Akman, Murat and Lewis, John and Vogel, Andrew (2019) 'Note on an Eigenvalue problem for an ODE originating from a homogeneous pharmonic function.' Algebra i Analiz, 31 (2). pp. 7587. ISSN 02340852

Text
ALV Eigenvalue.pdf  Submitted Version Download (358kB)  Preview 
Abstract
We discuss what is known about homogeneous solutions $ u $ to the pLaplace equation, $ p $ fixed, $1 < p < \infty,$ when $u$ is an entire pharmonic function in $\mathbb{R}^{n}$, </ul> or $ u > 0 $ is pharmonic in the cone, \[ K(\alpha) = \{ x = (x_1, \dots, x_n ) : x_1 > \cos \alpha \,  x \} \subset \mathbb{R}^n, \, n \geq 2, \] with continuous boundary value zero on $ \partial K (\alpha) \setminus \{0\} $ when $ \alpha \in (0, \pi]. $ We also outline a proof of our new result concerning the exact value, $ \lambda = 1  (n1)/p, $ for an eigenvalue problem in an ODE associated with $u$ when $ u$ is phamonic in $ K ( \pi ) $ and $ p > n  1. $ Generalizations of this result are stated for $ \lambda \leq n  1. $ Our result complements work of Krol'Maz'ya in [KM] for $ 1 < p \leq n  1. $
Item Type:  Article 

Uncontrolled Keywords:  pLaplacian; boundary Harnack inequalities; homogeneous pharmonic functions; eigenvalue problem 
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 13:52 
Last Modified:  26 Feb 2022 10:41 
URI:  http://repository.essex.ac.uk/id/eprint/25014 
Actions (login required)
View Item 