Constraint maps. Overview and recent developments

I will discuss, at a heuristic level, constraint maps that minimize Dirichlet energy, potentially with a forcing term. Specifically, I will focus on their optimal smoothness and the resulting free boundary. Although the study of these maps began in the 1970s and saw further development in the 1980s, it was later neglected. Our recent research has revived this theory, offering deeper insights and uncovering intriguing connections between solution singularities and free boundaries.
This is based on joint works with A. Figalli (ETH), A. Guerra (ETH), and S. Kim (Uppsala).