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σ-Finiteness of elliptic measures for quasilinear elliptic PDE in space

Akman, Murat and Lewis, John and Vogel, Andrew (2017) 'σ-Finiteness of elliptic measures for quasilinear elliptic PDE in space.' Advances in Mathematics, 309. pp. 512-557. ISSN 0001-8708

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Abstract

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 in [3] when the PDE is the p-Laplacian and the second result generalizes the well known theorem of Wolff in [24] when p=2 and n=2.

Item Type: Article
Additional Information: 35 pages, 3 figures, shortened title and some minor changes
Uncontrolled Keywords: Hausdorff Dimension of a Borel measure; Hausdorff measure; Hausdorff dimension; The four-corner Cantor set; Quasilinear elliptic PDEs
Divisions: Faculty of Science and Health
Faculty of Science and Health > Mathematical Sciences, Department of
SWORD Depositor: Elements
Depositing User: Elements
Date Deposited: 12 Sep 2019 15:48
Last Modified: 06 Jan 2022 14:01
URI: http://repository.essex.ac.uk/id/eprint/25312

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