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The Dirichlet problem for second-order elliptic equations in non-divergence form with continuous coefficients

Hongjie Dong, Dong-ha Kim, Seick Kim

TL;DR

This work addresses the Dirichlet problem for second-order elliptic equations in non-divergence form with continuous coefficients, under Dini mean oscillation of the principal part. It establishes two-sided Green's function estimates in balls, proves that regular boundary points for the operator $L$ coincide with those for the Laplacian via a Wiener-type criterion, and deduces unique solvability of the Dirichlet problem in regular domains for continuous boundary data. The authors also construct Green's functions in regular domains with sharp pointwise bounds and develop a potential-theoretic framework, including capacitary measures and a capacity comparison with the Laplacian, thereby extending classical estimates beyond $C^{1,1}$ domains. Collectively, these results deepen the understanding of non-divergence form elliptic operators with lower-order terms and provide robust tools for potential theory in irregular domains.

Abstract

This paper investigates the Dirichlet problem for a non-divergence form elliptic operator $L$ in a bounded domain of $\mathbb{R}^d$. Under certain conditions on the coefficients of $L$, we first establish the existence of a unique Green's function in a ball and derive two-sided pointwise estimates for it. Utilizing these results, we demonstrate the equivalence of regular points for $L$ and those for the Laplace operator, characterized via the Wiener test. This equivalence facilitates the unique solvability of the Dirichlet problem with continuous boundary data in regular domains. Furthermore, we construct the Green's function for $L$ in regular domains and establish pointwise bounds for it. This advancement is significant, as it extends the scope of existing estimates to domains beyond $C^{1,1}$, contributing to our understanding of elliptic operators in non-divergence form.

The Dirichlet problem for second-order elliptic equations in non-divergence form with continuous coefficients

TL;DR

This work addresses the Dirichlet problem for second-order elliptic equations in non-divergence form with continuous coefficients, under Dini mean oscillation of the principal part. It establishes two-sided Green's function estimates in balls, proves that regular boundary points for the operator coincide with those for the Laplacian via a Wiener-type criterion, and deduces unique solvability of the Dirichlet problem in regular domains for continuous boundary data. The authors also construct Green's functions in regular domains with sharp pointwise bounds and develop a potential-theoretic framework, including capacitary measures and a capacity comparison with the Laplacian, thereby extending classical estimates beyond domains. Collectively, these results deepen the understanding of non-divergence form elliptic operators with lower-order terms and provide robust tools for potential theory in irregular domains.

Abstract

This paper investigates the Dirichlet problem for a non-divergence form elliptic operator in a bounded domain of . Under certain conditions on the coefficients of , we first establish the existence of a unique Green's function in a ball and derive two-sided pointwise estimates for it. Utilizing these results, we demonstrate the equivalence of regular points for and those for the Laplace operator, characterized via the Wiener test. This equivalence facilitates the unique solvability of the Dirichlet problem with continuous boundary data in regular domains. Furthermore, we construct the Green's function for in regular domains and establish pointwise bounds for it. This advancement is significant, as it extends the scope of existing estimates to domains beyond , contributing to our understanding of elliptic operators in non-divergence form.
Paper Structure (16 sections, 38 theorems, 191 equations)

This paper contains 16 sections, 38 theorems, 191 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Preliminary
  4. Assumptions on the coefficients
  5. Dirichlet problem in a ball
  6. Supersolutions, subsolutions, and Perron's method
  7. Green's function in a ball
  8. Potential and Capacity
  9. Wiener test
  10. Green's function in regular domains
  11. Proof of Lemma \ref{['lem2334sun']}
  12. Proof of Lemma \ref{['lem1033thu']}
  13. Proof of Lemma \ref{['lem1034thu']}
  14. Proof of Lemma \ref{['lem1655sun']}
  15. Proof of Lemma \ref{['lem1013sat']}
  16. ...and 1 more sections

Key Result

Theorem 1

Assume that Conditions cond1 and cond2 hold. Let $\Omega \subset \mathbb{R}^d$ be a bounded open set. A point $x_0 \in \partial \Omega$ is a regular point for $L$ (See Definition def_regpt) if and only if $x_0$ is a regular point for the Laplace operator.

Theorems & Definitions (88)

  • Theorem 1: Theorem \ref{['cor0800sat']}
  • Theorem 2: Theorem \ref{['thm0802sat']}
  • Theorem 3: Theorem \ref{['thm1127sat']}
  • Proposition 3.3
  • proof
  • Lemma 3.6
  • proof
  • Lemma 3.12
  • proof
  • Definition 3.17
  • ...and 78 more