Biharmonic Green Functions on Homogeneous Trees

Abstract

The study of biharmonic functions under the ordinary (Euclidean) Laplace operator on the open unit disk $${\mathbb{D}}$$ in $${\mathbb{C}}$$ arises in connection with plate theory, and in particular, with the biharmonic Green functions which measure, subject to various boundary conditions, the deflection at one point due to a load placed at another point. A homogeneous tree T is widely considered as a discrete analogue of the unit disk endowed with the Poincare metric. The usual Laplace operator on T corresponds to the hyperbolic Laplacian. In this work, we consider a bounded metric on T for which T is relatively compact and use it to define a flat Laplacian which plays the same role as the ordinary Laplace operator on $${\mathbb{D}}$$ . We then study the simply-supported and the clamped biharmonic Green functions with respect to both Laplacians.