2025/05/29 by Andrea Merlo, Mihalis Mourgoglou, Merlo, Andrea +3 · 1 citation
Computer Science · Mathematics · #28A75 #33C55 #35J15 #42B20 #42B37 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2505.23478
openalex publication_date 2025/05/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
For n ≥ 2, we consider the operator LA = -div (A(⋅)∇), where A is a uniformly elliptic (n+1)×(n+1) matrix with variable coefficients, a Radon measure μ on ℝn+1, and the associated gradient of the single layer potential operator Tμ. 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 Tμ. 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 L2 boundedness of the gradient of Tμ -- we show the following: if the support of μ lies very close to an n-plane in B, and Tμ1 is nearly constant on B in the L2 sense, then there exists a uniformly n-rectifiable set Γ such that μ(B ∩ Γ) \gtrsim μ(B). 2) A Tb theorem for suppressed Tμ, 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.