On the Landis conjecture in 2d

The goal of this talk is to present the beautiful ideas of the work of A. Logunov, E. Malinnikova, N. Nadirashvili and F. Nazarov proving the Landis conjecture in the plane (https://arxiv.org/abs/2007.07034). The Landis conjecture states that if a real valued function $u$ satisfies $|\Delta u | \leq |u |$ in $\mathbb{R}^d$ and decays faster than $\exp(-C |x|)$ at infinity for all $C$, then the function $u$ vanishes everywhere. Almost equivalently, if $-\Delta u = Vu$ in a ball of radius 2 for some potential $V$, then the norm of $u$ in the ball of radius 1 can be bounded by $\exp(C \| V \|_{L^\infty}^{1/2})$ times the norm of $u$ in a neighborhood of the sphere of radius $2$.

To solve this problem up a logarithm loss, several new ideas are proposed:

- to create a network of holes in the domain to make the Poincaré constant small in the newly created domain.

- to use this small Poincaré constant to absorb the potential through the use of a multiplier function and a quasi-conformal transform.

- to suitably combine a Carleman estimate with Harnack’s inequality.

If time allows, I will also briefly explain how these ideas can be adapted to handle the case of a non-trivial source term and how it can be applied to a control problem for a semilinear elliptic equation in the spirit of an open problem pointed out by Enrique Fernandez-Cara and Enrique Zuazua in 2000, which is the content of a joint work with Kévin LeBalc’h.