Unique continuation and recent progress on Landis’ conjecture
Time and place
1 PM on Thursday, December 4th, 2014; NAC 6/113
Blair Davey (University of Minnesota)
Abstract
Let h be a holomorphic function and suppose h is equal to zero on some open set in the complex plane. Then h must be identically zero. In fact, if holomorphic h vanishes to infinite order at a point in the complex plane, then h is identically zero. Harmonic functions, solutions to the Laplace equation in R^n, behave in the same way. What about solutions to other partial differential equations (PDEs)? Do they have similar uniqueness properties? After reviewing some classical notions of unique continuation for PDEs, we are led to Landis' conjecture from the late 1960s: If u, V are bounded functions, and u is a solution to \Delta u + V u = 0 in R^n that decays faster than exponential in |x|, then u must be identically zero. We will explore the history of this conjecture along with recent progress. In particular, we show precisely when Landis' conjecture fails to be true and discuss the techniques involved. Positive results will also be presented, along with future research directions.
