The Governing Equations

Abstract

Abstract This chapter is devoted to the study of the governing partial differential equations for the theory of mappings of finite distortion. We have already met a number of these equations, principally the Cauchy-Riemann system and the Beltrami system. Here, the goal is to put these equations in a more general framework and discuss them from a number of perspectives. For instance, when lifted to the level of exterior algebra, the Beltrami system takes on a somewhat nicer form indeed, in special cases it linearizes. Studying these equations at the level of differential forms has led to major new advances in the theory of quasiregular mappings and conformal geometry. Also there is a close connection between mappings of finite distortion and the minima of certain variational integrals. We shall outline this connection and set up the associated Euler—Lagrange equations. This leads to the non-linear potential theory which we mention only briefly; the book [134] contains a detailed study of this area.