键词：
Length;area;modulus of continuity;Poisson's equation
摘要：
In this paper, we discuss the modulus of continuity of solutions to Poisson’s equation, and give bounds of length and area distortion for some classes of K-quasiconformal mappings satisfying Poisson’s equations. The obtained results are the extension of the corresponding classical results.
