Unique continuation estimates for Baouendi--Grushin equations on cylinders

Abstract

We prove time-pointwise quantitative unique continuation estimates for the evolution operators associated to (fractional powers of) the Baouendi--Grushin operators on the cylinder ${\mathbb{R}}^d\times {\mathbb{T}}^d$. Spectral inequalities are deduced, relating for functions from spectral subspaces associated to finite energy intervals their $L^2$-norm on the whole cylinder ${\mathbb{R}}^d\times {\mathbb{T}}^d$ to the $L^2$-norm on so-called thick subsets of ${\mathbb{R}}^d\times {\mathbb{T}}^d$. As a byproduct, we also obtain results on exact and approximate null-controllabilty for the corresponding evolution systems. This extends and complements results obtained recently by the authors and by Jaming and Wang.

arXiv comment32 pages

References in corpus (11)

• Null controllability of Grushin-type operators in dimension two
• Non null controllability of the Grushin equation in 2D
• Characterization by observability inequalities of controllability and stabilization properties
• An abstract Logvinenko-Sereda type theorem for spectral subspaces
• Sharp estimates and homogenization of the control cost of the heat equation on large domains
• Control of the Grushin equation: non-rectangular control region and minimal time
• Scale-free unique continuation estimates and Logvinenko-Sereda Theorems on the torus
• Observability of the heat equation, geometric constants in control theory, and a conjecture of Luc Miller
• Uncertainty principle for Hermite functions and null-controllability with sensor sets of decaying density
• Control problem for quadratic parabolic differential equations with sparse sensor sets of finite volume or anisotropically decaying density
• Spectral inequality with sensor sets of decaying density for Schrödinger operators with power growth potentials