On $p$-harmonic map heat flows for {$1\leq p< \infty$} and their finite element approximations

TL;DR

This paper studies the heat flow of generalized p-harmonic maps into spheres for all p in [1, ∞), establishing existence of solutions and proposing a finite element approximation with proven convergence.

Contribution

It introduces a unified analytical framework for p-harmonic map heat flows across the entire p spectrum, including the 1-harmonic case, and develops a convergent finite element method.

Findings

Existence of global weak solutions for all p in [1, ∞).

Introduction of a BV-solution concept for 1-harmonic map heat flow.

Development and convergence proof of a finite element approximation.

Abstract

Motivated by emerging applications from imaging processing, the heat flow of a generalized $p$-harmonic map into spheres is studied for the whole spectrum, $1\leq p<\infty$, in a unified framework. The existence of global weak solutions is established for the flow using the energy method together with a regularization and a penalization technique. In particular, a $BV$-solution concept is introduced and the existence of such a solution is proved for the 1-harmonic map heat flow. The main idea used to develop such a theory is to exploit the properties of measures of the forms $\cA\cdot\nab\bv$ and $\cA\wedge\nab\bv$; which pair a divergence-$L^1$, or a divergence-measure, tensor field $\cA$, and a $BV$-vector field $\bv$. Based on these analytical results, a practical fully discrete finite element method is then proposed for approximating weak solutions of the $p$-harmonic map heat flow, and the convergence of the proposed numerical method is also established.