On Minimisers of Lp-mean Distortion

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6 Scopus citations

Abstract

We study the Lp-mean distortion functionals, (Formula presented.) for Sobolev self homeomorphisms of the unit disk (Formula presented.) with prescribed boundary values (Formula presented.) and pointwise distortion function (Formula presented.). Here we discuss aspects of the existence, regularity and uniqueness questions for minimisers and discuss the diffeomorphic critical points of (Formula presented.) presenting results we know and making some conjectures. Remarkably, smooth minimisers of the (Formula presented.)-mean distortion functionals have inverses which are harmonic with respect to a metric induced by the distortion of the mapping. From this we are able to deduce that the complex conjugate Beltrami coefficient of a smooth minimiser is locally quasiregular and we identify the quasilinear equation it solves. This has other consequences such as a maximum principle for the distortion.

Original languageEnglish (US)
Pages (from-to)399-416
Number of pages18
JournalComputational Methods and Function Theory
Volume14
Issue number2-3
DOIs
StatePublished - Oct 31 2014

Keywords

  • Calculus of variations
  • Harmonic mappings
  • Mean distortion minimisers

ASJC Scopus subject areas

  • Analysis
  • Computational Theory and Mathematics
  • Applied Mathematics

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