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Local potential and Hölder estimates for the linearized Monge-Ampère equation

Guoqing Cui, Ling Wang, Bin Zhou

Abstract

In this paper, we establish local potential estimates and Hölder estimates for solutions of linearized Monge-Ampère equations with the right-hand side being a signed measure, under suitable assumptions on the data. In particular, the interior Hölder estimate holds for an inhomogeneous linearized Monge-Ampère equation with right-hand side being the nonnegative divergence of a bounded vector field in all dimensions. As an application, we give a new approach for the interior estimate of the singular Abreu equation.

Local potential and Hölder estimates for the linearized Monge-Ampère equation

Abstract

In this paper, we establish local potential estimates and Hölder estimates for solutions of linearized Monge-Ampère equations with the right-hand side being a signed measure, under suitable assumptions on the data. In particular, the interior Hölder estimate holds for an inhomogeneous linearized Monge-Ampère equation with right-hand side being the nonnegative divergence of a bounded vector field in all dimensions. As an application, we give a new approach for the interior estimate of the singular Abreu equation.

Paper Structure

This paper contains 13 sections, 17 theorems, 148 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lorentz spaces
  4. Monge-Ampère Sobolev inequality
  5. The homogeneous linearized Monge-Ampère equation
  6. Hölder continuity by growth of local integrals
  7. Riesz potential
  8. potential estimate
  9. Hölder regularity
  10. Hölder regularity when the right-hand side is a non-negative Radon measure
  11. Hölder regularity when the right-hand side is a signed Radon measure
  12. Proof of Theorem \ref{['thm:divF+f']}
  13. Application to Singular Abreu equations

Key Result

Theorem 1.1

Let $u\in C^2(\Omega)$ be a convex function satisfying condition, $v$ be a weak solution of eq: +- form, $x_0$ be a Lebesgue point of $v$ and $S_u(x_0,2h_0)\Subset\Omega$. Then for any $p>0$, there exists $C>0$ depending only on $n,\lambda,\Lambda$ and $p$ such that where $v_{+}=\max\{v,0\}$, $v_{-}=\max\{-v,0\}$ and $I_u^{\mu_\pm}$ is the Riesz potential with respect to $\mu_{\pm}$ (see eq:Riesz

Figures (1)

  • Figure 1: Boundary area controlled by volume

Theorems & Definitions (32)

  • Theorem 1.1: Local potential estimate
  • Theorem 1.2: Interior Hölder estimate
  • Remark 1.3
  • Theorem 1.4: Interior Hölder estimate with right-hand side in divergence form
  • Remark 1.5
  • Definition 2.1
  • Lemma 2.2: Monge-Ampère Sobolev inequality
  • Lemma 2.3: Monge-Ampère Poincaré inequality
  • Lemma 2.4: Local boundedness
  • Lemma 2.5: Weak Harnack inequality
  • ...and 22 more