On Variational Approximations For Wave Maps

n this paper, we revisit the existence of global weak solutions of wave maps from ${\mathbb{R}}^n$ into the sphere ${\mathbb{S}}^{L-1}$, $\mathrm{\square }u\perp T_u{\mathbb{S}}^{L-1}$, by establishing it as a singular limit of maps from ${\mathbb{R}}^n\times {\mathbb{R}}_{+}$ to ${\mathbb{S}}^{L-1}$ that minimize elliptic regularized variational functionals that contain an exponential weight in the time direction with a small parameter $\epsilon$, 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.