Akman, Murat (2014) 'On the dimension of a certain Borel measure in the plane.' Annales Academiae Scientiarum Fennicae. Mathematica, 39 (1). pp. 187209. ISSN 1239629X

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Abstract
In this paper we study the Hausdorff dimension of a measure μ related to a positive weak solution, u, of a certain partial differential equation in ω ∩ N where ω ⊂ C is a bounded simply connected domain and N is a neighborhood of ∂ω . u has continuous boundary value 0 on ∂ω and is a weak solution to 2 ∑ i,j=1 ∂ / ∂ xi(fηiηj(∇u(z))uxj(z))=0 in ω ∩ N. Also f(η), η ∈ C is homogeneous of degree p and ∇f is δmonotone on C for some δ > 0. Put u ≡ 0 in N \ ω . Then μ is the unique positive finite Borel measure with support on ∂ω satisfying ∫c 〈∇f(∇u(z)),∇φ(z)〉dA = ∫∂ωφ(z)dμ for every φ ∈ C∞0(N) . Our work generalizes work of Lewis and coauthors when the above PDE is the p Laplacian (i.e., f(η) = ηp) and also for p = 2, the well known theorem of Makarov regarding the Hausdorff dimension of harmonic measure relative to a point in ω.
Item Type:  Article 

Additional Information:  mrclass: 35J25 (28A78 35D30) mrnumber: 3186813 
Uncontrolled Keywords:  Hausdorff dimension; dimension of a measure; pharmonic measure 
Divisions:  Faculty of Science and Health Faculty of Science and Health > Mathematical Sciences, Department of 
SWORD Depositor:  Elements 
Depositing User:  Elements 
Date Deposited:  25 Jul 2019 14:57 
Last Modified:  18 Aug 2022 13:26 
URI:  http://repository.essex.ac.uk/id/eprint/25016 
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