Biharmonic Green Functions on Homogeneous Trees |
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Authors: | Joel M. Cohen Flavia Colonna David Singman |
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Affiliation: | (1) Information Systems Group, Faculty of IE&M, The Technion, Haifa, 32000, Israel |
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Abstract: | The study of biharmonic functions under the ordinary (Euclidean) Laplace operator on the open unit disk mathbbD{mathbb{D}} in mathbbC{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 Poincaré 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 mathbbD{mathbb{D}}. We then study the simply-supported and the clamped biharmonic Green functions with respect to both Laplacians. |
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