2023/11/01 by Casey, Emily, Max Goering, Goering, Max +4
Mathematics · #28A75 #42B20 #42B37 #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Morphological variations and asymmetry #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2311.00589
openalex publication_date 2023/11/01 · openalex created_date 2023/11/04 · openalex updated_date 2026/08/01
Characterizing rectifiability of Radon measures in Euclidean space has led to fundamental contributions to geometric measure theory. Conditions involving existence of principal values of certain singular integrals \citemattila1995rectifiable and the existence of densities with respect to Euclidean balls \citepreiss1987geometry have given rise to major breakthroughs. We study similar questions in a rough elliptic setting where Euclidean balls B(a,r) are replaced by ellipses BΛ(a,r) whose eccentricity and principal axes depend on a. Given Λ: ℝn → GL(n,ℝ), consider the family of ellipses BΛ(a,r) = a + Λ(a) B(0,r). We characterize m-rectifiability in terms of the almost everywhere existence of the densities θmΛ(a)(μ,a) = limr \downarrow 0 \fracμ(BΛ(a,r))rm ∈ (0, ∞). We characterize m-rectifiable measures in terms of the existence of the principal values-- and even under the weaker assumptions that limε\downarrow 0 ∫BΛ(a,εR) ∖ BΛ(a, εr) \fracΛ(a)-1(y-a)|Λ(a)-1(y-a)|m+1 d μ(y) = 0 ∀ 0 lt; r lt; R when 0 < θm*(μ,a) < ∞ almost everywhere. We apply the second result to characterize (n-1)-rectifiable measures in ℝn in terms of the behavior of the gradient of the single layer potential to the PDE LA u = - \textrmdiv(A ∇ u) under weak continuity assumptions on A.