Minimisation of mean exponential distortions and Teichmüller theory : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Albany, New Zealand
This thesis studies the Cauchy boundary value problem of minimising exponential integral averages of mappings of ﬁnite distortion. Direct methods in calculus of variations provide existence theorems and we derive the Euler-Lagrange equations for minimisers of
∫D exp(pK(z, f)) dz
for mappings of ﬁnite distortion f : D → D with prescribed boundary values. However, surprisingly, for these functionals some apriori regularity is needed before we can discuss these equations. We show by example how this can happen. We construct a mapping f : D → D with exponentially integrable distortion to exponent p which cannot perturbed by any diﬀeomorphism and still remain exponentially integrable with exponent p. Once enough apriori regularity is assumed for instance if a minimiser is locally quasiconformal, that is if the distortion function K(z, f) is locally bounded, then we use these equations to improve the regularity of the minimisers. In particular, we ﬁnd that minimisers with locally bounded distortions are diﬀeomorphisms. Then we analyse the two extreme cases (1) p → 0 and (2) p → ∞. In this way we see the p-exponential problem connects the L¹ ﬁnite distortion problem, which is closely related to the classical harmonic theory in case (1), and to the Teichmüller problem, which promoted the development of quasiconformal mappings, in case (2).