On Variational Approximations For Wave Maps

TL;DR

This paper investigates the existence of global weak solutions for wave maps into spheres by analyzing singular limits of elliptic regularized variational functionals with exponential weights, extending previous methods.

Contribution

It introduces a variational approximation approach for wave maps, generalizing De Giorgi's idea and applying it to sphere and $SO(m)$-target manifolds.

Findings

Established existence of weak solutions as singular limits of variational minimizers.

Extended the variational approximation method to $SO(m)$-target manifolds.

Connected the approach to earlier work by Serra and Tilli on nonlinear wave equations.

Abstract

n this paper, we revisit the existence of global weak solutions of wave maps from $\R^n$ into the sphere $\mathbb{S}^{L-1}$, $\Box u\perp T_u \mathbb{S}^{L-1}$, by establishing it as a singular limit of maps from $\R^n\times \R_+$ to $\mathbb S^{L-1}$ that minimize elliptic regularized variational functionals that contain an exponential weight in the time direction with a small parameter $\varepsilon$, where the initial data of the Cauchy problem serve as the boundary condition. The idea went back to De Giorgi \cite{Giorgi1996}, which has been implemented by Serra and Tilli \cite{Serra-Tilli2012, Serra-Tilli2016} for certain class of nonlinear wave equations. This approach is also applicable to the $SO(m)$-target manifold.