Harmonic maps and the vectorial obstacle problem

11:00-12:00 Monday, 31 March 2025Auditorium, Rectorate

Abstract

I will discuss some recent results obtained in collaboration with A. Figalli, S. Kim and H. Shahgholian. We consider minimizers of the Dirichlet energy among maps constrained to take values outside a smooth domain $\Omega \subset \mathbb{R}^m$. These minimizers can be thought of either as solutions of a vectorial obstacle problem, or as harmonic maps into the manifold-with-boundary given by the complement of $\Omega$. I will discuss results concerning the regularity of the minimizers, the location of their singularities, and the structure of the free boundary.