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

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.