A quantitative Carleman estimate for second-order elliptic operators

Abstract

We prove a Carleman estimate for elliptic second order partial differential expressions with Lipschitz continuous coefficients. The Carleman estimate is valid for any complex-valued function u ∈ W 2,2 with support in a punctured ball of arbitrary radius. The novelty of this Carleman estimate is that we establish an explicit dependence on the Lipschitz and ellipticity constants, the dimension of the space and the radius of the ball. In particular we provide a uniform and quantitative bound on the weight function for a class of elliptic operators given explicitly in terms of ellipticity and Lipschitz constant.

Scale-free unique continuation principle for spectral projectors, eigenvalue-lifting and Wegner estimates for random Schrödinger operators

“…Proposition 3.2 is a special case of the result obtained in [35] where general second order elliptic partial differential operators with Lipschitz continuous coefficients are considered. The estimate has been previously obtained; (1) in [3], but there without the gradient term on the left hand side; (2) in [11], but there without a quantitative statement of the admissible functions u.…”

“…We follow [3,35] and consider the case ρ = 1 and u ∈ C We follow the proof of [3,Lemma 3.15] until the estimate (8.2) in [3], i.e.…”

“…The second Carleman estimate is similar to the ones in [11,3]. However, none of the two is quite sufficient for our purposes, so we use a variant developed in [35], see also Appendix §A. Moreover, typically the diameter of the ambient manifold enters in the Carleman estimate.…”