Analysis Seminar: Xavier Lamy (Toulouse) — On relaxed harmonic maps with anisotropy

Consider maps $u:R^n\to R^k$ with values constrained in a fixed submanifold, and minimizing (locally) the energy $E(u)=\int W( abla u)$. Here $W$ is a positive definite quadratic form on matrices. Compared to the isotropic case $W( abla u)=| abla u|^2$ 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.