ERVEDOZA, Sylvain; LE BALC’H, Kévin

doi:10.1080/03605302.2023.2190526

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ERVEDOZA, Sylvain
Institut de Mathématiques de Bordeaux [IMB]

LE BALC’H, Kévin
Control And GEometry [CaGE]

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Article de revue

Ce document a été publié dans

Communications in Partial Differential Equations. 2023-04-10, vol. 48, n° 4, p. 623--677

Taylor & Francis

Résumé en anglais

The goal of this article is to obtain observability estimates for non-homogeneous elliptic equations in the presence of a potential, posed on a smooth bounded domain Ω in 2-d and observed from a non-empty open subset ω ⊂ Ω. More precisely, for every real-valued bounded potential V, our main result shows that, when Ω has a finite number of holes, the observability constant of the elliptic operator −∆ + V, with domain H^2 ∩ H^1_0(Ω), is of the form C exp (C |V|^{1/2} \log^{1/2}(|V|)) where C is a positive constant depending only on Ω and ω. Our methodology of proof is crucially based on the one recently developed by Logunov, Malinnikova, Nadirashvili, and Nazarov, in the context of the Landis conjecture on exponential decay of solutions to homogeneous elliptic equations in the plane. The main difference and additional difficulty is that the zero set of the solutions to elliptic equations with source term can be very intricate and should be dealt with carefully. As a consequence of these new observability estimates, we obtain new results concerning control of semi-linear elliptic equations in the spirit of Fernández-Cara, Zuazua's open problem concerning small-time global null-controllability of slightly super-linear heat equations.< Réduire

Mots clés en anglais

Quantitative unique continuation

Elliptic equations

Control

URI

https://oskar-bordeaux.fr/handle/20.500.12278/190396

DOI

http://dx.doi.org/10.1080/03605302.2023.2190526

Project ANR

Nouvelles directions en contrôle et stabilisation: Contraintes et termes non-locaux - ANR-20-CE40-0009

Origine

Importé de hal

Unités de recherche

Institut de Mathématiques de Bordeaux (IMB) - UMR 5251