Abstract 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 de- velop 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 be- lieve 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. Received by the editor Received by the editor July 27, 2000, and in revised form September 10, 2004. 2000 Mathematics Subject Classification. Primary 30C60, 35J15, 35J70. Key words and phrases. Beltrami Equation, quasiconformal, finite distortion, partial differ- ential equations, degenerate elliptic. The first author was supported in part by grants from the US National Science Foundation. The second author was supported in part by the NZ Marsden Fund, the Royal Society of NZ (James Cook Fellow) and the NSF.
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