On the exponential decay for real-valued solutions to elliptic equations with singular potentials in the plane

Balc’h, Kévin Le

Journées équations aux dérivées partielles (2024), Exposé no. 8, 14 p.

In this note, we prove that non-trivial real-valued solutions to $-\Delta u+Vu=0$ in ${ℝ}^2$, where $V\in L^p({ℝ}^2;ℝ)$ with $p\in (1,+\infty ]$, cannot decay faster than exponentially. The strategy builds crucially on the method introduced by Logunov, Malinnikova, Nadirashvili, and Nazarov, as well as some new arguments introduced by the author and Souza to solve the Landis conjecture in the plane for real-valued solutions to the Laplace equation perturbed by bounded lower-order terms.

Dans cette note, nous prouvons que les solutions non nulles à valeurs réelles de l’équation de Schrödinger elliptique avec potentiel singulier ne peuvent pas décroître plus rapidement qu’exponentiellement. La stratégie repose de manière cruciale sur la méthode introduite par Logunov, Malinnikova, Nadirashvili et Nazarov, ainsi que sur de nouveaux arguments introduits par l’auteur et Souza pour résoudre la conjecture de Landis dans le plan pour des solutions à valeurs réelles de l’équation de Laplace perturbée par des termes d’ordre inférieur bornés.

Classification : 35B60, 35J15, 30C62
Keywords: Landis’ conjecture, Schrödinger equation, quasiconformal mappings, vanishing order, Carleman estimates.
Mots-clés : Conjecture de Landis, équation de Schrödinger, applications quasi-conformes, ordre d’annulation, estimations de Carleman.

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