[Paper Review] Boundary regularity for solutions to the linearized Monge-Ampère equations

Nam Q. Le, Ovidiu Savin|arXiv (Cornell University)|Sep 26, 2011

Geometry and complex manifolds13 references4 citations

TL;DR

This paper establishes boundary Hölder gradient estimates and regularity for solutions to the linearized Monge-Ampère equation $ L_u v = g $, under natural assumptions on the domain, Hessian determinant bounds, and quadratic boundary separation. Using affine-invariant techniques, including localization theorems and rescaling, it proves that solutions gain $ C^{1,eta} $ regularity near the boundary even when the operator is degenerate or singular, extending Krylov's boundary estimates to the affine-invariant setting of Monge-Ampère equations.

ABSTRACT

We obtain boundary Holder gradient estimates and regularity for solutions to the linearized Monge-Ampere equations under natural assumptions on the domain, Monge-Ampere measures and boundary data. Our results are affine invariant analogues of the boundary Holder gradient estimates of Krylov.

Motivation & Objective

• To establish boundary Hölder gradient estimates for solutions to the linearized Monge-Ampère equation $ L_u v = g $ in convex domains.
• To extend Krylov's boundary regularity theory to the affine-invariant setting of Monge-Ampère equations.
• To address the challenge of degeneracy and singularity in $ L_u $, especially near the boundary, where eigenvalues of the cofactor matrix $ U^{ij} $ may vanish or blow up.
• To prove that solutions gain $ C^{1,eta} $ regularity under natural geometric and measure-theoretic assumptions on $ u $, $ \Omega $, and the data $ g $.

Proposed method

• Utilizes the Localization Theorem at the boundary for solutions to Monge-Ampère equations, which provides a geometric framework for rescaling near boundary points.
• Employs rescaling of solutions via affine transformations to normalize sections of $ u $, transforming them into standard ellipsoids $ E_h $, enabling uniform analysis.
• Applies weak Harnack inequalities and interior regularity estimates (e.g., Schauder-type) to the rescaled functions $ \tilde{v} $, which satisfy $ \tilde{U}^{ij}\tilde{v}_{ij} = \tilde{g} $.
• Relies on the quadratic separation condition: $ \rho |x - x_0|^2 \leq u(x) - u(x_0) - \nabla u(x_0)(x - x_0) \leq \rho^{-1} |x - x_0|^2 $ on $ \partial\Omega \cap B_\rho $, ensuring uniform control near the boundary.
• Uses the fact that $ U = (\det D^2 u)(D^2 u)^{-1} $ is positive semi-definite and divergence-free, making $ L_u $ a linear elliptic operator, possibly degenerate.
• Applies interior $ C^{2,\beta} $ and $ C^{1,\alpha} $ estimates to rescaled solutions and lifts them back to the original domain to obtain global Hölder bounds on $ \nabla v $.

Experimental results

Research questions

• RQ1 Under what geometric and measure-theoretic conditions does the linearized Monge-Ampère operator $ L_u $ admit boundary Hölder gradient estimates for solutions to $ L_u v = g $?
• RQ2 How can one achieve $ C^{1,\beta} $ regularity for $ v $ when $ L_u $ is degenerate or singular near the boundary, particularly when $ \det D^2 u $ is bounded but $ U^{ij} $ may blow up?
• RQ3 Can affine-invariant techniques, such as localization and rescaling, be used to extend Krylov-type boundary estimates to the Monge-Ampère setting?
• RQ4 What role does the quadratic separation of $ u $ from its tangent planes on $ \partial\Omega $ play in controlling the behavior of $ \nabla v $?
• RQ5 To what extent do the regularity results depend on the Hessian determinant bounds $ \lambda \leq \det D^2 u \leq \Lambda $, and how do they behave under $ C^\beta $ or continuous data?

Key findings

• Solutions $ v $ to $ L_u v = g $ satisfy $ \|\nabla v\|_{C^{\alpha}(\overline{\Omega})} \leq C $ for some universal $ \alpha \in (0,1) $, provided $ \Omega $ satisfies interior ball conditions and $ u $ has bounded Hessian determinant and quadratic boundary separation.
• The boundary Hölder gradient estimate is affine-invariant and extends Krylov's classical results to the Monge-Ampère setting, even when $ L_u $ is not uniformly elliptic.
• The degeneracy of $ L_u $ is handled via localization: sections of $ u $ are rescaled into standard ellipsoids $ E_h $, allowing uniform analysis.
• The singularity of $ U^{ij} $ near the boundary is controlled by showing that $ \|U\| $ grows at most logarithmically, and this is compensated via rescaling and weak Harnack inequalities.
• For $ f = \det D^2 u \in C^\beta(\overline{\Omega}) $, the rescaled solution $ \tilde{u} $ satisfies $ \|D^2 \tilde{u}\|_{C^\beta(\tilde{S}_{1/2})} \leq K $, leading to $ \|D^2 u(y)\| \leq K |\log r|^2 $, which is integrable and leads to $ C^{1,\alpha} $ regularity.
• The results are robust: even if $ f \in C(\overline{\Omega}) $, the $ C^{1,\alpha} $ bound for $ v $ still holds, relying on $ C^{1,\alpha} $ estimates from Gutiérrez and Nguyen.