CHAPTER 1 Introduction The interplay between Partial Differential Equations (PDEs) and the theory of mappings has a long and distinguished history. Gauss' practical work of geodesic surveying stimulated him to develop the theory of conformal transformations, for mapping figures from one surface to another. For conformal transformation from plane to plane, he used a pair of equations, apparently derived by d'Alembert who first related the derivatives of the real and imaginary part of a complex function in 1746 in his work on hydrodynamics [15] pg 497. These equations have become known as the Cauchy-Riemann equations. Gauss developed the differential geome- try of surfaces (1827) emphasizing the intrinsic geometry, with Gaussian curvature defined by measurements within the surface. Gauss also considered geodesic curves within surfaces. In 1829, Lobachevsky constructed a surface (the horosphere) within his non- Euclidean space, such that the intrinsic geometry within that surface is Euclidean, with geodesic curves being called Euclidean lines. For the converse process, he could only suggest tentatively that, within Euclidean space, the intrinsic geometry of a sphere of imaginary radius was Lobachevskian. But imaginary numbers were then regarded with justifiable suspicion, and he did not propose that as an acceptable model of his geometry within Euclidean space. The most famous work of Beltrami is [16] from 1867. There he showed that Lobachevsky's geometry is the intrinsic geometry of a surface of constant negative curvature, with geodesic curves being called lines in this geometry. Beltrami illustrated various surfaces with constant negative curvature, the simplest of which is the pseudosphere generated by revolv- ing a tractrix around its axis. Beltrami's paper convinced most mathematicians that the geometries of Euclid and of Lobachevsky are logically equivalent. In that work Beltrami used a differential equation corresponding to Gauss's equation. This has come to be known as the Beltrami equation. This major work of Beltrami's appeared 3 years after he took up the geodesy professorship at Pisa after it was turned down by Riemann. Beltrami went on to become one of Italy's foremost mathematicians, primarily as a geometer but making significant contributions in analysis, the theory of heat, and to mechanics. Of course since those early times, the theory of conformal mappings, analytic functions, Riemann surfaces and so forth has expanded in many different directions, far too numerous to relate, and now lie at the foundation of much of modern mathematics. Moreover practical applications, such as in fluid flow, hydrodynamics and more modern areas of control theory and robotics etc, abound. In this paper we present the most recent developments concerning certain linear and non-linear equations in the plane, solving Beltrami's equation at the critical point, where the uniform eUipticity bounds are lost. These provide far reaching l
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