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A Green's function approach to linearized Monge-Ampère equations in divergence form and application to singular Abreu type equations

Chong Gu, Nam Q. Le

TL;DR

This work addresses regularity for linearized Monge–Ampère equations in divergence form, where the coefficient matrix arises from the cofactor of the Hessian of a convex potential with Hessian determinant bounded away from zero. The authors develop sharp Lorentz-space estimates for the Green's function and its gradient, and leverage these to obtain interior and global Harnack and Hölder estimates for solutions to $L_u v = \mathrm{div}\mathbf F + \mu$, under data in Lorentz/L^p classes. A central contribution is the transfer of Green-function bounds into robust regularity results, including global Hölder estimates up to the boundary and, crucially, solvability results for singular Abreu-type fourth-order equations in all dimensions. The results extend previous affine-invariant regularity theory to all dimensions, handle measure-right-hand sides with controlled growth, and enable solvability of challenging Abreu-type problems by providing a priori Hessian bounds and higher-order estimates.

Abstract

In this paper, we establish local and global regularity estimates for linearized Monge-Ampère equations in divergence form via critical Lorentz space estimates for the Green's function of the linearized Monge-Ampère operator and its gradient. These estimates hold under suitable conditions on the data and the convex Monge-Ampère potential is assumed to have Hessian determinant bounded between two positive constants. As an application, we obtain the solvability in all dimensions of the second boundary value problem for a class of singular fourth-order Abreu type equations that arise from the approximation analysis of variational problems subject to convexity constraints.

A Green's function approach to linearized Monge-Ampère equations in divergence form and application to singular Abreu type equations

TL;DR

This work addresses regularity for linearized Monge–Ampère equations in divergence form, where the coefficient matrix arises from the cofactor of the Hessian of a convex potential with Hessian determinant bounded away from zero. The authors develop sharp Lorentz-space estimates for the Green's function and its gradient, and leverage these to obtain interior and global Harnack and Hölder estimates for solutions to , under data in Lorentz/L^p classes. A central contribution is the transfer of Green-function bounds into robust regularity results, including global Hölder estimates up to the boundary and, crucially, solvability results for singular Abreu-type fourth-order equations in all dimensions. The results extend previous affine-invariant regularity theory to all dimensions, handle measure-right-hand sides with controlled growth, and enable solvability of challenging Abreu-type problems by providing a priori Hessian bounds and higher-order estimates.

Abstract

In this paper, we establish local and global regularity estimates for linearized Monge-Ampère equations in divergence form via critical Lorentz space estimates for the Green's function of the linearized Monge-Ampère operator and its gradient. These estimates hold under suitable conditions on the data and the convex Monge-Ampère potential is assumed to have Hessian determinant bounded between two positive constants. As an application, we obtain the solvability in all dimensions of the second boundary value problem for a class of singular fourth-order Abreu type equations that arise from the approximation analysis of variational problems subject to convexity constraints.

Paper Structure

This paper contains 13 sections, 21 theorems, 178 equations.

Table of Contents

  1. Introduction
  2. Statement of the main results
  3. Lorentz space estimates for the Green's function
  4. Applications of Lorentz space estimates to the linearized Monge--Ampère equations
  5. Comments
  6. Preliminaries and a key estimate
  7. Lorentz space estimates for the Green's function
  8. Maximum principles for linearized Monge--Ampère equations
  9. Interior regularity for linearized Monge--Ampère equations
  10. Interior Harnack inequality
  11. Interior Hölder estimates
  12. Global Hölder estimates for linearized Monge--Ampère equations
  13. Application to singular Abreu type equations

Key Result

Theorem 2.1

Let $u\in C^3(\Omega)$ be a convex function satisfying MAu, where $\Omega\subset \mathbb{R}^n$. Assume $S_u(x_0, 2h) \Subset \Omega$ where $x_0\in\Omega$ and $h>0$. Let $g_{S_h}(\cdot, y)$ be the Green's function of the linearized Monge--Ampère operator $D_i(U^{ij}D_j)$ in $S_u(x_0, h)$ with pole $y where $C = C(n, \lambda, \Lambda) > 0$. Consequently, for all $y \in S_u(x_0, h)$,

Theorems & Definitions (45)

  • Definition 1.1: Sections
  • Definition 1.2: Green's function of the linearized Monge--Ampère operator
  • Definition 1.3: Proper sets
  • Definition 1.4: Lorentz spaces
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Remark 2.5
  • Remark 2.6
  • ...and 35 more