[Paper Review] Decay estimates for Rivière's equation, with applications to regularity and compactness

Ben Sharp, Peter M. Topping|arXiv (Cornell University)|Feb 3, 2011

Advanced Harmonic Analysis Research11 references4 citations

TL;DR

This paper establishes sharp decay estimates for solutions to Rivière's critical equation on the unit disc in ℝ², generalizing harmonic and almost-harmonic map theory. It proves that weak solutions lie in $W^{2,p}_{\text{loc}}$ for $p \in (1,2)$ under small $L^2$ connection $\Omega$, and establishes compactness in $W^{1,2}$ under $L\ln L$ bounds on the inhomogeneous term, extending results on regularity and energy concentration in geometric PDEs.

ABSTRACT

We derive a selection of energy estimates for a generalisation of a critical equation on the unit disc in $\mathbb{R}^2$ introduced by Rivière. Applications include sharp regularity results and compactness theorems which generalise a large amount of previous geometric PDE theory, including some of the theory of harmonic and almost-harmonic maps from surfaces.

Motivation & Objective

• To establish sharp $L^p$ regularity estimates for solutions to Rivière's equation $-\Delta u = \Omega \cdot \nabla u + f$ on the unit disc in $\mathbb{R}^2$.
• To extend regularity theory beyond the classical $W^{2,2}$ and $C^{1,\alpha}$ regimes, particularly in the critical $L^2$-connection setting.
• To prove compactness of solution sequences under minimal integrability assumptions on the inhomogeneous term $f$, specifically $f \in L\ln L$.
• To rule out energy concentration in $W^{1,2}$ while allowing potential concentration in higher-order norms, and to characterize such behavior via $f$-dependence.

Proposed method

• Derives $L^p$ energy estimates for the generalized Rivière equation using weighted Sobolev and Lorentz space techniques.
• Applies a smallness condition $\|\Omega\|_{L^2} \leq \eta_0$ to control the non-elliptic structure of the equation and obtain $W^{2,p}$ bounds.
• Uses the Rellich-Kondrachov theorem to deduce precompactness in $W^{1,t}$ for $t < \frac{2p}{2-p}$ from $W^{2,p}$ control.
• Employs a novel absorption lemma (Lemma A.7) to control oscillatory behavior and derive decay estimates.
• Analyzes concentration phenomena via Radon measure theory, showing that $L\ln L$ bounds on $f_n$ are sharp for compactness.
• Applies the theory of $BV$ functions and differentiation of measures to control the limit of $|V_n|^2$ and $|\nabla V_n|$.

Experimental results

Research questions

• RQ1 Can sharp $W^{2,p}$ regularity estimates be established for solutions to Rivière's equation when $f \in L^p$, $p < 2$, and $\|\Omega\|_{L^2}$ is small?
• RQ2 What is the optimal integrability of the inhomogeneous term $f$ that still allows compactness of solution sequences in $W^{1,2}$?
• RQ3 Can compactness be preserved even when second derivatives concentrate, provided $f_n$ is bounded in $L\ln L$?
• RQ4 Is the $L\ln L$ space sharp for compactness, or can it be replaced by a larger space like $h^1$?

Key findings

• For $f \in L^p(B_1)$, $p \in (1,2)$, and $\|\Omega\|_{L^2} \leq \eta_0$, solutions satisfy $\|u\|_{W^{2,p}(U)} \leq C(\|f\|_{L^p} + \|u\|_{L^1})$ for any $U \subset\subset B_1$, proving $W^{2,p}_{\text{loc}}$ regularity.
• The solution $u$ lies in $W^{1,q}_{\text{loc}}$ for all $q < \infty$, and in $C^{0,2(1-1/p)}$ for $f \in L^p$, recovering a result of Rupflin.
• Compactness holds in $W^{1,2}(B_{1/2})$ for sequences $u_n$ with $\|\Omega_n\|_{L^2} \leq \eta_2$, $\|u_n\|_{L^1} + \|f_n\|_{L\ln L} \leq \Lambda$, implying convergence after passing to a subsequence.
• The result fails if $f_n$ is bounded in $h^1$ instead of $L\ln L$, showing $L\ln L$ is sharp for the compactness result.
• The estimate (3) fails without the smallness condition on $\Omega$, as demonstrated by sequences with bounded $\|\Omega_k\|_{L^2}$ and unbounded $\|u_k\|_{W^{2,p}}$.
• Concentration of $\|\nabla^2 u_n\|_{L^1}$ is possible only if $f_n$ concentrates in $L\ln L$, linking higher-order concentration to the inhomogeneous term.