Elliptic PDEs and Diffusion: Short Proofs by Counting Particles and Applications

Friday, December 2, 2016 - 2:30pm - 3:30pm
Lind 305
Stefan Steinerberger (Yale University)
We describe a simple trick in the study of elliptic PDEs: introduce time and interpret the solution of the elliptic pde as a fixed point of the evolution of the parabolic equation. We illustrate this technique by giving extremely short proofs of some classical results, an alternative interpretation of the Filoche-Mayboroda landscape function and improvements of classical results of Makai, Hayman and E. Lieb (originally conjectured by Polya & Szego). Finally, we discuss applications in data science (related to properties of eigenfunctions of Graph Laplacians on graphs constructed from real-life data).