Carleman estimates for Baouendi–Grushin operators with applications to quantitative uniqueness and strong unique continuation

Abstract

Dedicated to Sergio Vessella, on his 65-th birthday Contents 1. Introduction 1 2. Notations and preliminary results 6 3. Proof of Theorem 1.1 9 4. Quantitative uniqueness 12 5. Strong unique continuation for sublinear equations 16 References 18

Carleman estimates for a class of variable coefficient degenerate elliptic operators with applications to unique continuation

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In this paper, we obtain new Carleman estimates for a class of variable coefficient degenerate elliptic operators whose constant coefficient model at one point is the so called Baouendi-Grushin operator. This generalizes the results obtained by the two of us with Garofalo in [10] where similar estimates were established for the "constant coefficient" Baouendi-Grushin operator. Consequently, we obtain: (i) a Bourgain-Kenig type quantitative uniqueness result in the variable coefficient setting; (ii) and a strong unique continuation property for a class of degenerate sublinear equations.

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“…As a consequence of the estimate (1.9), we deduce the following quantitative uniqueness result for "C 1 " type potentials V by repeating the arguments as in the proof of Theorem 1.3 in [10]. 8).…”

“…L 2 − L 2 Carleman estimates with singular weights for the general Baouendi-Grushin operators B γ which are analogous to the ones in [2] have been established very recently by two of us with Garofalo in [10] by using elementary arguments based on integration by parts and by an appropriate application of Rellich type identity. In the same paper, quantitative uniqueness result of Bourgain-Kenig type (see [14]) and a strong unique continuation for a class of sublinear equations of the type (1.11) (when A ≡ I) are also proven.…”

“…In the present work, we generalize the results in [10] to variable coefficient principal part where the matrix valued function A is assumed to be Lipschitz continuous with respect to a suitable pseudo-distance associated to the system of vector fields {X i }. We refer to (H) below for the precise assumptions.…”

Space like strong unique continuation for sublinear parabolic equations

“…We would also like to refer to a recent work by two of us with Garofalo as in [7] where the result of Ruland has been extended to sublinear equations associated to degenerate elliptic Baouendi–Grushin operators defined by The method in [7] also slightly simplifies the proof of Ruland when the principal part is and moreover our proof of the sublinear Carleman estimate as stated in () is also inspired in parts by the ideas in [7].…”