Anisotropic harmonic maps and Ginzburg-Landau type relaxation.

Abstract: Consider maps $u:R^n\to R^k$ with values constrained in a fixed submanifold, and minimizing (locally) the energy $E(u)=\int W(\nablau)$. Here $W$ is a positive definite quadratic form on matrices.

Compared to the isotropic case $W(\nabla u)=|\nabla u|^2$ (harmonic maps) this may look like a harmless generalization, but the regularity theory for general $W$’s is widely open. I will explain why, and describe results with Andres Contreras on a relaxed problem, where the manifold-valued constraint is replaced by an integral penalization.

Posted on Mar 15, 2021 in Differential Equations, Seminars