On partially segregated harmonic maps: optimal regularity and structure of the free boundary

We consider triplets of densities $(u_1,u_2,u_3)$ minimizing the Dirichlet energy $$\sum_{j=1}^3 \int_{\Omega} |\nabla u_j|^2\,dx$$ over a bounded domain $\Omega\subset \mathbb{R}^N$, subject to the partial segregation condition: $$u_1\,u_2\,u_3 \equiv 0 \ \text{in $\Omega$.}$$ We prove optimal regularity of the minimizers in spaces of H\"older continuous functions of exponent $3/4$; furthermore we prove that the free boundary is a collection of a locally finite number of smooth codimension one manifolds up to a residual set of Hausdorff dimension at most $N-2$. Finally we prove uniform-in-$\beta$ a priori bounds for minimal solutions to the penalized energy: $$J_\beta(\mathbf{u}, \Omega) = \int_{\Omega} \sum_{i=1}^3 |\nabla u_i|^2 \,dx+ \beta \int_{\Omega} \prod_{j=1}^3 u_j^2\,dx,$$ in spaces of H\"older continuous functions of exponent less than $3/4$. The proofs make use of an Almgren-type monotonicity formula, blow-up analysis together with some new Liouville-type theorems.