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Fabes-Stroock approach to higher integrability of Green's functions and ABP estimates with $L_d$ drift

Pilgyu Jung, Kwan Woo

TL;DR

The paper studies higher integrability of Green's functions for the linear elliptic operator $\mathcal L u = a^{ij}D_{ij}u + b^iD_i u$ with drift $b \in L_d(\Omega)$ in a bounded domain. It adopts a Fabes–Stroock-type analytic framework to prove a Gehring-type inequality for Green's functions, exploiting Aleksandrov solutions and Monge–Ampère convex corrections to handle the critical drift and a scaling-invariance property of the $L_d$ norm of $b$. The main contributions are an ABP-type estimate refined to the $L_d$ drift case and a higher integrability result: there exist $q>d/(d-1)$ and constants $N$ such that $\sup_{x\in\Omega} \left( \int_{\Omega} G_{\Omega}(x,y)^q \, dy \right)^{1/q} \le N\mathrm{diam}(\Omega)^{2-d/p}$ with $1/p+1/q=1$. This provides an analytic alternative to Krylov's probabilistic proofs and strengthens the $L_p$-theory for elliptic equations with critical drift, enabling refined ABP-type estimates and potential applications in stochastic analysis.

Abstract

We explore the higher integrability of Green's functions associated with the second-order elliptic equation $a^{ij}D_{ij}u + b^i D_iu = f$ in a bounded domain $Ω\subset \mathbb{R}^d$, and establish an enhanced version of Aleksandrov's maximum principle. In particular, we consider the drift term $b=(b^1, \ldots, b^d)$ in $L_d$ and the source term $f \in L_p$ for some $p < d$. This provides an alternative and analytic proof of a result by N. V. Krylov (\textit{Ann. Probab.}, 2021) concerning $L_d$ drifts. The key step involves deriving a Gehring-type inequality for Green's functions by using the Fabes-Stroock approach (\textit{Duke Math. J.}, 1984).

Fabes-Stroock approach to higher integrability of Green's functions and ABP estimates with $L_d$ drift

TL;DR

The paper studies higher integrability of Green's functions for the linear elliptic operator with drift in a bounded domain. It adopts a Fabes–Stroock-type analytic framework to prove a Gehring-type inequality for Green's functions, exploiting Aleksandrov solutions and Monge–Ampère convex corrections to handle the critical drift and a scaling-invariance property of the norm of . The main contributions are an ABP-type estimate refined to the drift case and a higher integrability result: there exist and constants such that with . This provides an analytic alternative to Krylov's probabilistic proofs and strengthens the -theory for elliptic equations with critical drift, enabling refined ABP-type estimates and potential applications in stochastic analysis.

Abstract

We explore the higher integrability of Green's functions associated with the second-order elliptic equation in a bounded domain , and establish an enhanced version of Aleksandrov's maximum principle. In particular, we consider the drift term in and the source term for some . This provides an alternative and analytic proof of a result by N. V. Krylov (\textit{Ann. Probab.}, 2021) concerning drifts. The key step involves deriving a Gehring-type inequality for Green's functions by using the Fabes-Stroock approach (\textit{Duke Math. J.}, 1984).
Paper Structure (7 sections, 14 theorems, 112 equations)

This paper contains 7 sections, 14 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Fabes-Stroock approach
  3. Preliminaries and main results
  4. Auxiliary estimates
  5. Proofs of the main results
  6. Key estimate for Gehring-type lemma
  7. Proofs of Theorem \ref{['cor_main']} and Theorem \ref{['thm_main']}

Key Result

Theorem 2.2

Let $\Omega$ be a bounded domain. There is a constant $d_0 = d_0(d, \delta, \|b\|) \in (d/2, d)$ such that the following holds: If $u \in W_{p, \mathrm{loc}}^2(\Omega) \cap C(\overline{\Omega})$ with $p \in [d_0, \infty)$, then where $N$ depends only on $d$, $\delta$, and $\|b\|$.

Theorems & Definitions (28)

  • Remark 2.1: Unique existence of the Green's function
  • Theorem 2.2
  • Remark 2.3: Neumann problem on half balls
  • Theorem 2.4
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • proof : Proof of Lemma \ref{['test estimate']}.
  • Theorem 3.3
  • Corollary 3.4
  • ...and 18 more