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The doubling property of the elliptic measure, for elliptic operators with drifts satisfying an average diverging condition

Aritro Pathak

TL;DR

This paper addresses doubling of the elliptic measure for a divergence-form elliptic operator with a drift term that satisfies an average Carleson-type smallness condition on Whitney cubes within a $1$-sided chord-arc domain. The main strategy combines Hardy inequalities, dyadic/Whitney decompositions, boundary Hölder regularity, Bourgain-type estimates, and Green function analysis to relate the elliptic measure to the Green function and establish doubling. The cumulative results include local Hardy and boundary Hölder inequalities, two-sided Green function bounds, Bourgain and boundary Harnack properties, and the doubling of the elliptic measure, extending previous pointwise smallness results to averaged smallness and spanning general domains, including the half-space. These findings underpin rough Dirichlet solvability under Carleson-measured lower-order terms and pave the way for further Dirichlet/Neumann/Regularity results with drifts under Carleson constraints.

Abstract

We show doubling of the elliptic measure corresponding to the operator with an elliptic principal term and a drift that diverges, on average on Whitney cubes, like the inverse distance to the boundary, with a small constant. Essentially a small Carleson constant assumption on the drift, this generalizes earlier results with the hypothesis of pointwise smallness of such a drift. This relates to recent perturbative results of rough Dirichlet solvability in domains with drifts or potentials that satisfy a Carleson measure condition, which have also been considered earlier by Hofmann-Lewis and Kenig-Pipher. While we work in 1-sided chord arc domains, these results are new even for the half-space. In the process, we also prove Hardy inequalities in such domains with Alhfors-David regular boundary, using a stopping time argument.

The doubling property of the elliptic measure, for elliptic operators with drifts satisfying an average diverging condition

TL;DR

This paper addresses doubling of the elliptic measure for a divergence-form elliptic operator with a drift term that satisfies an average Carleson-type smallness condition on Whitney cubes within a -sided chord-arc domain. The main strategy combines Hardy inequalities, dyadic/Whitney decompositions, boundary Hölder regularity, Bourgain-type estimates, and Green function analysis to relate the elliptic measure to the Green function and establish doubling. The cumulative results include local Hardy and boundary Hölder inequalities, two-sided Green function bounds, Bourgain and boundary Harnack properties, and the doubling of the elliptic measure, extending previous pointwise smallness results to averaged smallness and spanning general domains, including the half-space. These findings underpin rough Dirichlet solvability under Carleson-measured lower-order terms and pave the way for further Dirichlet/Neumann/Regularity results with drifts under Carleson constraints.

Abstract

We show doubling of the elliptic measure corresponding to the operator with an elliptic principal term and a drift that diverges, on average on Whitney cubes, like the inverse distance to the boundary, with a small constant. Essentially a small Carleson constant assumption on the drift, this generalizes earlier results with the hypothesis of pointwise smallness of such a drift. This relates to recent perturbative results of rough Dirichlet solvability in domains with drifts or potentials that satisfy a Carleson measure condition, which have also been considered earlier by Hofmann-Lewis and Kenig-Pipher. While we work in 1-sided chord arc domains, these results are new even for the half-space. In the process, we also prove Hardy inequalities in such domains with Alhfors-David regular boundary, using a stopping time argument.

Paper Structure

This paper contains 8 sections, 10 theorems, 99 equations, 1 figure.

Table of Contents

  1. Introduction.
  2. Main result.
  3. Notation.
  4. Local boundary Hardy inequality.
  5. Boundary Hölder regularity and Bourgain property.
  6. Green's function estimates
  7. Doubling of the elliptic measure.
  8. Acknowledgements.

Key Result

Lemma 1

( Existence and properties of the "dyadic grid") DS1DS2, Ch. Suppose that $E\subset \mathbb{R}^{n}$ is closed $n-1$-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 is where $\mathfrak{I}_k$ denotes some (possibly finite) index set depending on $k$, satisfying

Figures (1)

  • Figure 1: The Whitney region $U_{Q_0}$ is shown, along with the ball $B_x$ with center $x=x_{Q_0}$ which without loss of generality we have also taken to be the center of $U_{Q_0}$. The union of the dark regions is the set $\Omega_{B}'=\cup_i 5P_i$, and $f=1$ on the set $B_x\setminus \Omega'_B$ and $0$ elsewhere. We have the point-wise bound on the drift term, in $U_{Q_0}\setminus \Omega_{B}$ with $\Omega_B=\cup P_i$. Here the $P_i$ are the set of bad cubes contained in $U_{Q_{0}}$, obtained from the Calderon Zygmund decomposition for the function $|\mathcal{B}|^{2}1_{U_{Q_0}}$.

Theorems & Definitions (22)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Lemma 1
  • Theorem 2
  • Theorem 3
  • proof : Proof of \ref{['Hardyy']}
  • Lemma 4
  • Theorem 5
  • ...and 12 more