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The Lamm-Rivière system II: energy identity

Chang-Yu Guo, Wen-Juan Qi, Zhao-Min Sun, Changyou Wang

TL;DR

This work establishes angular energy quantization for the four-dimensional inhomogeneous Lamm-Rivière system, demonstrating that energy concentration occurs solely via finitely many bubbles and that the total energy splits between a limiting map and bubble energies. The authors develop a robust framework combining $\varepsilon$-regularity, energy gap, removable singularity, and weak compactness, with sharp neck estimates in Lorentz spaces to control tangential derivatives. They first prove the $L^p$-based angular-energy identity and then extend the result to the borderline case $f\in L\log L$, using $L\log L$ extensions and Calderón–Zygmund theory. The results generalize prior energy-identity results for biharmonic and Lamm-Rivière-type systems to the inhomogeneous, critical 4D setting and provide a precise bubble-tree decomposition underpinning the energy balance. The techniques leverage Lorentz-space tools and a detailed neck-analysis to offer sharp control of concentration phenomena in high-order elliptic systems with geometric structure.

Abstract

In this paper, we establish an angular energy quantization for the following fourth order inhomogeneous Lamm-Rivière system $$ Δ^2u=Δ(V\cdot\nabla u)+\text{div}(w\nabla u)+W\cdot\nabla u+f $$ in dimension four, with an inhomogeneous term $f\in L\log L$.

The Lamm-Rivière system II: energy identity

TL;DR

This work establishes angular energy quantization for the four-dimensional inhomogeneous Lamm-Rivière system, demonstrating that energy concentration occurs solely via finitely many bubbles and that the total energy splits between a limiting map and bubble energies. The authors develop a robust framework combining -regularity, energy gap, removable singularity, and weak compactness, with sharp neck estimates in Lorentz spaces to control tangential derivatives. They first prove the -based angular-energy identity and then extend the result to the borderline case , using extensions and Calderón–Zygmund theory. The results generalize prior energy-identity results for biharmonic and Lamm-Rivière-type systems to the inhomogeneous, critical 4D setting and provide a precise bubble-tree decomposition underpinning the energy balance. The techniques leverage Lorentz-space tools and a detailed neck-analysis to offer sharp control of concentration phenomena in high-order elliptic systems with geometric structure.

Abstract

In this paper, we establish an angular energy quantization for the following fourth order inhomogeneous Lamm-Rivière system in dimension four, with an inhomogeneous term .

Paper Structure

This paper contains 7 sections, 18 theorems, 171 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminaries and auxiliary results
  3. Function spaces
  4. Auxiliary regularity and compactness results
  5. Estimate of the tangential derivative in annular region
  6. Proof of Corollary \ref{['thm:main theorem']}
  7. Proof of Theorem \ref{['thm:main theorem for LlogL']}

Key Result

Theorem 1.1

Let $\{u_k\}\subset W^{2,2}(B_1,\mathbb{R}^n)$ be a sequence of weak solutions of with Assume that there exists a constant $\Lambda>0$ such that for all $k\in\mathbb{N}$, Then there exists a subsequence, still denoted by $u_k, V_k, w_k, \omega_k, F_k$ and $f_k$, such that $u_k\rightharpoonup u_\infty$ weakly in $W^{2,2}(B_1)$, $V_k\rightharpoonup V_\infty$ in $W^{1,2}(B_1)$, $w_k\rightharpoonup

Theorems & Definitions (30)

  • Theorem 1.1
  • Corollary 1.2
  • Proposition 2.1: lorentz3
  • Lemma 2.2: Sharp-Topping-2013-TAMS
  • Theorem 2.3: $\varepsilon$-regularity
  • Theorem 2.4: Energy gap
  • Theorem 2.5: Removable singularity
  • proof
  • Theorem 2.6: Weak compactness
  • proof
  • ...and 20 more