Accepted Paper

Inserted: 1 dec 2024
Last Updated: 29 may 2026

Journal: Arch. Ration. Mech. Anal.
Year: 2026

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.

Keywords: harmonic maps, Triple Junction, singular set, free boundary, optimal partitions, epiperimetric inequality, free interface

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