Regularity theory for vectorial constraint maps with free boundaries

The project aims to address fundamental regularity issues on constraint maps, which are minimizers of the Dirichlet energy under image constraints.

The complement of the image constraint can be considered as an obstacle, which naturally gives rise to a free boundary as the interface between the contact and non-contact region.

The key characteristics of constraint maps are twofold: (i) they may exhibit discontinuous singularities due to topological disparities between the domain and the target constraint; (ii) they may exhibit branch points either due to singularity in their image or a singular parametrization of a regular image.

Both characteristics have been extensively studied in the theory of harmonic maps and minimal surfaces.

However, neither of them was studied in-depth in relation to free boundaries until the recent breakthroughs established by the proposer and his internationally renowned collaborators.This project is situated at the forefront of this newly emerging area, with its focus on (i) the structure of the singular set near free boundaries of constraint maps and (ii) the structure of their free boundaries near the set of branch points.

The anticipated outcomes of this research carry profound implications for both analysis and geometry, as they signify the role of free boundaries in the study of irregular objects and produce robust and novel approaches to be applied to a wider class of vectorial problems.