The generic failure of lower-semicontinuity for the linear distortion functional : a thesis presented in partial fulfilment of the requirments for the degree of Doctor of Philosophy (PhD) in Mathematics, Institute of Advance Studies (NZIAS) Massey University of Auckland, New Zealand
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2022
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Massey University
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My research is primarily concerned with the convexity properties of distortion functionals (particularly the linear distortion) defined for quasiconformal homeomorphisms of domains in Euclidean n-spaces, though we will mainly stick to three-dimensions. The principal application is in studying the lower semi-continuity of distortion on uniformly convergent limits of sequences of quasiconformal mappings. For example, given the curve family or analytic definitions of quasiconformality - discussed in Chapter 3 - it is known that if {fₙ}∞ₙ₌₁ is a sequence of K-quasiconformal mappings (and here K depends on the particular distortion but is the same for every element of the sequence) which converges to a function f, then the limit function is also K-quasiconformal.
Despite a widespread belief that this was also true for the geometric definition of quasiconformality (via the linear distortion H(f) defined below) Tadeusz Iwaniec gave a specific surprising example to show that the linear distortion function is not lower semicontinuous. The main aim of this thesis is to show that this failure of lower semicontinuity is actually far more common, perhaps even generic in the sense that it is true that under mild restrictions on a quasiconformal f, there may be a sequence {fₙ}∞ₙ₌₁ with fₙ → f uniformly and with limsupₙ→∞ H(fₙ) < H(f). The main result of this thesis is to show this is true for a wide class of linear mappings and give bounds for the maximal jump down in the limit.
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Quasiconformal mappings, Algebras, Linear, Convex functions