The Dirichlet problem on a tree
We study the Dirichlet problem related to a weighted Laplacian in a tree. The Laplace operator is given so that the mean value formula has weight $\beta$ for the father of each vertex $x$, and its successors have weight $(1-\beta)/m(x)$, where $m(x)$ is the offspring number and could depend on x. Under certain conditions on $\beta$ and the boundary condition, we show that the Dirichlet problem has a unique solution. Moreover, we have an explicit formula for the solution. For the special case of the regular tree (offspring distribution constant $m$), we introduce two "natural" versions of a Dirichlet-to-Neumann map and establish some properties of this map. If we have time, we will comment on ongoing work for the case where the tree is a Galton-Watson tree and how to compare the solution in this environment with a regular tree (when the mean of the offspring distribution is an integer $m$).
Nicolas Frevenza de Universidad de la Republica, Uruguay