Regularity in two-dimensional variational problems with perimeter penalties (Q2746849)

Let \(\Omega \subset \mathbb R^2\) denote an open set and consider an energy like \(I_\Omega (u, A) = \int\limits_\Omega \sigma_A |\nabla u|^2 \;dx + P_\Omega (A)\) defined for functions \(u \in H^1 (\Omega)\) and measurable sets \(A \subset \Omega\) having finite perimeter \(P_\Omega (A)\). \(\sigma_A\) is a positive two-valued function taking the smaller value in \(A\) and the larger one in \(\Omega-A\). The main result, strongly relying on the properties of blow-up limits, can be formulated as follows: Let \((u, A)\) denote a minimizing pair of \(I_\Omega\) (or even more general functionals). Then the components of \(A\) are smooth away from \(\partial \Omega\).