2015/09/23 by Murat Akman, Akman, Murat, John L. Lewis +4
Computer Science · Engineering · Mathematics · #28A78 #35J25 #35J70 #37F35 #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Stability and Controllability of Differential Equations #math.AP #msc:28A78 #msc:35J25 #msc:35J70 #msc:37F35
paper · pdf · doi:10.48550/arxiv.1509.07068
35 pages, 3 figures, shortened title and some minor changes
openalex publication_date 2015/09/23 · arxiv created 2015/09/30 · arxiv updated 2015/10/01 · openalex created_date 2024/04/10 · openalex updated_date 2026/07/28
In this paper we study the Hausdorff dimension of a elliptic measure μf in space associated to a positive weak solution to a certain quasilinear elliptic PDE in an open subset and vanishing on a portion of the boundary of that open set. We show that this measure is concentrated on a set of σ-finite n-1 dimensional Hausdorff measure for p>n and the same result holds for p=n with an assumption on the boundary. We also construct an example of a domain in space for which the corresponding measure has Hausdorff dimension ≤ n-1-δ for p≥ n for some δ which depends on various constants including p. The first result generalizes the authors previous work when the PDE is the p-Laplacian and the second result generalizes the well known theorem of Wolff when p=2 and n=2.