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Conservation law for a fourth order elliptic system in supercritical dimensions

Wen-Juan Qi, Zhao-Min Sun

Abstract

In this short note, we extend the conservation law of Lamm-Rivière [Comm. PDEs 2008] for a fourth order elliptic system to supercritical dimensions, under certain Lorentz-Sobolev integrability assumptions on the associated coefficient functions.

Conservation law for a fourth order elliptic system in supercritical dimensions

Abstract

In this short note, we extend the conservation law of Lamm-Rivière [Comm. PDEs 2008] for a fourth order elliptic system to supercritical dimensions, under certain Lorentz-Sobolev integrability assumptions on the associated coefficient functions.
Paper Structure (3 sections, 4 theorems, 45 equations)

This paper contains 3 sections, 4 theorems, 45 equations.

Table of Contents

  1. Introduction and main results
  2. Auxillary results
  3. Proof of Theorem \ref{['thm:main theorem']}

Key Result

Theorem 1.1

For any $n,m\geq 4$, there exist canstants $\epsilon_{n,m},C_{n,m}>0$ satisfying the following property. Suppose $V,w,F,\omega$ satisfy the smallness condition Then there exist $A\in W^{3,\frac{n}{3},1}\cap L^{\infty}(B^{n},Gl(m))$ and $B\in W^{1,\frac{n}{3},2}(B^{n},M(m)\otimes\wedge^{2}{\mathbb R}^{n})$ such that eq:fourth order for CL holds in $B^n$. Moreover, we have the following estimate

Theorems & Definitions (7)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • proof : Proof of Theorem \ref{['thm:main theorem']}