[Paper Review] Uniform rectifiability and elliptic operators satisfying a Carleson measure condition. Part II: The large constant case

Steve Hofmann, José María Martell+3 more|arXiv (Cornell University)|Aug 8, 2019

Advanced Harmonic Analysis Research34 references7 citations

TL;DR

This paper establishes the equivalence between uniform rectifiability of the boundary and absolute continuity of elliptic measure with respect to surface measure for divergence form elliptic operators satisfying a Carleson measure condition on the coefficient oscillation, completing the program initiated in the authors' prior small constant work. Using a novel extrapolation argument and a transference mechanism between domains and their sawtooth subdomains, the authors prove the result in the general 'large constant' case, resolving a long-standing question in elliptic PDEs and geometric measure theory.

ABSTRACT

The present paper, along with its companion [Hofmann, Martell, Mayboroda, Toro, Zhao, arXiv:1710.06157], establishes the correspondence between the properties of the solutions of a class of PDEs and the geometry of sets in Euclidean space. We settle the question of whether (quantitative) absolute continuity of the elliptic measure with respect to the surface measure and uniform rectifiability of the boundary are equivalent, in an optimal class of divergence form elliptic operators satisfying a suitable Carleson measure condition. The result can be viewed as a quantitative analogue of the Wiener criterion adapted to the singular $L^p$ data case. The first step in this direction was taken in our previous paper [Hofmann, Martell, Mayboroda, Toro, Zhao, arXiv:1710.06157], where we considered the case in which the desired Carleson measure condition on the coefficients holds with sufficiently small constant. In this paper we establish the final, general result, that is, the "large constant case". The key elements of our approach are a powerful extrapolation argument, which provides a general pathway to self-improve scale-invariant small constant estimates, as well as a new mechanism to transfer quantitative absolute continuity of elliptic measure between a domain and its subdomains.

Motivation & Objective

• To resolve the full equivalence between uniform rectifiability of the boundary and A∞ property of elliptic measure for divergence form operators with Carleson coefficient oscillation.
• To extend the prior small constant result in [HMMTZ] to the general large constant case.
• To establish a sharp, optimal condition on the coefficient matrix oscillation that guarantees mutual absolute continuity of elliptic and surface measures.
• To develop a general mechanism for transferring quantitative absolute continuity of elliptic measure between a domain and its sawtooth subdomains.
• To provide a quantitative analogue of the Wiener criterion for Lp data in the context of elliptic PDEs.

Proposed method

• Employing a powerful extrapolation argument to self-improve scale-invariant small constant estimates into large constant results.
• Introducing a new transference mechanism to propagate the A∞ property of elliptic measure from a domain to its sawtooth subdomains.
• Constructing sawtooth domains with discrete Carleson measures to localize and analyze the behavior of solutions and elliptic measures.
• Using a mollification procedure to define a modified coefficient matrix that satisfies the necessary Carleson conditions while preserving the elliptic measure properties.
• Applying the theory of weights and singular integrals to control the oscillation of the coefficient matrix and its impact on the elliptic measure.
• Leveraging the chord-arc domain structure and Ahlfors regularity to ensure uniformity in the geometric and analytic estimates.

Experimental results

Research questions

• RQ1 Is the A∞ property of elliptic measure with respect to surface measure equivalent to uniform rectifiability of the boundary for divergence form elliptic operators with Carleson coefficient oscillation?
• RQ2 Can the small constant result from [HMMTZ] be extended to the general large constant case?
• RQ3 What is the optimal condition on the coefficient matrix oscillation that guarantees mutual absolute continuity of elliptic and surface measures?
• RQ4 How can the A∞ property of elliptic measure be transferred from a domain to its sawtooth subdomains?
• RQ5 Is the Carleson condition on coefficient oscillation sharp for the equivalence between uniform rectifiability and A∞ elliptic measure?

Key findings

• The equivalence between uniform rectifiability of the boundary and the A∞ property of elliptic measure holds for all divergence form elliptic operators satisfying the Carleson condition on coefficient oscillation, regardless of the size of the constant.
• The large constant case is resolved via an extrapolation argument that self-improves small constant estimates into the general case.
• A new transference mechanism enables the propagation of the A∞ property from a domain to its sawtooth subdomains, which is crucial for the proof.
• The condition ∫∫_{B(q,r)∩Ω} osc(𝒜,X)² / δ(X) dX ≤ C r^{n-1} is sharp and optimal for the A∞ equivalence.
• The elliptic measure associated with the limiting operator L is singular with respect to Lebesgue measure, demonstrating the sharpness of the Carleson condition.
• Corollary 6.3 shows that the oscillation condition (6.4) is sufficient and necessary for the A∞ equivalence, and the result extends to chord-arc domains.