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Extremal Alexandrov estimates: singularities, obstacles, and stability

Tianling Jin, Xushan Tu, Jingang Xiong

TL;DR

This work advances the theory of Alexandrov-type estimates for convex functions by proving that the classical 1/n exponent is not optimal in the small-misfit regime when the background potential φ is non-degenerate. The authors identify two canonical MA extremizers—the isolated singularity and the linear obstacle—that govern the sharp, mass-discrepancy–dependent bounds, establishing a^2 (and a^2|log a| in 2D) scaling and, under higher regularity, precise pointwise asymptotics with explicit leading constants. A pointwise variational framework, a propagation estimate for small MA perturbations via the linearized MA operator, and a divergence-form identity reduce the problem to model configurations, yielding both global and interior sharp bounds and a rigidity phenomenon for almost-extremal cases in dimensions n≥3. The results extend to an obstacle/MA framework with consequences for strict convexity and regularity of solutions to MA equations with isolated singularities or multiple obstacles, highlighting how near-extremal configurations concentrate mass near the evaluation point and quantifying stability in terms of the MA measure.

Abstract

The classical Alexandrov estimate controls the oscillation of a convex function by the mass of its associated Monge-Ampère measure and yields, for two convex functions of $n$ variables with the same boundary values, a sup-norm bound with exponent $1/n$ in the measure discrepancy. We show that this exponent is not optimal in the small-discrepancy regime once one of the functions is non-degenerate in the sense of having Monge-Ampère density bounded above and below by two positive constants. We prove sharp quantitative estimates comparing two convex functions by the total variation of the difference of their Monge-Ampère measures: in dimensions $n\ge 3$ the optimal dependence is quadratic in the natural mass scale, while in dimension $n=2$ the optimal dependence contains a logarithmic correction. These rates are shown to be optimal for all small discrepancies. A key structural ingredient is a characterization of extremizers. We identify the pointwise minimizers and maximizers in the admissible class and prove that they are realized, respectively, by solutions to Monge-Ampère equations with an isolated singularity and by solutions to Monge-Ampère equations with a linear obstacle. This extremal description reduces the sharp estimates to a precise asymptotic analysis of these two model configurations. Assuming further that the domain and the non-degenerate reference function are $C^{2,α}$ and uniformly convex, we obtain sharp pointwise two-sided asymptotics at interior points with explicit leading constants. Finally, in dimensions $n\ge 3$ we establish a stability phenomenon: if the pointwise estimate is nearly saturated, then the measure discrepancy must concentrate near the point at the natural scale, quantifying rigidity of almost-extremal configurations.

Extremal Alexandrov estimates: singularities, obstacles, and stability

TL;DR

This work advances the theory of Alexandrov-type estimates for convex functions by proving that the classical 1/n exponent is not optimal in the small-misfit regime when the background potential φ is non-degenerate. The authors identify two canonical MA extremizers—the isolated singularity and the linear obstacle—that govern the sharp, mass-discrepancy–dependent bounds, establishing a^2 (and a^2|log a| in 2D) scaling and, under higher regularity, precise pointwise asymptotics with explicit leading constants. A pointwise variational framework, a propagation estimate for small MA perturbations via the linearized MA operator, and a divergence-form identity reduce the problem to model configurations, yielding both global and interior sharp bounds and a rigidity phenomenon for almost-extremal cases in dimensions n≥3. The results extend to an obstacle/MA framework with consequences for strict convexity and regularity of solutions to MA equations with isolated singularities or multiple obstacles, highlighting how near-extremal configurations concentrate mass near the evaluation point and quantifying stability in terms of the MA measure.

Abstract

The classical Alexandrov estimate controls the oscillation of a convex function by the mass of its associated Monge-Ampère measure and yields, for two convex functions of variables with the same boundary values, a sup-norm bound with exponent in the measure discrepancy. We show that this exponent is not optimal in the small-discrepancy regime once one of the functions is non-degenerate in the sense of having Monge-Ampère density bounded above and below by two positive constants. We prove sharp quantitative estimates comparing two convex functions by the total variation of the difference of their Monge-Ampère measures: in dimensions the optimal dependence is quadratic in the natural mass scale, while in dimension the optimal dependence contains a logarithmic correction. These rates are shown to be optimal for all small discrepancies. A key structural ingredient is a characterization of extremizers. We identify the pointwise minimizers and maximizers in the admissible class and prove that they are realized, respectively, by solutions to Monge-Ampère equations with an isolated singularity and by solutions to Monge-Ampère equations with a linear obstacle. This extremal description reduces the sharp estimates to a precise asymptotic analysis of these two model configurations. Assuming further that the domain and the non-degenerate reference function are and uniformly convex, we obtain sharp pointwise two-sided asymptotics at interior points with explicit leading constants. Finally, in dimensions we establish a stability phenomenon: if the pointwise estimate is nearly saturated, then the measure discrepancy must concentrate near the point at the natural scale, quantifying rigidity of almost-extremal configurations.
Paper Structure (23 sections, 35 theorems, 316 equations)

This paper contains 23 sections, 35 theorems, 316 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Ideas of the proof
  4. Notation and Conventions
  5. Reduction by normalization
  6. Reduction by the comparison principle
  7. Reduction to one-signed measures
  8. Reduction to interior estimates
  9. Reduction to one-sided boundary inequalities
  10. The extremal configurations in Alexandrov estimates
  11. The isolated singularity problem and Alexandrov estimates revisited
  12. The obstacle problem
  13. Proof of Theorem \ref{['thm:abp extremal']} and further discussions
  14. An optimal Alexandrov estimate
  15. Proof of estimate \ref{['eq:abp gene 2/n']}
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $\Omega \subset \mathbb{R}^n$, $n \geq 2$, be a convex domain satisfying $B_1(0) \subset \Omega \subset B_n(0)$, and let $\varphi \in C(\overline{\Omega})$ be a convex function satisfying eq:varphi equation in $\Omega$. Then there exists a positive constant $C$ depending only on $n,\lambda$ and where If we additionally assume $\varphi$ is strictly convex and satisfies eq:varphi equation in a

Theorems & Definitions (76)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 3.1
  • Remark 3.2
  • Theorem 3.3
  • proof
  • Lemma 3.4
  • proof
  • Remark 3.5
  • ...and 66 more