Uniqueness of constraint maps that span small images
Marçal Font i Elias · KTH Publication Database DiVA (KTH Royal Institute of Technology) · 2026
Constraint maps as solution of the obstacle problem in a vectorial setting generally do not exhibit uniqueness, except in some specific situations. This thesis focuses on the uniqueness of energy minimizing constraint maps that span small images, despite the main active area of research in constraint maps being the study of the free boundaries and their regularity. The main uniqueness result takes the idea given by Figalli et al. in 2024, in which the uniqueness of constraint maps whose image is constrained in a small ball was proved when the obstacle was the unitary ball, and extends it to a convex obstacle with smooth boundary (in the case the minimizers are globally Lipschitz). The proof mimics the uniqueness proof of the scalar obstacle problem, taking advantage of the fact that in the vectorial case we can only guarantee both the convexity of the class of admissible functions and the strict convexity of the energy functional when the minimizers span small images, a case in which the curvature of the boundary can be assumed to be very small.