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Regularity of elliptic equations in double divergence form and applications to Green's function estimates

Jongkeun Choi, Hongjie Dong, Dong-ha Kim, Seick Kim

TL;DR

The paper advances regularity theory for elliptic equations in double divergence form with coefficients of Dini mean oscillation, proving interior and boundary differentiability of weak solutions on zero level and on $C^{1,\alpha}$ boundaries, respectively. It develops a two-tier strategy: first, a simple-case Campanato-type contraction to obtain differentiability, then a generalization to nonzero data via careful averaging and moduli of continuity $\omega_{\rm coef}$ and $\omega_{\rm dat}$, with sharp estimates that depend on DMO. These interior and boundary results feed into global Green's function estimates for non-divergence form operators with DMO coefficients, yielding optimal pointwise bounds that reflect both interior and boundary distances. The methods hinge on decompositions, boundary flattening, DMO control of coefficients, and Campanato-type arguments, and open the door to refined Green's function behavior and potential-theoretic applications in domains with mild coefficient regularity.

Abstract

We investigate the regularity of elliptic equations in double divergence form, where the leading coefficients satisfying the Dini mean oscillation condition. We prove that the solutions are differentiable on the zero level set and derive a pointwise bound for the derivative, which substantially improve a recent result by Leitão, Pimentel, and Santos (Anal. PDE 13(4):1129--1144, 2020). As an application, we establish global pointwise estimates for the Green's function of second-order uniformly elliptic operators in non-divergence form, considering Dini mean oscillation coefficients in bounded $C^{1,α}$ domains. This result extends a recent work by Chen and Wang (Electron. J. Probab. 28(36):54 pp, 2023).

Regularity of elliptic equations in double divergence form and applications to Green's function estimates

TL;DR

The paper advances regularity theory for elliptic equations in double divergence form with coefficients of Dini mean oscillation, proving interior and boundary differentiability of weak solutions on zero level and on boundaries, respectively. It develops a two-tier strategy: first, a simple-case Campanato-type contraction to obtain differentiability, then a generalization to nonzero data via careful averaging and moduli of continuity and , with sharp estimates that depend on DMO. These interior and boundary results feed into global Green's function estimates for non-divergence form operators with DMO coefficients, yielding optimal pointwise bounds that reflect both interior and boundary distances. The methods hinge on decompositions, boundary flattening, DMO control of coefficients, and Campanato-type arguments, and open the door to refined Green's function behavior and potential-theoretic applications in domains with mild coefficient regularity.

Abstract

We investigate the regularity of elliptic equations in double divergence form, where the leading coefficients satisfying the Dini mean oscillation condition. We prove that the solutions are differentiable on the zero level set and derive a pointwise bound for the derivative, which substantially improve a recent result by Leitão, Pimentel, and Santos (Anal. PDE 13(4):1129--1144, 2020). As an application, we establish global pointwise estimates for the Green's function of second-order uniformly elliptic operators in non-divergence form, considering Dini mean oscillation coefficients in bounded domains. This result extends a recent work by Chen and Wang (Electron. J. Probab. 28(36):54 pp, 2023).
Paper Structure (10 sections, 9 theorems, 243 equations)

This paper contains 10 sections, 9 theorems, 243 equations.

Table of Contents

  1. Introduction and main results
  2. Proof of Theorem \ref{['thm-main01']}: A simple case
  3. Proof of Theorem \ref{['thm-main01']}: General case
  4. Proof of Theorem \ref{['thm-main02']}
  5. Special case: $\partial\Omega$ is a flat near $x^o$.
  6. General case.
  7. Applications to Green's function estimates
  8. Appendix
  9. Proof of Lemma \ref{['lem01']}
  10. Proof of Lemma \ref{['lem2059mon']}

Key Result

Theorem 1.6

Let $\Omega \subset \mathbb{R}^d$ be a domain, and assume the conditions ellipticity-nd--cond_dat hold. Let $u \in L^\infty_{\rm loc}(\Omega)$ be a weak solution of If $u(x^o)=0$ for some $x^o\in\Omega$, then $u$ is differentiable at $x^o$. More precisely, there exist moduli of continuity $\varrho_{\rm coef}$ and $\varrho_{\rm dat}$ determined by $\omega_{\rm coef}$ and $\omega_{\rm dat}$ in omeg

Theorems & Definitions (23)

  • Theorem 1.6
  • Theorem 1.7
  • Remark 1.8
  • Remark 2.1
  • Remark 2.10
  • Lemma 2.25
  • proof
  • Lemma 2.31
  • proof
  • Remark 2.37
  • ...and 13 more