Structure of the free interfaces near triple junction singularities in harmonic maps and optimal partition problems

TL;DR

This paper proves the regularity and geometric structure of free interfaces near triple junction points in energy-minimizing harmonic maps into trees and applies these results to spectral optimal partition problems, revealing smooth manifolds meeting at 120-degree angles.

Contribution

It introduces a new epiperimetric inequality to analyze the structure of free interfaces near triple junctions in harmonic maps and optimal partition problems.

Findings

Free interfaces near triple junctions are composed of three smooth manifolds.

These manifolds meet along a common boundary at 120-degree angles.

Results apply to both harmonic maps into trees and spectral optimal partition problems.

Abstract

We consider energy-minimizing harmonic maps into trees and we prove the regularity of the singular part of the free interface near triple junction points. Precisely, by proving a new epiperimetric inequality, we show that around any point of frequency $3/2$, the free interface is composed of three $C^{1,\alpha}$-smooth $(d-1)$-dimensional manifolds (composed of points of frequency $1$) with common $C^{1,\alpha}$-regular boundary (made of points of frequency $3/2$) that meet along this boundary at 120 degree angles. Our results also apply to spectral optimal partition problems for the Dirichlet eigenvalues.