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

Abstract

We consider triplets of densities $(u_1,u_2,u_3)$ minimizing the Dirichlet energy \[\sum_{j=1}^3 \int_Ω |\nabla u_j|^2\,dx \] over a bounded domain $\hat{I}\text{c◯}\subset {\mathbb{R}}^N$, subject to the partial segregation condition: \[ u_1\,u_2\,u_3 \equiv 0 \ \text{in $\hat{I}\text{c◯}$.} \] We prove optimal regularity of the minimizers in spaces of Hölder 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-${\hat{I}}^2$ a priori bounds for minimal solutions to the penalized energy: \[ J_β(\mathbf{u}, Ω) = \int_Ω \sum_{i=1}^3 |\nabla u_i|^2 \,dx+ β\int_Ω \prod_{j=1}^3 u_j^2\,dx, \] in spaces of Hölder 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.