Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Boundary estimates for non-divergence equations in $C^1$ domains

Clara Torres-Latorre

TL;DR

The paper addresses boundary behavior for solutions to non-divergence elliptic equations in $C^1$ domains by establishing a quantitative Hopf-Oleinik-type nondegeneracy estimate and a boundary regularity result with an explicit modulus of continuity. The approach combines a novel barrier construction based on a locally regularized distance with a dyadic analysis to bridge Lipschitz and $C^1$ boundary regimes, yielding results that persist under bounded measurable coefficients and extend to $C^{1,\mathrm{Dini}}$ boundaries. When the boundary modulus is Dini, the framework recovers classical Lipschitz regularity and, for $C^1$ domains with Dini-continuous coefficients, provides explicit moduli of continuity for solutions up to the boundary. The work thus unifies interior and boundary regularity phenomena within a single argument and has potential applications to free boundary problems.

Abstract

We obtain boundary nondegeneracy and regularity estimates for solutions to non-divergence equations in $C^1$ domains, providing an explicit modulus of continuity. Our results extend the classical Hopf-Oleinik lemma and boundary Lipschitz regularity for domains with $C^{1,\mathrm{Dini}}$ boundaries, while also recovering the known $C^{1-\varepsilon}$ regularity for flat Lipschitz domains, unifying both theories with a single proof.

Boundary estimates for non-divergence equations in $C^1$ domains

TL;DR

The paper addresses boundary behavior for solutions to non-divergence elliptic equations in domains by establishing a quantitative Hopf-Oleinik-type nondegeneracy estimate and a boundary regularity result with an explicit modulus of continuity. The approach combines a novel barrier construction based on a locally regularized distance with a dyadic analysis to bridge Lipschitz and boundary regimes, yielding results that persist under bounded measurable coefficients and extend to boundaries. When the boundary modulus is Dini, the framework recovers classical Lipschitz regularity and, for domains with Dini-continuous coefficients, provides explicit moduli of continuity for solutions up to the boundary. The work thus unifies interior and boundary regularity phenomena within a single argument and has potential applications to free boundary problems.

Abstract

We obtain boundary nondegeneracy and regularity estimates for solutions to non-divergence equations in domains, providing an explicit modulus of continuity. Our results extend the classical Hopf-Oleinik lemma and boundary Lipschitz regularity for domains with boundaries, while also recovering the known regularity for flat Lipschitz domains, unifying both theories with a single proof.

Paper Structure

This paper contains 15 sections, 11 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Background
  4. The boundary point lemma
  5. Boundary regularity
  6. Parallels with interior regularity
  7. Strategies of proof
  8. Connection to free boundary problems
  9. Acknowledgements.
  10. Plan of the paper
  11. Preliminaries
  12. Setting
  13. Technical tools
  14. Barriers
  15. Quantitative nondegeneracy and regularity in C1 domains

Key Result

Theorem 1.1

Let $\mathcal{L}$ be as in eq:non-divergence_operator, let $\Omega$ satisfy the interior $C^1$ condition at $0$ with modulus $\omega$ in the sense of Definition defn:interiorC1, and let $u$ be a nonnegative solution to $\mathcal{L} u = 0$ in $\Omega$. Then, for every $0 < \rho < r < r_0$, where $C$ and $r_0$ are positive and depend only on $\omega$, the dimension, and ellipticity constants.

Theorems & Definitions (28)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Lemma 1.4
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • ...and 18 more