Sharp resolvent estimate for the Baouendi-Grushin operator and applications

Abstract

In this article we study the semiclassical resolvent estimate for the non-selfadjoint Baouendi-Grushin operator on the two-dimensional torus ${\mathbb{T}}^2={\mathbb{R}}^2/(2\pi \mathbb{Z}{)}^2$ with Hölder dampings. The operator is subelliptic degenerating along the vertical direction at $x=0$. We exhibit three different situations: (i) the damping region verifies the geometric control condition with respect to both the non-degenerate Hamiltonian flow and the vertical subelliptic flow; (ii) the undamped region contains a horizontal strip; (iii) the undamped part is a line. In all of these situations, we obtain sharp resolvent estimates. Consequently, we prove the optimal energy decay rate for the associated damped waved equations. For (i) and (iii), our results are in sharp contrast to the Laplace resolvent since the optimal bound is governed by the quasimodes in the subelliptic regime. While for (ii), the optimality is governed by the quasimodes in the elliptic regime, and the optimal energy decay rate is the same as for the classical damped wave equation on ${\mathbb{T}}^2$. Our analysis contains the study of adapted two-microlocal semiclassical measures, construction of quasimodes and refined Birkhoff normal-form reductions in different regions of the phase-space. Of independent interest, we also obtain the propagation theorem for semiclassical measures of quasimodes microlocalized in the subelliptic regime.

arXiv comment91 pages, to appear in Comm. Math. Physics

References in corpus (12)

• Pseudospectra of semi-classical (pseudo)differential operators
• Decay rates for the damped wave equation on the torus
• Null controllability of Grushin-type operators in dimension two
• Spectral asymptotics for sub-Riemannian Laplacians. I: quantum ergodicity and quantum limits in the 3D contact case
• Non null controllability of the Grushin equation in 2D
• Time optimal observability for Grushin Schrödinger equation
• Control of the Grushin equation: non-rectangular control region and minimal time
• Stabilisation of wave equations on the torus with rough dampings
• Stabilization rates for the damped wave equation with Hölder-regular damping
• Observability of Baouendi-Grushin-Type Equations Through Resolvent Estimates
• Two-microlocal regularity of quasimodes on the torus
• Semiclassical asymptotics for nonselfadjoint harmonic oscillators