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

Abstract

For $n\ge 2$, 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 $\hat{I}¼$ on ${\mathbb{R}}^{n+1}$, and the associated gradient of the single layer potential operator $T_{\hat{I}}¼$. Under a Dini-type assumption on the mean oscillation of the matrix $A$, we establish the following results: 1) A rectifiability criterion for $\hat{I}¼$ in terms of $T_{\hat{I}}¼$. Under quantitative geometric and analytic assumptions within a ball $B$ -- including an upper $n$-growth condition on $\hat{I}¼$ 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 $T_{\hat{I}}¼$ -- we show the following: if the support of $\hat{I}¼$ lies very close to an $n$-plane in $B$, and $T_{\hat{I}}¼1$ is nearly constant on $B$ in the $L^2$ sense, then there exists a uniformly $n$-rectifiable set $\hat{I}“$ such that $\hat{I}¼(B\cap \hat{I}“)\gtrsim \hat{I}¼(B)$. 2) A $Tb$ theorem for suppressed $T_{\hat{I}}¼$, 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.

80 pages