Joint BCAM-UPV/EHU Analysis and PDE seminar: Observability inequalities for elliptic equations with potentials in 2D and applications to control theory

LOCATION: Online

Abstract
The goal of the talk is to present new observability estimates for non homogeneous elliptic equations, posed on a domain in R2, and observed from a subdomain. More precisely, for a real-valued bounded potential V , we show that the constant of observability of the operator −∆ + V , is of order exp(|V |1/2+ε∞ ). The method of proof is inspired by a recent article of Logunov, Malinnikova, Nadirashvili, Nazarov dealing with the Landis conjecture in the plane. I will present the three main ideas of the proof: a perforation process based on the nodal set of the solution that transforms the domain to a perforated domain with small Poincar ́e constant, a quasiconformal transformation to reduce the elliptic equation into an harmonic equation and Carleman estimates conjugated with Harnack inequalities. Finally, I will present new results for controlling semi-linear elliptic equations in the spirit of Fernandez-Cara, Zuazua's open problem concerning small-time global null-controllability of slightly super-linear heat equations.
This is a joint work with Sylvain Ervedoza.


Link to the session: 
https://us06web.zoom.us/j/99649860282?pwd=SE0vemtYMFlwbFBNTXQyOTBONG0vZz09


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