Corpus ID: 236034157

The Teichm\"uller problem for $L^p$-means of distortion

  title={The Teichm\"uller problem for \$L^p\$-means of distortion},
  author={Gaven J. Martin and Cong Yao},
Teichmüller’s problem from 1944 is this: Given x ∈ [0, 1) find and describe the extremal quasiconformal map f : D → D, f |∂D = identity and f(0) = −x ≤ 0. We consider this problem in the setting of minimisers of Lp-mean distortion. The classical result is that there is an extremal map of Teichmüller type with associated holomorphic quadratic differential having a pole of order one at x, if x 6= 0. For the Lp-norm, when p = 1 it is known that there can be no locally quasiconformal minimiser… Expand
1 Citations
Energy-minimal Principles in Geometric Function Theory.
We survey a number of recent developments in geometric analysis as they pertain to the calculus of variations and extremal problems in geometric function theory following the NZMRI lectures given byExpand


The classical Teichmuller problem asks one to identify the deformation of a disk which holds the boundary fixed, moves the origin to a given point and which minimises the maxi- mal conformalExpand
Extremal mappings of finite distortion
The theory of mappings of finite distortion has arisen out of a need to extend the ideas and applications of the classical theory of quasiconformal mappings to the degenerate elliptic setting whereExpand
The $L^p$ Teichm\"uller theory: Existence and regularity of critical points
We study minimisers of the $p$-conformal energy functionals, \[ \mathsf{E}_p(f):=\int_\ID \IK^p(z,f)\,dz,\quad f|_\IS=f_0|_\IS, \] defined for self mappings $f:\ID\to\ID$ with finite distortion andExpand
Extremal mappings of finite distortion and the Radon-Riesz property
We consider Sobolev mappings f ∈ W (Ω,C), 1 < q < ∞, between planar domains Ω ⊂ C. We analyse the Radon-Riesz property for convex functionals of the form f 7→ ∫ Ω Φ(|Df(z)|, J(z, f)) dz and show thatExpand
Non-variational extrema of exponential Teichmüller spaces
The exponential Teichmuller spaces $E_p$, $0\leq p \leq \infty$, interpolate between the classical Teichmuller space ($p=\infty$) and the space of harmonic diffeomorphisms $(p=0)$. In this article weExpand
Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane (Pms-48)
This book explores the most recent developments in the theory of planar quasiconformal mappings with a particular focus on the interactions with partial differential equations and nonlinear analysis.Expand
Deformations of Annuli with Smallest Mean Distortion
We determine the extremal mappings with smallest mean distortion for mappings of annuli. As a corollary, we find that the Nitsche harmonic maps are Dirichlet energy minimizers among allExpand
Geometric Function Theory and Non-linear Analysis
0. Introduction and Overview 1. Conformal Mappings 2. Stability of the Mobius Group 3. Sobolev Theory and Function Spaces 4. The Liouville Theorem 5. Mappings of Finite Distortion 6. Continuity 7.Expand
Mappings of finite distortion: The zero set of the Jacobian
This paper is part of our program to establish the fundamentals of the theory of mappings of finite distortion [6], [1], [8], [13], [14], [7] which form a natural generalization of the class ofExpand
Quasiregular Families Bounded in $$L^p$$ and Elliptic Estimates
We prove that a family of quasiregular mappings of a domain $\Omega$ which are uniformly bounded in $L^p$ for some $p>0$ form a normal family. From this we show how an elliptic estimate on aExpand