Mathematical Physics and Harmonic Analysis Seminar

Date: October 28, 2022

Time: 09:00AM - 10:00AM

Location: Zoom

Speaker: Francesco Tudisco, Gran Sasso Science Institute, Italy

Title: Nodal domain count for the generalized graph p-Laplacian

Abstract: In recent years, there has been a surge in interest towards nonlinear extensions of graph operators such as the p-Laplacian and the generalized p-Laplacian (or p-Schrödinger) operators. This interest is prompted by applications connected to data clustering and semisupervised learning, where the limiting cases p=1 and p=\infty are especially noteworthy. In particular, similarly to the linear case, an important relation connects the nodal domains of the p-Laplacian and the k-th order isoperimetric constant of the graph.

In this talk, we consider a set of variational eigenvalues of the generalized p-Laplacian operator and present new results that characterize several properties of these eigenvalues, with particular attention to the nodal domain count of their eigenfunctions. Just like the one-dimensional continuous p-Laplacian, we prove that the variational spectrum of the discrete generalized p-Laplacian on forests is the entire spectrum. Moreover, we show how to transfer Weyl’s inequalities for the Laplacian operator to the nonlinear case and thus we prove new upper and lower bounds on the number of nodal domains of every eigenfunction of the generalized p-Laplacian on graphs, including those corresponding to variational eigenvalues. When applied to the linear case p=2, the new results imply well-known properties of the linear Schrödinger operator as well as novel ones.

Based on a joint work with P.Deidda and M.Putti.