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Estimates on the Laplace Operator in Heat Flows of Harmonic Maps

Qingtong Wu

Abstract

In this paper we investigate estimates about the Laplace operator in heat flows of harmonic maps, focusing outside the singularities through spherical coordinates. These estimates can be used in the general Ericksen--Leslie system to obtain higher-order estimates. We consider the problem subject to the $\mathbb{T}^2$ and $\mathbb{T}^3$ boundary conditions.

Estimates on the Laplace Operator in Heat Flows of Harmonic Maps

Abstract

In this paper we investigate estimates about the Laplace operator in heat flows of harmonic maps, focusing outside the singularities through spherical coordinates. These estimates can be used in the general Ericksen--Leslie system to obtain higher-order estimates. We consider the problem subject to the and boundary conditions.

Paper Structure

This paper contains 5 sections, 4 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Estimates
  3. Proof.
  4. Proof.
  5. Proof.

Key Result

Lemma 1.1

Let $\Omega \subset \mathbb{R}^n$ be a bounded domain with sufficient smoothness, and let $u \in H^m(\Omega) \cap L^p(\Omega)$ for some $m \in \mathbb{N}$ and $1 \leq p \leq \infty$. Then there exists a constant $C$ depending only on $\Omega$, $m$, $p$, and $n$, such that where $\theta = \frac{n \left( \frac{1}{p} - \frac{1}{q} \right) + m}{n + m}$ and $q$ is the Sobolev conjugate exponent of $p$

Theorems & Definitions (8)

  • Remark 1.1
  • Lemma 1.1: Gagliardo–Nirenberg Inequality ref6
  • Theorem 2.1
  • Remark 2.1
  • Theorem 2.2
  • Remark 2.2
  • Theorem 2.3
  • Remark 2.3