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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 inand observed from a non-empty open subsetMore precisely, forour main result shows that, whenhas a finite number of holes, the observability constant of the elliptic operatorwith domainis of the formwhere  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 [ 1 ], in the context of the Landis conjecture on exponential decay of solutions to homogeneous elliptic equations in the planeThe main difference and additional difficulty compared to [ 1 ] 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.

Cost of observability inequalities for elliptic equations in 2-d with potentials and applications to control theory