Event description
Proposed by Mumford and Shah in 1989 to study image segmentation with a variational approach, the Mumford-Shah functional naturally arises in physical contexts involving the competition between bulk and surface energy. The regularity of its minimizers poses a significant challenge in geometric measure theory. The original functional does not include any boundary conditions, leading to homogeneous Neumann boundary conditions. In this setting, the regularity at the boundary is known. Conversely, the regularity at the boundary for minimizers of the Mumford-Shah functional with Dirichlet boundary conditions is an open problem. The goal of this talk is to present recent developments on this topic. The main results include a partial classification of blow-ups of minimizers at boundary points and an epsilon-regularity theorem.