Biharmonic extensions on trees without positive potentials

• Published in: Journal of mathematical analysis and applications 2011-06, Vol.378 (2), p.710-722
• Main Authors: Bajunaid, Ibtesam O., Cohen, Joel M., Colonna, Flavia, Singman, David
• Format: Article
• Language: English
• Subjects: Algebra; Biharmonic; Combinatorics; Combinatorics. Ordered structures; Exact sciences and technology; Graph theory; Harmonic; Mathematical analysis; Mathematics; Number theory; Sciences and techniques of general use; Trees
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/j.jmaa.2010.12.026

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 = β K + B + L , where β 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.