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Interior $BMO$ regularity for elliptic equations in divergence form

Yuanyuan Lian

TL;DR

The paper proves interior $BMO$ regularity for weak solutions to uniformly elliptic divergence-form equations with lower-order terms under near-optimal smallness assumptions. It employs a perturbation strategy around the De Giorgi class, leveraging the De Giorgi class’s Hölder regularity and a scaling/stability framework to obtain pointwise $BMO$ control. The main result shows that, when the data satisfy critical-space smallness (e.g., $m{b} o L^n$, $c o L^{n/2}$, $m{d} o L^n$, and $f,m{f}$ with suitable smoothness), the solution lies in $BMO_2$ (locally, with quantitative bounds). The approach highlights a Schauder-type perspective in the elliptic setting and demonstrates near-optimality of assumptions, contributing a robust method for borderline regularity in divergence-form elliptic equations.

Abstract

In this note, we establish the interior $BMO$ regularity of weak solutions to uniformly elliptic equations in divergence form. Moreover, the assumptions on the coefficients are nearly optimal.

Interior $BMO$ regularity for elliptic equations in divergence form

TL;DR

The paper proves interior regularity for weak solutions to uniformly elliptic divergence-form equations with lower-order terms under near-optimal smallness assumptions. It employs a perturbation strategy around the De Giorgi class, leveraging the De Giorgi class’s Hölder regularity and a scaling/stability framework to obtain pointwise control. The main result shows that, when the data satisfy critical-space smallness (e.g., , , , and with suitable smoothness), the solution lies in (locally, with quantitative bounds). The approach highlights a Schauder-type perspective in the elliptic setting and demonstrates near-optimality of assumptions, contributing a robust method for borderline regularity in divergence-form elliptic equations.

Abstract

In this note, we establish the interior regularity of weak solutions to uniformly elliptic equations in divergence form. Moreover, the assumptions on the coefficients are nearly optimal.
Paper Structure (2 sections, 7 theorems, 77 equations)

This paper contains 2 sections, 7 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. BMO regularity

Key Result

Theorem 1.1

Let $u\in W^{1,2}(B_1)$ be a weak solution of e.div-0. Suppose that where $\delta_0>0$ (small) is universal. Then $u\in BMO_2(0)$ and where $C$ depends only on $n,\lambda, \Lambda$ and $\bm{b}$.

Theorems & Definitions (16)

  • Definition 1
  • Definition 2
  • Remark 1
  • Remark 2
  • Theorem 1.1: Pointwise $BMO$ regularity
  • Remark 3
  • Remark 4
  • Remark 5
  • Corollary 1: Local $BMO$ regularity
  • Corollary 2: Local $BMO$ regularity
  • ...and 6 more