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A weak energy identity for $(n+α)$-harmonic maps with a free boundary in a sphere

Dorian Martino, Katarzyna Mazowiecka, Rémy Rodiac

TL;DR

This work addresses the compactness and energy distribution of free-boundary $(n{+}\alpha)$-harmonic maps into spheres as $\alpha\to 0$. The authors develop boundary epsilon-regularity, a detailed bubble-tree analysis, and energy-comparison tools to prove a weak energy identity for sequences $(u_k)$, with bubble energies weighted by coefficients $\lambda_*^i$; they perform a generation-by-generation extraction of bubbles at boundary concentration points and show termination after finitely many generations due to a fixed quantum energy $\varepsilon_b$. In the special case $d=n$, they establish a boundary degree identity connecting the degrees of $u_k$ to those of the bubbles along the boundary. The results extend Sacks--Uhlenbeck type bubbling analysis to free-boundary, non-linear $n$-harmonic maps and illuminate energy quantization and neck-rupture phenomena at the boundary, with potential implications for optimizations involving Steklov eigenvalues and homogeneous targets.

Abstract

In this article, we show that sequences of $(n+α)$-harmonic maps with a free boundary in $\mathbb S^{d-1}$, where $α$ is a parameter tending to zero, converge to a bubble tree. For such sequences, we prove in detail that the limiting energy is equal to the energy of the macroscopic limit plus the sum of the energies of certain ``bubbles'', each multiplied by a corresponding coefficient.

A weak energy identity for $(n+α)$-harmonic maps with a free boundary in a sphere

TL;DR

This work addresses the compactness and energy distribution of free-boundary -harmonic maps into spheres as . The authors develop boundary epsilon-regularity, a detailed bubble-tree analysis, and energy-comparison tools to prove a weak energy identity for sequences , with bubble energies weighted by coefficients ; they perform a generation-by-generation extraction of bubbles at boundary concentration points and show termination after finitely many generations due to a fixed quantum energy . In the special case , they establish a boundary degree identity connecting the degrees of to those of the bubbles along the boundary. The results extend Sacks--Uhlenbeck type bubbling analysis to free-boundary, non-linear -harmonic maps and illuminate energy quantization and neck-rupture phenomena at the boundary, with potential implications for optimizations involving Steklov eigenvalues and homogeneous targets.

Abstract

In this article, we show that sequences of -harmonic maps with a free boundary in , where is a parameter tending to zero, converge to a bubble tree. For such sequences, we prove in detail that the limiting energy is equal to the energy of the macroscopic limit plus the sum of the energies of certain ``bubbles'', each multiplied by a corresponding coefficient.

Paper Structure

This paper contains 11 sections, 13 theorems, 235 equations.

Table of Contents

  1. Introduction
  2. Compactness outside concentration points
  3. Properties of the bubbles
  4. Some energy comparison lemmas
  5. Bubbling decomposition
  6. Step 1: Local strong convergence away from a finite set
  7. Step 2: Extraction of the first-generation bubbles
  8. Step 3: No-neck energy property for the first-generation bubble at a given point $x_i\in S$.
  9. Step 4: Extraction of a possible second-generation bubble at a given point $x_i\in S$
  10. Step 5: Induction for the extraction of all the bubbles at a given point of concentration
  11. The case $d=n$: proof of \ref{['prop:degrees']}

Key Result

Theorem 1.3

Let $n,d\geq 2$ and $(\Sigma^n,g)$ be a compact manifold with non-empty boundary. Let $(p_k)_{k\in{\mathbb N}} \subset [n,n+1]$ be a sequence of numbers such that $p_k\to n$ as $k\to \infty$. Let $M>0$ and let $(u_k)_{k\in{\mathbb N}} \subset W^{1,n}(\Sigma^n;\mathbb{R}^d)$ be a sequence of maps tha Then there exist such that

Theorems & Definitions (57)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Proposition 1.4
  • Proposition 2.1
  • Lemma 2.2: MazowieckaRodiacSchikorra
  • proof : Proof of Proposition \ref{['prop:uniform_epsilon_reg']}
  • Lemma 2.3
  • proof
  • Claim 1
  • ...and 47 more