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

Roberto Ognibene, Bozhidar Velichkov

TL;DR

The paper addresses the structure of the singular set for energy-minimizing harmonic maps into singular targets and for spectral optimal partition problems. It introduces a novel epiperimetric inequality for the $3/2$-Weiss energy and classifies all $3/2$-homogeneous blow-ups as a Y-configuration, which enables a sharp regularity description of the free interface near triple junction points. The main results show that the $3/2$-stratum of the singular set, $\mathcal{F}_{3/2}$, is locally a $(d-2)$-dimensional $C^{1,\alpha}$ manifold and that near any such point the nodal set decomposes into three $(d-1)$-dimensional sheets meeting at $120^\circ$ along $\mathcal{F}_{3/2}$. The work yields a frequency gap, a no-holes lemma, and rate-of-convergence results for blow-ups, leading to applications to locally finite-tree valued harmonic maps and to spectral partition problems; in particular, it solves the Bishop-Friedland-Hayman min-max conjecture for $p=+\infty$ in all $d\ge3$ by establishing the unique Y-minimizer for the triple-partition on spheres.

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 $C^{1,α}$-smooth $(d-1)$-dimensional manifolds (composed of points of frequency $1$) with common $C^{1,α}$-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.

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

TL;DR

The paper addresses the structure of the singular set for energy-minimizing harmonic maps into singular targets and for spectral optimal partition problems. It introduces a novel epiperimetric inequality for the -Weiss energy and classifies all -homogeneous blow-ups as a Y-configuration, which enables a sharp regularity description of the free interface near triple junction points. The main results show that the -stratum of the singular set, , is locally a -dimensional manifold and that near any such point the nodal set decomposes into three -dimensional sheets meeting at along . The work yields a frequency gap, a no-holes lemma, and rate-of-convergence results for blow-ups, leading to applications to locally finite-tree valued harmonic maps and to spectral partition problems; in particular, it solves the Bishop-Friedland-Hayman min-max conjecture for in all by establishing the unique Y-minimizer for the triple-partition on spheres.

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 , the free interface is composed of three -smooth -dimensional manifolds (composed of points of frequency ) with common -regular boundary (made of points of frequency ) that meet along this boundary at 120 degree angles. Our results also apply to spectral optimal partition problems for the Dirichlet eigenvalues.

Paper Structure

This paper contains 15 sections, 19 theorems, 170 equations.

Table of Contents

  1. Introduction
  2. Two consequences of \ref{['t:main-main']}
  3. The Bishop-Friedland-Hayman's conjecture for the min-max
  4. Preliminaries
  5. Lipschitz continuity
  6. Almgren and Weiss monotonicity formulas
  7. Blow-up analysis
  8. Singular blow-ups of minimal frequency
  9. A topological lemma
  10. Classification of the blow-ups of frequency $3/2$
  11. Linearization around triple junctions
  12. Epiperimetric inequality at points of frequency $3/2$
  13. Regularity of the free boundary around points of frequency $3/2$
  14. Rate of convergence of the blow-up sequence
  15. Regularity of $\mathcal{F}_{3/2}$

Key Result

Theorem 1.1

Let $u:B_1\to\Sigma_N$ be an energy-minimizing map in $B_1\subset\mathbb{R}^d$, for some $d\ge 2$ and $N\ge 1$. Then, the singular set of $\mathrm{Sing}{(u)}$ of the free interface $\mathcal{F}(u)$ can be decomposed as where $\delta_d>0$ is dimensional constant. The set $\mathcal{F}_{3/2}(u)$ is an open subset of $\mathrm{Sing}(u)$ and, locally, a $(d-2)$-dimensional $C^{1,\alpha}$-smooth manifol

Theorems & Definitions (36)

  • Theorem 1.1
  • Theorem 1.2: Epiperimetric inequality
  • Theorem 1.3
  • Theorem 1.4
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 26 more