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 $\Delta(\Delta f)=g$ $(g\in \mathcal{C}(\overline{\mathbb{D}}))$, (2) the boundary condition $\Delta f=\varphi$ ($\varphi\in\mathcal{C}(\mathbb{T})$ 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 $\|\varphi\|_{\infty}$ are small enough. Moreover, the estimates are asymptotically sharp as $K\to 1$, $\|g\|_{\infty}\to 0$ and $\|\varphi\|_{\infty}\to 0$, and thus, such a mapping $f$ behaves almost like a rotation for sufficiently small $K$, $\|g\|_{\infty}$ and $\|\varphi\|_{\infty}$.Comment: 26 pages. Indiana University Mathematics Journal, 202

Computation of the Memory Functions in the Generalized Langevin Models for Collective Dynamics of Macromolecules

We present a numerical method to compute the approximation of the memory functions in the generalized Langevin models for collective dynamics of macromolecules. We first derive the exact expressions of the memory functions, obtained from projection to subspaces that correspond to the selection of coarse-grain variables. In particular, the memory functions are expressed in the forms of matrix functions, which will then be approximated by Krylov-subspace methods. It will also be demonstrated that the random noise can be approximated under the same framework, and the fluctuation-dissipation theorem is automatically satisfied. The accuracy of the method is examined through several numerical examples

Coefficient estimates and the Fekete-Szeg\H o problem for certain classes of polyharmonic mappings

We give coefficient estimates for a class of close-to-convex harmonic mappings, and discuss the Fekete-Szeg\H{o} problem of it. We also introduce two classes of polyharmonic mappings $\mathcal{HS}_{p}$ and $\mathcal{HC}_{p}$, consider the starlikeness and convexity of them, and obtain coefficient estimates on them. Finally, we give a necessary condition for a mapping $F$ to be in the class $\mathcal{HC}_{p}$.Comment: 14 pages, 2 figure

On lengths, areas and Lipschitz continuity of polyharmonic mappings

In this paper, we continue our investigation of polyharmonic mappings in the complex plane. First, we establish two Landau type theorems. We also show a three circles type theorem and an area version of the Schwarz lemma. Finally, we study Lipschitz continuity of polyharmonic mappings with respect to the distance ratio metric.Comment: 18 pages, 1 figur

Starlikeness and convexity of polyharmonic mappings

In this paper, we first find an estimate for the range of polyharmonic mappings in the class $HC_{p}^{0}$. Then, we obtain two characterizations in terms of the convolution for polyharmonic mappings to be starlike of order $\alpha$, and convex of order $\beta$, respectively. Finally, we study the radii of starlikeness and convexity for polyharmonic mappings, under certain coefficient conditions

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: $\Delta(\Delta f)=g$, 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.Comment: 22pages, To appear in J. Geom. Ana

On polyharmonic univalent mappings

In this paper, we introduce a class of complex-valued polyharmonic mappings, denoted by $HS_{p}(\lambda)$, and its subclass $HS_{p}^{0}(\lambda)$, where $\lambda\in [0,1]$ is a constant. These classes are natural generalizations of a class of mappings studied by Goodman in 1950's. We generalize the main results of Avci and Z{\l}otkiewicz from 1990's to the classes $HS_{p}(\lambda)$ and $HS_{p}^{0}(\lambda)$, showing that the mappings in $HS_{p}(\lambda)$ are univalent and sense preserving. We also prove that the mappings in $HS_{p}^{0}(\lambda)$ are starlike with respect to the origin, and characterize the extremal points of the above classes

Linear connectivity, Schwarz-Pick lemma and univalency criteria for planar harmonic mappings

In this paper, we first establish the Schwarz-Pick lemma of higher-order and apply it to obtain a univalency criteria for planar harmonic mappings. Then we discuss distortion theorems, Lipschitz continuity and univalency of planar harmonic mappings defined in the unit disk with linearly connected images.Comment: 12 page

On bi-Lipschitz continuity of solutions of hyperbolic Poisson's equation

In this paper, we investigate solutions of the hyperbolic Poisson equation $\Delta_{h}u(x)=\psi(x)$, where $\psi\in L^{\infty}(\mathbb{B}^{n}, \mathbb{R}^n)$ and $$\Delta_{h}u(x)= (1-|x|^2)^2\Delta u(x)+2(n-2)(1-|x|^2)\sum_{i=1}^{n} x_{i} \frac{\partial u}{\partial x_{i}}(x)$$ is the hyperbolic Laplace operator in the $n$-dimensional space $\mathbb{R}^n$ for $n\ge 2$. We show that if $n\geq 3$ and $u\in C^{2}(\mathbb{B}^{n},\mathbb{R}^n) \cap C(\overline{\mathbb{B}^{n}},\mathbb{R}^n )$ is a solution to the hyperbolic Poisson equation, then it has the representation $u=P_{h}[\phi]-G_{ h}[\psi]$ provided that $u\mid_{\mathbb{S}^{n-1}}=\phi$ and $\int_{\mathbb{B}^{n}}(1-|x|^{2})^{n-1} |\psi(x)|\,d\tau(x)<\infty$. Here $P_{h}$ and $G_{h}$ denote Poisson and Green integrals with respect to $\Delta_{h}$, respectively. Furthermore, we prove that functions of the form $u=P_{h}[\phi]-G_{h}[\psi]$ are bi-Lipschitz continuous.Comment: 32 page

On the Lipschitz continuity of certain quasiregular mappings between smooth Jordan domains

We first investigate the Lipschitz continuity of $(K, K')$-quasiregular $C^2$ mappings between two Jordan domains with smooth boundaries, satisfying certain partial differential inequalities concerning Laplacian. Then two applications of the obtained result are given: As a direct consequence, we get the Lipschitz continuity of $\rho$-harmonic $(K, K')$-quasiregular mappings, and as the other application, we study the Lipschitz continuity of $(K,K')$-quasiconformal self-mappings of the unit disk, which are the solutions of the Poisson equation $\Delta w=g$. These results generalize and extend several recently obtained results by Kalaj, Mateljevi\'{c} and Pavlovi\'{c}.Comment: 20 page