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arXiv · 2412.00781

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

Roberto Ognibene·Bozhidar Velichkov
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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.

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Roberto Ognibene, Bozhidar Velichkov. 2024-12-01. Structure of the free interfaces near triple junction singularities in harmonic maps and optimal partition problems. https://doi.org/10.1007/s00205-026-02221-4

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