A new approximation of relaxed energies for harmonic maps and the Faddeev model

TL;DR

This paper introduces a new approximation method for relaxed energies in harmonic maps and the Faddeev model, demonstrating convergence to minimizers and establishing their regularity and existence without Cartesian currents.

Contribution

It presents a novel approximation approach for relaxed energies, proving convergence and regularity of minimizers for harmonic maps and the Faddeev model.

Findings

Minimizers of the approximating functionals converge to minimizers of the relaxed energy.

The method proves the partial regularity of minimizers without Cartesian currents.

Existence of minimizers for the relaxed Faddeev energy with Hopf degree ±1 is established.

Abstract

We propose a new approximation for the relaxed energy $E$ of the Dirichlet energy and prove that the minimizers of the approximating functionals converge to a minimizer $u$ of the relaxed energy, and that $u$ is partially regular without using the concept of Cartesian currents. We also use the same approximation method to study the variational problem of the relaxed energy for the Faddeev model and prove the existence of minimizers for the relaxed energy $\tilde{E}_F$ in the class of maps with Hopf degree $\pm 1$.