Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

On Minimisers of $$L^p$$ L p -mean Distortion

On Minimisers of $$L^p$$ L p -mean Distortion We study the $$L^p$$ L p -mean distortion functionals, $$\begin{aligned} \mathcal{E}_p[f] = \int \int \limits _{\mathbb {D}}{\mathbb {K}}^p(z,f) \; \mathrm{d}z,\quad f_{|{\mathbb {S}}}=f_0 \end{aligned}$$ E p [ f ] = ∫ ∫ D K p ( z , f ) d z , f | S = f 0 for Sobolev self homeomorphisms of the unit disk $${\mathbb {D}}$$ D with prescribed boundary values $$f_0:{\mathbb {S}}\rightarrow {\mathbb {S}}$$ f 0 : S → S and pointwise distortion function $${\mathbb {K}}={\mathbb {K}}(z,f)$$ K = K ( z , f ) . Here we discuss aspects of the existence, regularity and uniqueness questions for minimisers and discuss the diffeomorphic critical points of $$\mathcal{E}_p$$ E p presenting results we know and making some conjectures. Remarkably, smooth minimisers of the $$L^p$$ L p -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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

On Minimisers of $$L^p$$ L p -mean Distortion

Download PDF
18 pages

Loading next page...
 
/lp/springer-journals/on-minimisers-of-l-p-l-p-mean-distortion-Ogm9qDN3rh
Publisher
Springer Journals
Copyright
Copyright © 2014 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/s40315-014-0063-1
Publisher site
See Article on Publisher Site

Abstract

We study the $$L^p$$ L p -mean distortion functionals, $$\begin{aligned} \mathcal{E}_p[f] = \int \int \limits _{\mathbb {D}}{\mathbb {K}}^p(z,f) \; \mathrm{d}z,\quad f_{|{\mathbb {S}}}=f_0 \end{aligned}$$ E p [ f ] = ∫ ∫ D K p ( z , f ) d z , f | S = f 0 for Sobolev self homeomorphisms of the unit disk $${\mathbb {D}}$$ D with prescribed boundary values $$f_0:{\mathbb {S}}\rightarrow {\mathbb {S}}$$ f 0 : S → S and pointwise distortion function $${\mathbb {K}}={\mathbb {K}}(z,f)$$ K = K ( z , f ) . Here we discuss aspects of the existence, regularity and uniqueness questions for minimisers and discuss the diffeomorphic critical points of $$\mathcal{E}_p$$ E p presenting results we know and making some conjectures. Remarkably, smooth minimisers of the $$L^p$$ L p -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.

Journal

Computational Methods and Function Theory – Springer Journals

Published: Apr 1, 2014

References

  • On mappings with integrable dilatation
    Iwaniec, T; Šverák, V
  • Bilipschitz homeomorphisms of the circle and nonlinear Beltrami equations
    Astala, K; Iwaniec, T; Martin, GJ
  • Extremal mappings of finite distortion
    Astala, K; Iwaniec, T; Martin, GJ; Onninen, J
  • Fully non-linear models for the deformation of cellular structures
    Hussan, J.; Martin, G.
  • The boundary correspondence under quasiconformal mappings
    Beurling, A; Ahlfors, LV
  • Canadian Mathematical Society Monographs
    Nag, S
  • Unique extremality
    Bozin, V; Lakic, N; Markovic, V; Mateljevic, M
  • Is there always an extremal Teichmüller mapping?
    Yao, G
  • Is there always a unique extremal Teichmüller mapping
    Strebel, K
  • Ergebnisse der Mathematik und ihrer Grenzgebiete (3)
    Strebel, K
  • Princeton Mathematical Series
    Astala, K; Iwaniec, T; Martin, GJ
  • Boundary correspondence under harmonic quasiconformal homeomorphisms of the unit disk
    Pavlovich, M
  • On univalent harmonic maps between surfaces
    Schoen, R; Yau, ST
  • Differential equations on Riemannian manifolds and their geometric applications
    Cheng, SY; Yau, ST
  • Uniqueness and regularity of proper harmonic maps
    Li, P; Tam, L-F
  • Mappings of least Dirichlet energy and their Hopf differentials
    Iwaniec, T; Onninen, J
  • The Teichmüller theory of harmonic maps
    Wolf, M
  • Constant mean curvature surface, harmonic maps and universal Teichmüller space
    Wan, TYH
  • Quasi-convexity and the lower semicontinuity of multiple integrals
    Morrey, CB
  • Oxford Mathematical Monographs
    Iwaniec, T; Martin, G
  • Regularity of the inverse of a planar Sobolev homeomorphism
    Hencl, S; Koskela, P
  • Deformations with smallest weighted $$L^p$$ L p average distortion and Nitsche-type phenomena
    Martin, G.J.; McKubre-Jordens, M.
  • The Teichmüller problem for mean distortion
    Martin, G.J.
  • Lipschitz regularity for inner variational equations
    Iwaniec, T; Kovalev, LV; Onninen, J
  • Hopf differentials and smoothing Sobolev homeomorphisms
    Iwaniec, T.; Kovalev, L.V.; Onninen, J.