2019/11/11 by Carmelo Puliatti, Puliatti, Carmelo · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Numerical methods in inverse problems #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1911.04421
We identify a set of sufficient local conditions under which a significant portion of a Radon measure μ on ℝn+1 with compact support can be covered by an n-uniformly rectifiable set at the level of a ball B⊂ ℝn+1 such that μ(B)≈ r(B)n. This result involves a flatness condition, formulated in terms of the so-called β1-number of B, and the L2(μ|B)-boundedness, as well as a control on the mean oscillation on the ball, of the operator Tμf(x)=∫ ∇xE(x,y)f(y) dμ(y). Here E(⋅,⋅) is the fundamental solution for a uniformly elliptic operator in divergence form associated with an (n+1)×(n+1) matrix with Hölder continuous coefficients. This generalizes a work by Girela-Sarrión and Tolsa for the n-Riesz transform. The motivation for our result stems from a two-phase problem for the elliptic harmonic measure.