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# Structure of the Free Interfaces Near Triple Junction Singularities in Harmonic Maps and Optimal Partition Problems

## Details

**Authors** Ognibene and Velichkov

**Year** 2026

**Publisher** Archive for Rational Mechanics and Analysis

**Discipline** Mathematics

[Read it at the publisher](https://doi.org/10.1007/s00205-026-02221-4) 
10.1007/s00205-026-02221-4

## In authors' words

### What they found (results)

The paper proves that near frequency-3/2 triple-junction points of energy-minimizing harmonic maps into trees, the free interface decomposes into exactly three smooth (d-1)-dimensional surfaces meeting along a common regular boundary at 120-degree angles, a result that also applies to optimal spectral partition problems.

## Commentary

### In short

The theorem characterizes how a domain's whole free-interface structure is built from distinct constituent surface-parts meeting at a precisely characterized shared boundary, a system of parts governed by a rigorous distinction.

**Patterns it shows** D, S

**Added** 2026-08-25

**How to cite this** Ognibene and Velichkov (2026). Structure of the Free Interfaces Near Triple Junction Singularities in Harmonic Maps and Optimal Partition Problems. Archive for Rational Mechanics and Analysis.