[Paper Review] Non linear elliptic theory and the Monge-Ampere equation

Luis Caffarelli|ArXiv.org|Dec 1, 2002

Geometry and complex manifolds11 citations

TL;DR

This paper establishes foundational results in nonlinear elliptic PDE theory, focusing on the Monge-Ampère equation as a central model. It unifies variational, elliptic, and geometric perspectives by proving Hölder regularity of solutions via De Giorgi-type theorems and applying them to global Liouville theorems, homogenization, and optimal transport, revealing deep connections across analysis, geometry, and mathematical physics.

ABSTRACT

The Monge-Ampere equation, plays a central role in the theory of fully non linear equations. In fact we will like to show how the Monge-Ampere equation, links in some way the ideas comming from the calculus of variations and those of the theory of fully non linear equations.

Motivation & Objective

• To unify the theory of fully nonlinear elliptic equations with the calculus of variations and geometric analysis through the Monge-Ampère equation.
• To establish Hölder regularity of solutions to the Monge-Ampère equation using De Giorgi-type theorems for non-divergence form equations.
• To investigate global solutions of the Monge-Ampère equation in periodic and random media, leading to homogenization and Liouville-type theorems.
• To explore connections between the Monge-Ampère equation and optimal transportation, particularly in the context of vorticity dynamics and dimension-free concentration inequalities.
• To extend the geometric and analytic understanding of optimal transport maps beyond the quadratic cost case, including the Lagrangian formulation of the Euler-Lagrange equation.

Proposed method

• Applies De Giorgi's theorem to non-divergence form equations with bounded measurable coefficients $ A_{ij}(x) = F_{ij}(D^2u) $, proving Hölder continuity of first derivatives.
• Uses the comparison principle and monotonicity of $ F(M) = ext{trace}(M) $ or $ ext{det}(M) $ for positive definite $ M $ to establish regularity for the Monge-Ampère equation.
• Employs rescaling and quadratic transformations $ u_0001 = \varepsilon^2 u(x/\varepsilon) $ to analyze asymptotic behavior and derive Liouville-type theorems.
• Applies homogenization techniques to periodic or random right-hand sides $ f(x) $, proving existence of correctors $ w $ such that $ \det D^2(P + w) = f(x) $.
• Analyzes vorticity transport in 2D via the equation $ \det(I + D^2\psi) = \rho $, linking stream functions to velocity fields and conservation laws.
• Extends optimal transport theory to non-quadratic cost functions by studying $ \det(I + D(F_j(\nabla\psi))) $, recovering the Laplacian in the linearized limit.

Experimental results

Research questions

• RQ1 Under what conditions does a global convex solution of $ \det D^2u = f(x) $ with periodic $ f $ decompose into a quadratic polynomial plus a periodic corrector?
• RQ2 Can the Monge-Ampère equation in non-divergence form with bounded measurable coefficients $ A_{ij}(x) $ still yield Hölder continuous solutions, as in De Giorgi's theorem?
• RQ3 What is the long-time behavior of vorticity patches governed by $ \rho_t + \text{div}(v\rho) = 0 $, where $ v = -(\psi_y, \psi_x) $ and $ \det(I + D^2\psi) = \rho $?
• RQ4 How does optimal transport with a strictly convex cost $ C(X-Y) $ generalize the classical quadratic case, and what is the corresponding Euler-Lagrange equation?
• RQ5 What are the implications of the Monge-Ampère equation for concentration of measure and dimension-free inequalities in high-dimensional probability?

Key findings

• Any global convex solution $ u $ of $ \det D^2u = f(x) $ with periodic $ f $ of average $ a $ must be of the form $ u = P + w $, where $ P $ is a quadratic polynomial with $ \det D^2P = a $, and $ w $ is a periodic corrector.
• For periodic $ f $, there exists a unique periodic function $ w $ such that $ \det D^2(P + w) = f(x) $, establishing a homogenization result for the Monge-Ampère equation.
• Solutions to $ D_i A_{ij}(x) D_j w = 0 $ with bounded measurable $ A_{ij}(x) $ are Hölder continuous, generalizing De Giorgi’s theorem to non-divergence form equations.
• The vorticity equation $ \det(I + D^2\psi) = \rho $ with $ \rho $ periodic generates a velocity field $ v = -(\psi_y, \psi_x) $ that conserves $ \rho $, and stationary solutions exist when $ \rho = F(\psi) $.
• Linearizing the Monge-Ampère equation with cost $ C(X-Y) $ yields $ \det(I + \varepsilon D(F_j(\nabla\psi))) = 1 + \varepsilon \text{div}(F_j(\nabla\psi)) + O(\varepsilon^2) $, recovering the Laplacian in the limit.
• Optimal transport with strictly convex cost leads to a Lagrangian formulation of the Euler-Lagrange equation, generalizing the classical quadratic case and enabling infinite-dimensional change-of-variables formulas.