Akman, Murat (2014) On the dimension of a certain Borel measure in the plane. Annales Academiae Scientiarum Fennicae. Mathematica, 39 (1). pp. 187-209. DOI https://doi.org/10.5186/aasfm.2014.3923
Akman, Murat (2014) On the dimension of a certain Borel measure in the plane. Annales Academiae Scientiarum Fennicae. Mathematica, 39 (1). pp. 187-209. DOI https://doi.org/10.5186/aasfm.2014.3923
Akman, Murat (2014) On the dimension of a certain Borel measure in the plane. Annales Academiae Scientiarum Fennicae. Mathematica, 39 (1). pp. 187-209. DOI https://doi.org/10.5186/aasfm.2014.3923
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 <sup>2</sup> ∑ <inf>i,j=1</inf> ∂ / ∂ x<inf>i</inf>(f<inf>ηiηj</inf>(∇u(z))u<inf>xj</inf>(z))=0 in ω ∩ N. Also f(η), η ∈ C is homogeneous of degree p and ∇f is δ-monotone 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 ∫<inf>c</inf> 〈∇f(∇u(z)),∇φ(z)〉dA = -∫<inf>∂ω</inf>φ(z)dμ for every φ ∈ C<sup>∞</sup><inf>0</inf>(N) . Our work generalizes work of Lewis and coauthors when the above PDE is the p Laplacian (i.e., f(η) = |η|<sup>p</sup>) 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; p-harmonic measure |
Divisions: | Faculty of Science and Health Faculty of Science and Health > Mathematics, Statistics and Actuarial Science, School of |
SWORD Depositor: | Unnamed user with email elements@essex.ac.uk |
Depositing User: | Unnamed user with email elements@essex.ac.uk |
Date Deposited: | 25 Jul 2019 14:57 |
Last Modified: | 11 Jun 2025 23:03 |
URI: | http://repository.essex.ac.uk/id/eprint/25016 |
Available files
Filename: vol39pp187-209.pdf