## Two questions of Landis and their applications

Eugenia Malinnikova

NTNU; von Neumann Fellow, School of Mathematics

December 14, 2018

We discuss two old questions of Landis concerning behavior of solutions of second order elliptic equations. The first one is on

propagation of smallness for solutions from sets of positive measure,

we answer this question and as a corollary prove an estimate for eigenfunctions of Laplace operator conjectured by Donnelly and Fefferman. The second question is on the decay rate of solutions of the Schrödinger equation, we give the answer in dimension two. The talk is based on joint work with A. Logunov, N. Nadirashvili, and F. Nazarov.

propagation of smallness for solutions from sets of positive measure,

we answer this question and as a corollary prove an estimate for eigenfunctions of Laplace operator conjectured by Donnelly and Fefferman. The second question is on the decay rate of solutions of the Schrödinger equation, we give the answer in dimension two. The talk is based on joint work with A. Logunov, N. Nadirashvili, and F. Nazarov.