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BMO solvability with singular drifts on ample sawtooth domains implies $L^p$ solvability

Aritro Pathak

TL;DR

This work shows that for a divergence-form elliptic operator $L=-\mathrm{div}(A\nabla\cdot)+\mathcal{B}\cdot\nabla(\cdot)$ with a drift $\mathcal{B}$ obeying a Carleson measure condition and a pointwise bound $|\mathcal{B}(x)|\le M/\delta_{\Omega}(x)$, there exist ample sawtooth subdomains $\Omega_{\eta}$ of the unit ball such that $BMO$ solvability on $\Omega_{\eta}$ forces the elliptic measure $\omega_L$ to be in weak-$A_\infty$ relative to surface measure, yielding $L^p$ solvability for some $p>1$. The approach extends HL18 by incorporating a Markov property for the elliptic measure and handling the drift via an extrapolation of Carleson measures, using a constructive decomposition into good and bad Whitney cubes and a carefully controlled stopping-time argument. The main contributions are the construction of ample sawtooth domains under a drift-driven Carleson condition, the adaptation of Bourgain-type and boundary regularity estimates to this setting, and the deduced quantitative absolute continuity leading to $L^p$ solvability. These results lay groundwork for extensions to more general bounded Lipschitz domains and invite exploration of the converse implications and vanishing Carleson regimes.

Abstract

For a linear elliptic operator with a singular drift that satisfies a finite Carleson measure condition, we prove that there exist `ample' sawtooth domains of the unit ball $B(0,1)\subset \R^{n+1}$ so that a BMO solvability assumption in these sawtooth subdomains implies that the elliptic measure satisfies the weak $A_\infty$ condition with respect to the surface measure on this `ample' sawtooth domain. This is a quantifiable absolute continuity condition, which is equivalent to saying the $L^p$ Dirichlet problem is solvable for some $1<p<\infty$. Such singular drifts have been considered in the literature in the context of perturbative $L^p$ Dirichlet solvability problems, by Hofmann-Lewis and Kenig-Pipher. By an ample sawtooth domain, we mean a sawtooth domain whose boundary coincides with the boundary of the unit ball, except for an arbitrarily small fraction. The methods can be naturally extended to show the result for more general bounded Lipschitz domains.

BMO solvability with singular drifts on ample sawtooth domains implies $L^p$ solvability

TL;DR

This work shows that for a divergence-form elliptic operator with a drift obeying a Carleson measure condition and a pointwise bound , there exist ample sawtooth subdomains of the unit ball such that solvability on forces the elliptic measure to be in weak- relative to surface measure, yielding solvability for some . The approach extends HL18 by incorporating a Markov property for the elliptic measure and handling the drift via an extrapolation of Carleson measures, using a constructive decomposition into good and bad Whitney cubes and a carefully controlled stopping-time argument. The main contributions are the construction of ample sawtooth domains under a drift-driven Carleson condition, the adaptation of Bourgain-type and boundary regularity estimates to this setting, and the deduced quantitative absolute continuity leading to solvability. These results lay groundwork for extensions to more general bounded Lipschitz domains and invite exploration of the converse implications and vanishing Carleson regimes.

Abstract

For a linear elliptic operator with a singular drift that satisfies a finite Carleson measure condition, we prove that there exist `ample' sawtooth domains of the unit ball so that a BMO solvability assumption in these sawtooth subdomains implies that the elliptic measure satisfies the weak condition with respect to the surface measure on this `ample' sawtooth domain. This is a quantifiable absolute continuity condition, which is equivalent to saying the Dirichlet problem is solvable for some . Such singular drifts have been considered in the literature in the context of perturbative Dirichlet solvability problems, by Hofmann-Lewis and Kenig-Pipher. By an ample sawtooth domain, we mean a sawtooth domain whose boundary coincides with the boundary of the unit ball, except for an arbitrarily small fraction. The methods can be naturally extended to show the result for more general bounded Lipschitz domains.

Paper Structure

This paper contains 4 sections, 5 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and statement of result.
  3. Proofs of Claim 1 and Claim 2.
  4. Conclusion

Key Result

Lemma 1

( Existence and properties of the "dyadic grid") DS1DS2, Ch90. Suppose that $E\subset \mathbb{R}^{n+1}$ is a closed $n$-dimensional ADR set. Then there exist constants $a_0>0,\, \gamma>0$ and $C_*<\infty$, depending only on dimension and the ADR constant, such that for each $k \in \mathbb{Z},$ there where $\mathfrak{I}_k$ denotes some (possibly finite) index set depending on $k$, satisfying

Theorems & Definitions (22)

  • Lemma 1
  • Definition 1: Average smallness assumption(ASA)
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7: Ample sawtooth domains
  • Definition 8
  • Remark 1
  • ...and 12 more