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Quantitative bounds for Hölder exponents in the Krylov--Safonov and Evans--Krylov theories

Jongmyeong Kim, Se-Chan Lee

TL;DR

This work provides explicit quantitative bounds on Hölder regularity in the Krylov--Safonov and Evans--Krylov theories for fully nonlinear elliptic equations when the ellipticity ratio is near one. It delivers a concrete Hölder exponent criterion $(\Lambda/\lambda)-1$-dependent on the dimension, via an Ishii--Lions approach for KS and a Bernstein/Schauder-type perturbation for EK, with parallel parabolic extensions. The results yield explicit C^{\alpha} and C^{2,\alpha} regularity criteria, plus corollaries for special operators and gradient-regime equations, and they extend to non-smooth operators through mollification. Collectively, the findings provide sharp, actionable thresholds that connect ellipticity contrast to regularity in a quantitative, constructive manner.

Abstract

We establish quantitative bounds for Hölder exponents in the Krylov--Safonov and Evans--Krylov theories when the ellipticity ratio is close to one. Our analysis relies on the Ishii--Lions method for the Krylov--Safonov theory and a Schauder-type perturbation argument for the Evans--Krylov theory.

Quantitative bounds for Hölder exponents in the Krylov--Safonov and Evans--Krylov theories

TL;DR

This work provides explicit quantitative bounds on Hölder regularity in the Krylov--Safonov and Evans--Krylov theories for fully nonlinear elliptic equations when the ellipticity ratio is near one. It delivers a concrete Hölder exponent criterion -dependent on the dimension, via an Ishii--Lions approach for KS and a Bernstein/Schauder-type perturbation for EK, with parallel parabolic extensions. The results yield explicit C^{\alpha} and C^{2,\alpha} regularity criteria, plus corollaries for special operators and gradient-regime equations, and they extend to non-smooth operators through mollification. Collectively, the findings provide sharp, actionable thresholds that connect ellipticity contrast to regularity in a quantitative, constructive manner.

Abstract

We establish quantitative bounds for Hölder exponents in the Krylov--Safonov and Evans--Krylov theories when the ellipticity ratio is close to one. Our analysis relies on the Ishii--Lions method for the Krylov--Safonov theory and a Schauder-type perturbation argument for the Evans--Krylov theory.
Paper Structure (4 sections, 20 theorems, 134 equations)

This paper contains 4 sections, 20 theorems, 134 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quantitative Hölder exponent bound in Krylov--Safonov theory
  4. Quantitative Hölder exponent bound in Evans--Krylov theory

Key Result

Theorem 1.1

If $u\in C(B_1)$ is a viscosity solution of for some constant $K\ge 0$, then $u\in C^{\alpha}(\overline B_{1/2})$ for some $\alpha=\alpha(n,\lambda,\Lambda)\in(0,1)$.

Theorems & Definitions (36)

  • Theorem 1.1: KS79KS80
  • Theorem 1.2: LY24
  • Theorem 1.3: Hölder exponent bound in the Krylov--Safonov theory
  • Theorem 1.4: Eva82Kry82Kry83
  • Theorem 1.5: WN23
  • Theorem 1.6: Hölder exponent bound in the Evans--Krylov theory
  • Definition 2.1: Viscosity solutions
  • Theorem 2.2: Ishii-Jensen's lemma; elliptic
  • Theorem 2.3: Ishii-Jensen's lemma; parabolic
  • Lemma 2.4: GT01
  • ...and 26 more