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The extremal problem for weighted combined energy and the generalization of Nitsche inequality

Xiaogao Feng, Ruyue Tang, Ting Peng

Abstract

We consider the existence and uniqueness of a minimizer of the extremal problem for weighted combined energy between two concentric annuli and obtain that the extremal mapping is a certain radial mapping. Meanwhile, this in turn implies a Nitsche type phenomenon and we get a $\frac{1}{|w|^λ}-$Nitsche type inequality ($λ\neq1$). As an application, on the basis of the relationship between weighted combined energy and weighted combined distortion, we also investigate the extremal problem for weighted combined distortion on annuli. This extends the result obtained by Kalaj in \cite{Ka1}.

The extremal problem for weighted combined energy and the generalization of Nitsche inequality

Abstract

We consider the existence and uniqueness of a minimizer of the extremal problem for weighted combined energy between two concentric annuli and obtain that the extremal mapping is a certain radial mapping. Meanwhile, this in turn implies a Nitsche type phenomenon and we get a Nitsche type inequality (). As an application, on the basis of the relationship between weighted combined energy and weighted combined distortion, we also investigate the extremal problem for weighted combined distortion on annuli. This extends the result obtained by Kalaj in \cite{Ka1}.
Paper Structure (6 sections, 3 theorems, 120 equations)

This paper contains 6 sections, 3 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. weighted combined energy of radial mapping
  3. Proof of Theorem 1.1
  4. When $\lambda\neq1$
  5. When $\lambda=1$
  6. The extremal problem for weighted combined distortion

Key Result

Theorem 1.1

(I) When $\lambda\neq 1$ in (1.8), under condition and among all mapping $\mathfrak{H}(\mathbb{A}_{1},\mathbb{A}_{2})$, the infimum of is attained at the radial mapping where $\alpha$ satisfies That is to say, The minimizer is unique up to a rotation of annuli. (II)When $\lambda=1$ in (1.8), the weighted combined energy $\mathbb{E}_{\lambda}[h]$ attains its minimum for a radial mapping The m

Theorems & Definitions (3)

  • Theorem 1.1
  • Lemma 4.1
  • Theorem 4.2