环同胚与素端
Ring homeomorphisms and prime ends
摘要：
证明了Beltrami方程$\overline{\partial}f=μ\,\partial f$在定义域$D\subseteq\Bbb C$中的每一个同胚$W^{1,1}_{\rm loc}$解$f$是所谓的环$Q-$与$Q(z)=K^T_μ(z, z_0)$同胚，其中$K^T_μ(z, z_0)$是方程关于任意点$z_0\in {\overline{D}}$的切（角）伸缩商。在这方面，我们发展了环$Q-$同胚关于素端点的边界行为的理论。在此基础上，我们证明了对于大类退化Beltrami方程$\overline{\partial}f=μ\,\partial f$，Dirichlet问题在$\Bbb C$中的任意单连通区域上存在正则解，在$\Bbb C$中的任意双连通区域上存在伪正则解和多叶解，且边界数据$φ$关于素端拓扑是连续的.
英文摘要：
We show that every homeomorphic $W^{1,1}_{\rm loc}$ solution $f$ of a Beltrami equation $\overline{\partial}f=μ\,\partial f$ in a domain $D\subseteq\Bbb C$ is the so--called ring $Q-$homeomorphism with $Q(z)=K^T_μ(z, z_0)$ where $K^T_μ(z, z_0)$ is the tangent (angular) dilatation quotient of the equation with respect to an arbitrary point $z_0\in {\overline{D}}$. In this connection, we develop the theory of the boundary behavior of the ring $Q-$homeomorphisms with respect to prime ends. On this basis, we show that, for wide classes of degenerate Beltrami equations $\overline{\partial}f=μ\,\partial f$, there exist regular solutions of the Dirichlet problem in arbitrary simply connected domains in $\Bbb C$ and pseudoregular and multivalent solutions in arbitrary finitely connected domains in $\Bbb C$ with boundary datum $φ$ that are continuous with respect to the topology of prime ends.
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