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Nonlinear analysis and quasiconformal mappings from the perspective of PDEs

Introduction. The general trend of the geometric function theory in R is to generalize certain topological aspects of the analytic functions of one complex variable. The category of mappings that one usually considers in higher dimensions are the mappings with finite distortion, thus, in particular, quasiconformal and quasiregular mappings. This program, whose origin can be traced back to the works of M. A. Lavrentiev (1938), L. V. Ahlfors (1954), F. W. Gehring (1961), J. Vaisala (1961) and Y. Reshetnyak (1966), was held by an important school of Finnish geometers in the 1970’s, led by O. Martio, S. Rickman and J. Vaisala. For a recent account see [Ri] and [Vu]. But in this drive towards generalizations of analytic functions, one aspect has been quite neglected. This is the fact that the mappings in question solve important first order systems of PDEs analogous in many respects to the Cauchy-Riemann equations.