Title
Gradients of single layer potentials for elliptic operators with coefficients of DMO-type and applications to elliptic measures
Abstract
We study a uniformly elliptic operator LA in divergence form, associated with an (n+1)×(n+1) matrix A with real, bounded, and possibly non-symmetric coefficients. Assuming that a suitable L1-mean oscillation of the coefficients of A satisfies a Dini-type condition, we establish a rectifiability criterion for Radon measures in terms of the operator
Tµf(x)=∫∇xΓA(x,y)f(y) dµ(y),
where ΓA(x,y) denotes the fundamental solution associated with LA.
In combination with a Tb theorem for Tµ, this criterion yields both qualitative and quantitative rectifiability results in the context of one- and two-phase free boundary problems for elliptic measures.
This is joint work with Andrea Merlo and Mihalis Mourgoglou.
