Polyharmonic Functions on Trees

Abstract

Biharmonic functions in the Euclidean space appear in the study of bending of plates or beams, a C ⁴- function u in a domain ω is biharmonic if Δ ² u = 0 in ω. Such functions can be considered in a network, but the operator Δ ² is unwieldy. Another way to define biharmonic functions is used here: start with a harmonic function h and call u a biharmonic function generated by h if Δu = h on X. This requires solving the Poisson equation Δg = f when f is known. Unable to solve this equation in a general infinite network X, the network is restricted to a tree T in which every non-terminal vertex has at least two non-terminal neighbours. Then, it is possible to define inductively an m-harmonic function (m ≥ 2) u on T as a solution of the equation Δu = v where v is an (m − 1)-harmonic function. This chapter is about the potential theory associated with m-harmonic functions: m-superharmonic functions, m-potentials, domination principle for m-potentials, existence of m-harmonic Green functions, Riquier problem and the Riesz-Martin representation for positive m-superharmonic functions.