Estimates on the Laplace Operator in Heat Flows of Harmonic Maps

TL;DR

This paper derives estimates for the Laplace operator in heat flows of harmonic maps, especially outside singularities, to aid in analyzing the Ericksen--Leslie system with toroidal boundary conditions.

Contribution

It provides new estimates on the Laplace operator in heat flows of harmonic maps, applicable to higher-order analysis in complex systems.

Findings

Estimates applicable outside singularities

Useful for higher-order analysis in Ericksen--Leslie system

Addresses boundary conditions on tori

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.