约化Beltrami方程的同胚解

约化Beltrami方程与拟正则映射的线性族

On reduced Beltrami equations and linear families of quasiregular mappings

本文研究了平面拟正则映射的线性类。对K的一个猜想给出了肯定的回答。Astala，T. Iwaniec和G. Martin（2009）关于简化的Beltrami方程。此外，我们还利用它证明了一般线性Beltrami系统的一个Wronsky型定理。这是证明线性拟正则族的相应Beltrami方程唯一性的关键。

拟共形映射在直线上的变分

Variation of quasiconformal mappings on lines

与标准Beltrami方程相比，我们得到了约化Beltrami方程同胚解的正则性的改进。这样的改进是不可能的持有人或Sobolev正则性，相反，我们的结果关注的广义变化的限制行。具体地说，我们证明了任何线段的限制有有限的p-变化的所有p>1，但不一定为p=1。

The Hodge Star Operator and the Beltrami Equation

An essentially unique homeomorphic solution to the Beltrami equation was found in the 1960s using the theory of Calderón-Zygmund and singular integral operators in $L^p(\mathbb C)$. We will present an alternative method to solve the Beltrami equation using the Hodge star operator and standard elliptic PDE theory. We will also discuss a different method to prove the regularity of the solution. This approach is partially based on work by Dittmar.

Toward the theory of Dirichlet problem for the degenerate Beltrami equations

In this article, first we give a general lemma on the existence of regular homeomorphic solutions $f$ with the hydrodynamic normalization $f(z)=z+o(1)$ as $z\to\infty$ to the degenerate Beltrami equations $\overline{\partial}f=μ\,\partial f$ in $\mathbb C$ whose complex coefficients $μ$ have compact supports. On this basis, we establish criteria for existence and representation of regular discrete open solutions for the Dirichlet problem with continuous data to degenerate Beltrami equations in arbitrary simply connected bounded domains $D$ in $\mathbb C$. Moreover, we obtain similar criteria f…

关于一般Beltrami方程的可积条件

On integral conditions for the general Beltrami equations

在积分限制条件下，证明了具有两个特征的退化Beltrami方程，特别是在数学物理的许多问题中起重要作用的第二类Beltrami方程和同样有重要应用的所谓约化Beltrami方程的存在性定理.

On compactness of one class of solutions of the Dirichlet problem

We consider the Dirichlet problem for the Beltrami equation in some simply connected domain. We consider the class of all homeomorphic solutions of such a problem with a normalization condition and set-theoretic constraints on their complex characteristics. We have proved the compactness of this class in terms of prime ends for an arbitrary continuous function in the Dirichlet condition.

Mixed boundary value problems for the Laplace-Beltrami equations

We investigate the mixed Dirichlet-Neumann boundary value problems for the Laplace-Beltrami equation on a smooth hypersurface $\mathcal{C}$ with the smooth boundary in non-classical setting in the Bessel potential spaces $\mathbb{H}^1_p(\mathcal{C})$ for $1<p<\infty$. To the initial BVP we apply quasilocalization and obtain model BVPs for the Laplacian. The model mixed BVP on the half plane is investigated by potential method and is reduced to an equivalent system of Mellin convolution equations in Bessel potential and Besov spaces. The symbol of the obtained system is written explicitly, whic…

具有强表面张力的Beltrami流中完全局域化的三维孤立水波

Fully localised three-dimensional solitary water waves on Beltrami flows with strong surface tension

本文研究了在Beltrami流中强表面张力条件下，三维孤立水波的局域化存在性及KP-I方程的扰动解

微分包含的唯一延拓性

Unique continuation for differential inclusions

本文研究梯度位于椭圆集拟共形包络内并在正测度集上接触该集合的 Sobolev 映射是否必为仿射，证明距离满足强唯一延拓性，并推出 Beltrami 与 Monge–Ampère 方程的新正则性结果。