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

TL;DR

This paper studies triplet densities minimizing energy under partial segregation, proving optimal regularity, structure of free boundary, and uniform bounds for penalized solutions using advanced mathematical tools.

Contribution

It establishes the optimal regularity and detailed structure of free boundaries for partially segregated harmonic maps, and provides uniform bounds for penalized solutions, advancing understanding of such variational problems.

Findings

Minimizers are Hölder continuous with exponent 3/4.

Free boundary consists of smooth manifolds except for a small residual set.

Uniform bounds for penalized solutions are established independently of the penalty parameter.

Abstract

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.