Seminario del Departamento de
Ecuaciones Diferenciales y Análisis Numérico

Fecha : 19 de diciembre de 2023
Hora  : 10:00
Lugar : Seminario del Departamento (Fac. de Matemáticas, 3a. planta, módulo 34)

Kévin Le Balc'h

(Laboratoire Jacques-Louis Lions – Sorbonne Université)

Resumen

In this talk, I will first present the Landis conjecture on exponential decay for solutions to second order elliptic equation in the Euclidean setting. While for complex-valued functions, the Landis conjecture was disproved by Meshkov in 1992, the question is still open for real-valued functions. I will present one way to tackle the conjecture, due to Bourgain and Kenig, that consists in establishing quantitative unique continuation results. Previous results in the two-dimensional setting by Kenig, Sylvestre, Wang in 2014 and more recently by Logunov, Malinnikova, Nadirashvili, Nazarov in 2020 will be recalled and explained. Then, the goal of the talk will be to prove that the qualitative and quantitative Landis conjecture hold for the Laplace operator, perturbed by lower order terms for real-valued solutions in the plane. Even if the strategy of the proof mainly follows the one of Logunov, Malinnikova, Nadirashvili, Nazarov, some new difficulties appear as new weak quantitative maximum principles, generalization of Stoilow factorization theorem, approximate type Poincare lemma in a perforated domain and a Carleman estimate to a non-homogeneous d_z bar equation with some non-local term. This is a joint work with Diego A. Souza (Universidad de Sevilla).