Energy estimates of harmonic maps between Riemannian manifolds

TL;DR

This paper investigates the regularity of minimizers of certain energy functionals related to harmonic maps between Riemannian manifolds, using majorization techniques instead of traditional Euler-Lagrange equations.

Contribution

It introduces a novel approach based on majorizations to analyze the regularity of minimizers of non-differentiable energy functionals.

Findings

Established regularity results for minimizers with non-smooth integrands

Developed a method that bypasses Euler equations using majorizations

Extended understanding of harmonic map energy estimates

Abstract

Let $\Omega \subset {R}^n,$ $n \geq 3,$ be a bounded open set, $x=(x_1,x_2,\ldots,x_n)$ a generic point which belongs to $\Omega,$ $u \colon \Omega \to {R}^N ,$ $N>1,$ and $ Du=(D_\alpha u^i)$, $D_\alpha = \partial/\partial x_\alpha, $ $\alpha =1,\ldots,n,\,$ $i=1,\ldots,N .\,$ Main goal is the study of regularity of the minima of nondifferentiable functionals $$ {\cal F} \,=\, \int_\Omega F(x,u,Du) dx. $$ having the integrand function different shapes of smoothness. The method is based on the use some majorizations for the functional, rather than the well known Euler equation associated to it.