Browsing by Author "Martin GJ"

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    The Generic Failure of Lower-Semicontinuity for the Linear Distortion Functional
    (Springer Nature, 2024-08-07) Hashemi M; Martin GJ
    We consider the convexity properties of distortion functionals, particularly the linear distortion, defined for homeomorphisms of domains in Euclidean n-spaces, n≥3. The inner and outer distortion functionals are lower semi-continuous in all dimensions and so for the curve modulus or analytic definitions of quasiconformality it ifollows that if {fn}n=1∞ is a sequence of K-quasiconformal mappings (here K depends on the particular distortion functional but is the same for every element of the sequence) which converges locally uniformly to a mapping f, then this limit function is also K-quasiconformal. Despite a widespread belief that this was also true for the geometric definition of quasiconformality (defined through the linear distortion H(fn)), T. Iwaniec gave a specific and surprising example to show that the linear distortion functional is not always lower-semicontinuous on uniformly converging sequences of quasiconformal mappings. Here we show that this failure of lower-semicontinuity is common, perhaps generic in the sense that under mild restrictions on a quasiconformal f, there is a sequence {fn}n=1∞ with fn→f locally uniformly and with lim supn→∞H(fn)