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Interior $C^{2,α}$ regularity for fully nonlinear uniformly elliptic equations in dimension two

Kai Zhang

TL;DR

The paper proves interior $C^{2,\bar{\alpha}}$ regularity for viscosity solutions of fully nonlinear uniformly elliptic equations in two dimensions, without requiring convexity/concavity of the operator or smoothness of F. It develops a framework of global a priori $C^{2,\alpha}$ estimates for smooth solutions, establishes existence of classical solutions via the method of continuity, and extends the regularity result to viscosity solutions through a robust approximation approach. The Hölder exponent and the estimates are universal, depending only on the ellipticity constants, making the results robust to perturbations and enabling general perturbation arguments. This work strengthens the Schauder-type regularity theory in 2D and provides explicit, self-contained proofs for non-smooth settings.

Abstract

In this note, we present the interior $C^{2,α}$ regularity for viscosity solutions of fully nonlinear uniformly elliptic equations in dimension two.

Interior $C^{2,α}$ regularity for fully nonlinear uniformly elliptic equations in dimension two

TL;DR

The paper proves interior regularity for viscosity solutions of fully nonlinear uniformly elliptic equations in two dimensions, without requiring convexity/concavity of the operator or smoothness of F. It develops a framework of global a priori estimates for smooth solutions, establishes existence of classical solutions via the method of continuity, and extends the regularity result to viscosity solutions through a robust approximation approach. The Hölder exponent and the estimates are universal, depending only on the ellipticity constants, making the results robust to perturbations and enabling general perturbation arguments. This work strengthens the Schauder-type regularity theory in 2D and provides explicit, self-contained proofs for non-smooth settings.

Abstract

In this note, we present the interior regularity for viscosity solutions of fully nonlinear uniformly elliptic equations in dimension two.

Paper Structure

This paper contains 4 sections, 7 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. A priori C2,a estimate
  3. Existence of classical solutions
  4. Proof of Theorem \ref{['th1.1']}

Key Result

Theorem 1.1

Suppose that $u\in C(\bar{B}_1)$ is a viscosity solution of where $F$ is uniformly elliptic and $F(0)=0$. Then there exists a universal constant $0<\bar{\alpha}<1$ such that $u\in C^{2,\bar{\alpha}}(\bar{B}_{1/2})$ and where $C$ is universal. A constant is called universal if it depends only on $\lambda$ and $\Lambda$.

Theorems & Definitions (8)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Remark 2.4
  • Theorem 3.1
  • Lemma 4.1