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Regularity of minimizing $p$-harmonic maps into spheres and sharp Kato inequality

Katarzyna Mazowiecka, Michał Miśkiewicz

Abstract

We study regularity of minimizing $p$-harmonic maps $u \colon B^3 \to \mathbb{S}^3$ for $p$ in the interval $[2,3]$. For a long time, regularity was known only for $p = 3$ (essentially due to Morrey) and $p = 2$ (Schoen-Uhlenbeck), but recently Gastel extended the latter result to $p \in [2,2+\frac{2}{15}]$ using a version of Kato inequality. Here, we establish regularity for a small interval $p\in [2.961,3]$ by combining Morrey's methods with Hardt and Lin's Extension Theorem. We also improve on the other result by obtaining regularity for $p \in [2,p_0]$ with $p_0 = \frac{3+\sqrt{3}}{2} \approx 2.366$. In relation to this, we address a question posed by Gastel and prove a sharp Kato inequality for $p$-harmonic maps in two-dimensional domains, which is of independent interest.

Regularity of minimizing $p$-harmonic maps into spheres and sharp Kato inequality

Abstract

We study regularity of minimizing -harmonic maps for in the interval . For a long time, regularity was known only for (essentially due to Morrey) and (Schoen-Uhlenbeck), but recently Gastel extended the latter result to using a version of Kato inequality. Here, we establish regularity for a small interval by combining Morrey's methods with Hardt and Lin's Extension Theorem. We also improve on the other result by obtaining regularity for with . In relation to this, we address a question posed by Gastel and prove a sharp Kato inequality for -harmonic maps in two-dimensional domains, which is of independent interest.
Paper Structure (12 sections, 16 theorems, 77 equations, 1 figure)

This paper contains 12 sections, 16 theorems, 77 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Weakly, stationary, and stable $p$-harmonic maps
  4. Tangent maps
  5. Motivation: the equatorial map
  6. Regularity for p close to 3
  7. Regularity for p close to 2
  8. Sharp Kato inequality
  9. Explicit constant estimates
  10. Hardt and Lin extension theorem with explicit constant
  11. Poincaré inequality on a sphere with explicit constant.
  12. Trace inequality with explicit constant

Key Result

Theorem 1.1

Any minimizing harmonic map $u \colon B^n \to {\mathbb S}^k$ is regular whenever $n \le d(k)$, where

Figures (1)

  • Figure 1: Computation of $Q_{a_0}$

Theorems & Definitions (38)

  • Theorem 1.1: SU3, Okayasu94
  • Theorem 1.2: Nakauchi01XY
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: HLp
  • Remark 2.4
  • Lemma 3.1
  • Theorem 3.2
  • Theorem 3.3: JK
  • ...and 28 more