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The Beltrami Equation

机译：Beltrami方程

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摘要

The "measurable Riemann Mapping Theorem" (or the existence theorem for quasiconformal mappings) has found a central role in a diverse variety areas such as holomorphic dynamics, Teichmuller theory, low dimensional topology and geometry, and the planar theory of PDEs. Anticipating the needs of future researchers we give an account of the "state of the art" as it pertains to this theorem, that is to the existence and uniqueness theory of the planar Beltrami equation, and various properties of the solutions to this equation. The classical theory concerns itself with the uniformly elliptic case (quasiconformal mappings). Here we develop the theory in the more general framework of mappings of finite distortion and the associated degenerate elliptic equations. We recount aspects of this classical theory for the uninitiated, and then develop the more general theory. Much of this is either new at the time of writing, or provides a new approach and new insights into the theory. Indeed, it is the substantial recent advances in non-linear harmonic analysis, Sobolev theory and geometric function theory that motivated our approach here. The concept of a principal solution and its fundamental role in understanding the natural domain of definition of a given Beltrami operator is emphasized in our investigations. We believe our results shed considerable new light on the theory of planar quasiconformal mappings and have the potential for wide applications, some of which we discuss.
机译：“可测量的黎曼映射定理”（或拟保形映射的存在性定理）已在诸如亚纯动力学，Teichmuller理论，低维拓扑和几何以及PDE的平面理论等各种领域中发挥了核心作用。预期未来研究人员的需求，我们将介绍与该定理有关的“最新技术”，即平面Beltrami方程的​​存在和唯一性理论以及该方程解的各种性质。经典理论涉及均匀椭圆的情况（拟保形映射）。在这里，我们在有限变形映射和相关的退化椭圆方程的更通用的框架中发展该理论。我们将这些经典理论的各个方面介绍给那些没有经验的人，然后再发展更为通用的理论。其中的大部分要么在撰写本文时是新手，要么提供了对该理论的新方法和新见解。确实，正是非线性谐波分析，Sobolev理论和几何函数理论的最新进展推动了我们的研究。我们的研究强调了主要解决方案的概念及其在理解给定Beltrami算子定义的自然域中的基本作用。我们相信我们的结果为平面拟形映射的理论提供了相当多的新思路，并且具有广泛的应用潜力，我们将讨论其中的一些。

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