Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Interior Harnack inequality and Hölder estimates for linearized Monge-Ampère equations in divergence form with drift

Young Ho Kim

TL;DR

This work develops interior regularity theory for linearized Monge-Ampère equations in divergence form with drift, proving a Harnack inequality and Hölder continuity in two dimensions and in higher dimensions under a Hessian integrability assumption on the Monge-Ampère potential. The proofs blend Moser iteration, Monge-Ampère Sobolev inequalities, and a robust rescaling via John’s lemma, avoiding Green’s function techniques. By extending Le’s results to equations with drift and a nonzero divergence-right-hand side, the paper advances interior estimates for problems arising in semigeostrophic theory and singular Abreu-type variational equations. The results provide quantitative interior regularity under precise structural and integrability conditions, with implications for convexity-constrained variational problems and related meteorological models.

Abstract

In this paper, we study interior estimates for solutions to linearized Monge-Ampère equations in divergence form with drift terms and the right-hand side containing the divergence of a bounded vector field. Equations of this type appear in the study of semigeostrophic equations in meteorology and the solvability of singular Abreu equations in the calculus of variations with a convexity constraint. We prove an interior Harnack inequality and Hölder estimates for solutions to equations of this type in two dimensions, and under an integrability assumption on the Hessian matrix of the Monge-Ampère potential in higher dimensions. Our results extend those of Le (Analysis of Monge-Ampère equations, Graduate Studies in Mathematics, vol.240, American Mathematical Society, 2024) to equations with drift terms.

Interior Harnack inequality and Hölder estimates for linearized Monge-Ampère equations in divergence form with drift

TL;DR

This work develops interior regularity theory for linearized Monge-Ampère equations in divergence form with drift, proving a Harnack inequality and Hölder continuity in two dimensions and in higher dimensions under a Hessian integrability assumption on the Monge-Ampère potential. The proofs blend Moser iteration, Monge-Ampère Sobolev inequalities, and a robust rescaling via John’s lemma, avoiding Green’s function techniques. By extending Le’s results to equations with drift and a nonzero divergence-right-hand side, the paper advances interior estimates for problems arising in semigeostrophic theory and singular Abreu-type variational equations. The results provide quantitative interior regularity under precise structural and integrability conditions, with implications for convexity-constrained variational problems and related meteorological models.

Abstract

In this paper, we study interior estimates for solutions to linearized Monge-Ampère equations in divergence form with drift terms and the right-hand side containing the divergence of a bounded vector field. Equations of this type appear in the study of semigeostrophic equations in meteorology and the solvability of singular Abreu equations in the calculus of variations with a convexity constraint. We prove an interior Harnack inequality and Hölder estimates for solutions to equations of this type in two dimensions, and under an integrability assumption on the Hessian matrix of the Monge-Ampère potential in higher dimensions. Our results extend those of Le (Analysis of Monge-Ampère equations, Graduate Studies in Mathematics, vol.240, American Mathematical Society, 2024) to equations with drift terms.
Paper Structure (10 sections, 18 theorems, 205 equations)

This paper contains 10 sections, 18 theorems, 205 equations.

Table of Contents

  1. Introduction and Statements of the Main Results
  2. Linearized Monge-Ampère Equations
  3. The Main Results
  4. Related Results for Equations in Divergence Form
  5. Methods of the proofs
  6. Preliminaries
  7. Global Estimates
  8. Harnack Inequality
  9. Interior Estimates
  10. Interior Hölder Estimates

Key Result

Theorem 1.1

Let $\varphi\in C^3(\Omega)$ be a convex function satisfying (maeqn). Suppose that $\mathbf{F},\mathbf{b},\mathbf{B}\in W_{\mathrm{loc}}^{1,n}(\Omega;\mathbb{R}^n) \cap L_{\mathrm{loc}}^\infty(\Omega;\mathbb{R}^n)$, $f\in L_{\mathrm{loc}}^n(\Omega)$, $n/2<r\leq n$, and $\mathop{\mathrm{div}}\nolim Then, there are positive constants $C$ and $\gamma$ such that Here the constants $C$ and $\gamma$

Theorems & Definitions (35)

  • Theorem 1.1: Harnack inequality
  • Corollary 1.2: Hölder estimates with $L^\infty$ norms
  • Theorem 1.3: Hölder estimates with $L^2$ norm
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Definition 2.1: Sections
  • Theorem 2.2: John's lemma John
  • Definition 2.3: Normalized convex sets
  • Theorem 2.4: Monge-Ampère Sobolev inequality
  • ...and 25 more