Regularity of the free boundary at density-one singular points

Characterize the regularity of the free boundary \(\partial\Omega_U\) near the set \(\mathrm{Sing}_2(\partial\Omega_U)\) of density-one singular points for minimizers of the vectorial Bernoulli functional \(J_\Lambda\).

Background

The paper studies minimizers of the vectorial Bernoulli functional $$J_\Lambda(W,D)=\int_D |\nabla W|^2\,dx+\Lambda|\Omega_W\cap D|$$, where $\Omega_W=\{W\neq 0\}$. The free boundary is divided according to the density of the positivity set, with $\mathrm{Sing}_2(\partial\Omega_U)$ consisting of points where the positivity set has density one.

The authors explain that understanding the structure of the free boundary near $\mathrm{Sing}_2(\partial\Omega_U)$ requires characterizing blow-up limits at such points. The paper advances this program by analyzing linear blow-ups and determining the associated global minimizers, but it does not resolve the broader regularity problem for the free boundary at these singular points.