Minimising weighted mean distortion : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Albany, New Zealand
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Date
2009
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Massey University
Abstract
There has been considerable recent interest in geometric function theory,
nonlinear partial differential equations, harmonic mappings, and the connection
of these to minimal energy phenomena. This work explores Nitsche's
1962 conjecture concerning the nonexistence of harmonic mappings between
planar annuli, cast in terms of distortion functionals. The connection between
the Nitsche problem and the famous Grötzsch problem is established
by means of a weight function. Traditionally, these kinds of problems are
investigated in the class of quasiconformal mappings, and the assumption is
usually made a priori that solutions preserve various symmetries. Here the
conjecture is solved in the much wider class of mappings of finite distortion,
symmetry-preservation is proved, and ellipticity of the variational equations
concerning these sorts of general problems is established. Furthermore, various
alternative interpretations of the weight function introduced herein lead
to an interesting analysis of a much wider variety of critical phenomena --
when the weight function is interpreted as a thickness, density or metric, the
results lead to a possible model for tearing or breaking phenomena in material
science. These physically relevant critical phenomena arise, surprisingly,
out of purely theoretical considerations.
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Keywords
Weighted mean distortion, Weight function, Mathematics