Analysis Seminar: Gradients of single layer potentials for elliptic operators with coefficients of DMO-type and applications to elliptic measure

Abstract: We study a uniformly elliptic operator L_A in divergence form, associated with an (n+1)×(n+1) matrix A with real, bounded, and possibly non-symmetric coefficients. Assuming that a suitable L^1-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

T_μ f(x) = ∫ ∇_x Γ_A(x, y) f(y) dμ(y),

where Γ_A(x, y) denotes the fundamental solution associated with L_A.

In combination with a Tb theorem for T_μ, 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.

Host: Alan Chang