Layer potentials for elliptic operators with DMO-type coefficients: big pieces $Tb$ theorem, quantitative rectifiability, and free boundary problems

TL;DR

The paper advances the theory of elliptic layer potentials with DMO-type coefficients by establishing a quantitative rectifiability criterion for measures via the gradient of the single-layer potential, and by proving a big pieces $Tb$ theorem for these gradients under Dini-type oscillation controls. It develops a robust suppression framework for singular integrals, constructs a good SIO class that includes the gradient of the elliptic SLP, and proves $L^2$ boundedness on large pieces of measures, which then yield uniform rectifiability results. The results enable qualitative and quantitative one- and two-phase free boundary problems for elliptic measure in Wiener-regular domains, generalizing prior harmonic- and elliptic-measure frameworks to coefficients in $\widetilde{\mathrm{DMO}}$. Overall, the work provides a comprehensive PDE–GM T framework to derive geometric regularity of boundaries from analytic bounds on elliptic operators with minimal regularity assumptions on the coefficients, with significant implications for elliptic measure and free boundary theory.

Abstract

For $n \geq 2$, we consider the operator $L_A = -\mathrm{div }(A(\cdot)\nabla)$, where $A$ is a uniformly elliptic $(n+1)\times(n+1)$ matrix with variable coefficients, a Radon measure $μ$ on $\mathbb{R}^{n+1}$, and the associated gradient of the single layer potential operator $T_μ$. Under a Dini-type assumption on the mean oscillation of the matrix $A$, we establish the following results: 1) A rectifiability criterion for $μ$ in terms of $T_μ$. Under quantitative geometric and analytic assumptions within a ball $B$ -- including an upper $n$-growth condition on $μ$ in $B$, a thin boundary condition, a scale-invariant decay condition expressed via a weighted sum of densities over dyadic dilations of $B$, and $L^2$ boundedness of the gradient of $T_μ$ -- we show the following: if the support of $μ$ lies very close to an $n$-plane in $B$, and $T_μ1$ is nearly constant on $B$ in the $L^2$ sense, then there exists a uniformly $n$-rectifiable set $Γ$ such that $μ(B \cap Γ) \gtrsim μ(B)$. 2) A $Tb$ theorem for suppressed $T_μ$, which extends a well-known theorem of Nazarov, Treil, and Volberg, and holds also for a broader class of singular integral operators. These results make it possible to prove both qualitative and quantitative one- and two-phase free boundary problems for elliptic measure, formulated in terms of (uniform) rectifiability, in bounded Wiener-regular domains.

Theorems & Definitions (116)

• Theorem A
• Theorem B
• Theorem C
• Theorem D
• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6