[Paper Review] Conservation laws for conformal invariant variational problems

Tristan Rivière|ArXiv.org|Mar 15, 2006

Geometric Analysis and Curvature Flows8 references4 citations

TL;DR

This paper establishes conservation laws for 2-dimensional conformally invariant variational problems, such as harmonic maps and prescribed mean curvature equations, by expressing the equations in divergence form. It proves that solutions to the prescribed bounded mean curvature equation in arbitrary target manifolds are continuous, resolving a conjecture by E. Heinz, using the structure of these conservation laws and Hardy space estimates for the nonlinearity.

ABSTRACT

We succeed in writing 2-dimensional conformally invariant non-linear elliptic PDE (harmonic map equation, prescribed mean curvature equations...etc) in divergence form. This divergence free quantities generalize to target manifolds without symmetries the well known conservation laws for harmonic maps into homogeneous spaces. From this form we can recover, without the use of moving frame, all the classical regularity results known for 2-dimensional conformally invariant non-linear elliptic PDE . It enable us also to establish new results. In particular we solve a conjecture by E.Heinz asserting that the solutions to the precribed bounded mean curvature equation in arbitrary manifolds are continuous.

Motivation & Objective

• To extend the theory of conservation laws beyond symmetric targets (e.g., spheres) to general Riemannian submanifolds.
• To establish regularity results for 2-dimensional conformally invariant nonlinear elliptic systems without relying on moving frames.
• To resolve E. Heinz's conjecture on the continuity of solutions to the prescribed bounded mean curvature equation in arbitrary target manifolds.
• To provide a unified framework for analyzing critical regularity and compactness in conformally invariant variational problems.

Proposed method

• Derives a divergence-form representation of the Euler-Lagrange equations for conformally invariant variational problems in 2D.
• Introduces a generalized conservation law structure analogous to the classical $ \mathrm{div}(u^i \nabla u^j - u^j \nabla u^i) = 0 $ for harmonic maps into spheres.
• Uses the fact that the nonlinearity lies in the local Hardy space $ \mathcal{H}^1_{\text{loc}} $, enabling improved regularity estimates.
• Applies Calderón-Zygmund theory and $ W^{2,1} $ estimates to show that the inverse Laplacian of the nonlinearity is continuous.
• Employs a continuity path argument via rescaling and a closedness/ openness argument in a parameter space to prove existence of solutions to the associated PDE system.
• Relies on elliptic regularity and Sobolev embedding theorems to control the $ W^{1,2} $ and $ W^{2,2} $ norms of the solution components.

Experimental results

Research questions

• RQ1 Can conservation laws be derived for 2-dimensional conformally invariant variational problems with arbitrary target manifolds lacking symmetry?
• RQ2 Does the nonlinearity in the prescribed mean curvature equation lie in a space that supports improved regularity estimates?
• RQ3 Are solutions to the prescribed bounded mean curvature equation in arbitrary target manifolds continuous?
• RQ4 Can the classical regularity results for harmonic maps be recovered and extended without using the moving frame method?

Key findings

• The paper constructs a divergence-form representation of the Euler-Lagrange equations for conformally invariant variational problems in 2D, generalizing the classical conservation laws for harmonic maps into spheres.
• It proves that the nonlinearity in the equation lies in the local Hardy space $ \mathcal{H}^1_{\text{loc}} $, which implies that the solution's second derivatives are in $ L^1 $, yielding the estimate $ \int_O |\nabla^2 u| \, dx < \infty $ for any open set $ O \subset B^2 $.
• The paper establishes that solutions to the prescribed bounded mean curvature equation in arbitrary target manifolds are continuous, thereby resolving E. Heinz's conjecture.
• It provides a new proof of the real analyticity of $ W^{1,2} $ solutions to the harmonic map equation in 2D, independent of the moving frame method.
• The method yields a compactness result for sequences of solutions with bounded energy, using the weak* compactness of $ \mathcal{H}^1 $ and the Rellich-Kondrachov embedding.
• The paper proves that for sufficiently small $ \varepsilon $, any $ \Omega \in W^{1,2}(D^2, \mathfrak{so}(m)) $ with $ \|\Omega\|_{L^2}^2 < \varepsilon $ lies in a closed, open, and path-connected set $ \mathcal{U}_{\varepsilon,C} $, enabling a continuity argument to solve the PDE.