Classification of boundary blow-ups for the Mumford-Shah functional with Dirichlet boundary conditions

We classify boundary blow-ups of minimizers of the Mumford-Shah functional on bounded planar domains with Dirichlet boundary conditions. At every point where the discontinuity set meets the boundary of the domain, we prove that the discontinuity set of each blow-up is the tangent line to the boundary at that point. This settles the conjecture formulated by the author in 2024.