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$L^\infty$-variational problems associated to measurable Finsler structures

Chang-Yu Guo, Chang-Lin Xiang, Dachun Yang

Abstract

We study $L^\infty$-variational problems associated to measurable Finsler structures in Euclidean spaces. We obtain existence and uniqueness results for the absolute minimizers.

$L^\infty$-variational problems associated to measurable Finsler structures

Abstract

We study -variational problems associated to measurable Finsler structures in Euclidean spaces. We obtain existence and uniqueness results for the absolute minimizers.

Paper Structure

This paper contains 9 sections, 9 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Finsler structure and its dual
  4. Intrinsic distance associated to an admissible Finsler structure
  5. Comparison of metric derivatives
  6. Weak coincidence of differential structure and distance structure
  7. Existence and Uniqueness
  8. Proof of existence
  9. Proof of uniqueness

Key Result

Theorem 1.1

Let $F:{\Omega}\times{\mathbb R}^n\to {\mathbb R}$ be an admissible Finsler structure on ${\Omega}$. Then for each open subset $U\subset\subset {\Omega}$ and each boundary data $f\in{\mathop\mathrm{\,Lip}}(\partial U)$, there exists a unique absolutely minimizing Lipschitz extension on $U$ with resp

Theorems & Definitions (18)

  • Theorem 1.1
  • Definition 2.1: Finsler structure
  • Definition 2.2: Admissible Finsler structure
  • Definition 2.3: Dual Finsler structure
  • Proposition 2.4: Basic properties of dual Finsler structures
  • Lemma 2.5
  • Proposition 3.1
  • proof
  • Theorem 4.1
  • Lemma 4.2
  • ...and 8 more