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Recent progress on second-order elliptic and parabolic equations in double divergence form

Seick Kim

TL;DR

The paper surveys recent advances in second-order elliptic equations in double divergence form and their parabolic counterparts, focusing on existence, regularity, and quantitative estimates under structured coefficient classes. It develops a coherent regularity theory using the transposition method, $\mathrm{DMO}$/Morrey-type conditions, and associated modulus of continuity to obtain continuity, higher integrability, and Harnack inequalities, including extensions to lower-order terms. Key contributions include sharp weak-solution frameworks, higher integrability under $\mathrm{DMO}$, global continuity results, and parabolic analogues with corresponding Harnack principles, enabling Green's-function-type bounds and robust boundary behavior. The results significantly broaden the PDE toolbox for double divergence operators, enabling finer qualitative understanding in non-smooth settings and across elliptic and parabolic regimes, with potential applications to diffusion processes and invariant measures.

Abstract

This article presents a comprehensive overview and supplement to recent developments in second-order elliptic partial differential equations formulated in double divergence form, along with an exploration of their parabolic counterparts.

Recent progress on second-order elliptic and parabolic equations in double divergence form

TL;DR

The paper surveys recent advances in second-order elliptic equations in double divergence form and their parabolic counterparts, focusing on existence, regularity, and quantitative estimates under structured coefficient classes. It develops a coherent regularity theory using the transposition method, /Morrey-type conditions, and associated modulus of continuity to obtain continuity, higher integrability, and Harnack inequalities, including extensions to lower-order terms. Key contributions include sharp weak-solution frameworks, higher integrability under , global continuity results, and parabolic analogues with corresponding Harnack principles, enabling Green's-function-type bounds and robust boundary behavior. The results significantly broaden the PDE toolbox for double divergence operators, enabling finer qualitative understanding in non-smooth settings and across elliptic and parabolic regimes, with potential applications to diffusion processes and invariant measures.

Abstract

This article presents a comprehensive overview and supplement to recent developments in second-order elliptic partial differential equations formulated in double divergence form, along with an exploration of their parabolic counterparts.

Paper Structure

This paper contains 14 sections, 13 theorems, 151 equations.

Table of Contents

  1. Introduction
  2. Elliptic equations in double divergence form
  3. Definition of weak solutions
  4. Existence of weak solutions
  5. Higher integrability of weak solutions
  6. Continuity
  7. Harnack inequality
  8. Elliptic equations with coefficients in Morrey classes
  9. Parabolic equations in double divergence form
  10. Definition of weak solutions
  11. Existence of weak solutions
  12. Higher integrability of weak solutions
  13. Continuity
  14. Parabolic Harnack inequality

Key Result

Theorem 2.4

Let $\Omega$ be a bounded $C^{1,1}$ domain in $\mathbb{R}^d$ with $d\ge 3$. Assume $\mathbf A \in \mathrm{VMO}$, $\boldsymbol b \in L_{p_0}(\Omega)$, $c \in L_{p_0/2}(\Omega)$ with $p_0>d$. Additionally, assume $c \ge 0$. Let $\mathbf f \in L_{q_0}(\Omega)$, $\boldsymbol g \in L_{d q_0/(d+q_0)}(\Ome satisfying the estimate eq1742sat. Moreover, the identity eq0958sat holds with any $\eta \in \mathr

Theorems & Definitions (16)

  • Definition 2.1
  • Theorem 2.4
  • Theorem 2.14
  • Theorem 2.15
  • Theorem 2.18
  • Theorem 2.19
  • Theorem 2.20
  • Theorem 2.21
  • Lemma 3.3
  • proof
  • ...and 6 more