Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quantitative stability of the spiral-stretch map

Zoltán M. Balogh, Károly J. Böröczky

TL;DR

This paper provides a sharp quantitative stability result for the minimization of a mean distortion functional in the spiral-stretch setting between annuli. By first proving a quantitative stability for the linear stretch on quadrilaterals and then transferring the result to annuli via exponential/logarithmic coordinates, it shows that a near-minimizer $g$ (in the sense of the spiral-stretch deficit $\delta^{SP}(g)$) is close to the extremal $g^{\ast}$ in $L^1$, with an optimal $\sqrt{\varepsilon}$ rate. The deficit $\delta^{SP}(g)$ measures deviation from equality in Feng–Hu–Shen-type inequalities, and strict convexity of $\varphi$ ensures uniqueness of the minimizer while the rate cannot be improved. The work thus extends quantitative stability paradigms from isoperimetric-type inequalities to the geometric analysis of quasiconformal distortion minimization in annuli, with explicit constructions demonstrating sharpness.

Abstract

In this note, we prove the quantitative statibility of the extremal spiral-stretch maps minimizing the mean distortion functional in the class of maps of finite distortion between two annuli with given boundary values.

Quantitative stability of the spiral-stretch map

TL;DR

This paper provides a sharp quantitative stability result for the minimization of a mean distortion functional in the spiral-stretch setting between annuli. By first proving a quantitative stability for the linear stretch on quadrilaterals and then transferring the result to annuli via exponential/logarithmic coordinates, it shows that a near-minimizer (in the sense of the spiral-stretch deficit ) is close to the extremal in , with an optimal rate. The deficit measures deviation from equality in Feng–Hu–Shen-type inequalities, and strict convexity of ensures uniqueness of the minimizer while the rate cannot be improved. The work thus extends quantitative stability paradigms from isoperimetric-type inequalities to the geometric analysis of quasiconformal distortion minimization in annuli, with explicit constructions demonstrating sharpness.

Abstract

In this note, we prove the quantitative statibility of the extremal spiral-stretch maps minimizing the mean distortion functional in the class of maps of finite distortion between two annuli with given boundary values.

Paper Structure

This paper contains 3 sections, 10 theorems, 119 equations.

Table of Contents

  1. Introduction and main results
  2. A quantitative stability version of the minimality of the linear stretch map
  3. A quantitative stability version of the minimality of the spiral-stretch map

Key Result

Theorem 1.1

If $g:A_1\to A_2$ is an orientation preserving homeomorphism in $W^{1,2}(A_1, A_2)$ with finite distortion such that $g=g^{\ast}$ on $\partial A_1$, and $\varphi:[1,\infty)\to[1,\infty)$ is increasing and strictly convex with $\varphi(1)=1$, then with equality if and only if $g=g^{\ast}$.

Theorems & Definitions (23)

  • Theorem 1.1: Feng-Hu-Shen
  • Definition 1.1
  • Theorem 1.2
  • Example 2.1
  • Proposition 2.1
  • Theorem 2.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 13 more