A combinatorial approach to the analysis of the Laplacian
A combinatorial approach to the analysis of the Laplacian
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Federico Glaudo, IAS
We introduce the harmonic decomposition for polynomials (and analytic听functions) and then we observe how some linear operators (multiplication,听differentiation, inverse Laplacian) act on it.听Unexpectedly, an efficient way to represent the action of these operators听is a graph with vertices indexed by $\mathbb Z^2$.听This representation transforms convergence issues into the study of听paths in such graphs.听We will apply these tools to the study of two PDEs involving the听Laplacian: the construction of a canonical fundamental solution for听a second-order differential operator, and a canonical bijection between听minimal graphs and harmonic functions.
This is an ongoing project with F. Franceschini.