André Guerra University of Cambridge

Harmonic maps and the vectorial obstacle problem

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 O in ℝ^m. These minimizers can be thought of as harmonic maps into the manifold-with-boundary given by the complement of O, while O acts as an obstacle. I will discuss results concerning the regularity of the minimizers, the location of their singularities, and the structure of the free boundary.