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

Pêdra D. S. Andrade, Clara Torres-Latorre

Abstract

We obtain boundary nondegeneracy and regularity estimates for solutions to non-divergence form parabolic equations in parabolic $C^1$ domains, providing explicit moduli 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 parabolic Lipschitz domains, unifying both regimes with a single proof.

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

Abstract

We obtain boundary nondegeneracy and regularity estimates for solutions to non-divergence form parabolic equations in parabolic domains, providing explicit moduli 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 parabolic Lipschitz domains, unifying both regimes with a single proof.

Paper Structure

This paper contains 10 sections, 15 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Background
  4. Plan of the paper
  5. Preliminaries
  6. Setting
  7. Auxiliary results
  8. Barriers
  9. Quantitative nondegeneracy and regularity in C1 domains
  10. Proof of Theorem \ref{['thm:interior_reg_modulus']}

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 Then, for every $0 < \rho < \frac{r}{4} < \frac{r_0}{4}$, where $C$ and $r_0$ are positive and depend only on $\omega$, the dimension, and ellipticity constants. $\blacktrianglele

Theorems & Definitions (33)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • ...and 23 more