PDE and applications

Title: Nodal sets, Quasiconformal mappings and how to apply them to Landis' conjecture.

The seminar will take place at Zoom.

Abstract: A while ago Nadirashvili proposed a beautiful idea how to attack problems on zero sets of Laplace eigenfunctions using quasiconformal mappings, aiming to estimate the length of nodal sets (zero sets of eigenfunctions) on closed two-dimensional surfaces. The idea did not work out as it was planned. However, it appears to be useful in relation to Landis' Conjecture. We will explain how to apply the combination of quasiconformal mappings and zero sets to quantitative properties of solutions to $\Delta u + V u =0 on the plane, where $V$ is a real, bounded function. The method reduces some questions about solutions to Schrodinger equation $\Delta u + V u =0$ on the plane to questions about harmonic functions. Based on a joint work with E.Malinnikova, N.Nadirashvili and F. Nazarov.