Biharmonic extensions on trees without positive potentials

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:378

Article

Bajunaid, Ibtesam O.³  Cohen, Joel M.²  Colonna, Flavia¹  Singman, David¹ 

[1] George Mason Univ, Fairfax, VA 22030 USA

[2] Univ Maryland, College Pk, MD 20742 USA

[3] King Saud Univ, Riyadh, Saudi Arabia

关键词: Biharmonic;    Trees;    Harmonic;   

DOI  :  10.1016/j.jmaa.2010.12.026

来源: Elsevier

【 摘 要 】

Let T be a tree rooted at e endowed with a nearest-neighbor transition probability that yields a recurrent random walk. We show that there exists a function K biharmonic off e whose Laplacian has potential theoretic importance and, in addition, has the following property: Any function f on T which is biharmonic outside a finite set has a representation, unique up to addition of a harmonic function, of the form f = beta K + B + L, where beta a constant, B is a biharmonic function on T, and L is a function, subject to certain normalization conditions, whose Laplacian is constant on all sectors sufficiently far from the root. We obtain a characterization of the functions biharmonic outside a finite set whose Laplacian has 0 flux similar to one that holds for a function biharmonic outside a compact set in R-n for n = 2,3, and 4 proved by Bajunaid and Anandam. Moreover, we extend the definition of flux and, under certain restrictions on the tree, we characterize the functions biharmonic outside a finite set that have finite flux in this extended sense. (C) 2010 Elsevier Inc. All rights reserved.

【 授权许可】

Free