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

Text
vol39pp187209.pdf  Published Version Download (274kB)  Preview 
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:  06 Jan 2022 14:01 
URI:  http://repository.essex.ac.uk/id/eprint/25016 
Actions (login required)
View Item 