A quantitative Carleman estimate for second order elliptic operators

Abstract

We prove a Carleman estimate for elliptic second order partial differential operators with Lipschitz continuous coefficients. The Carleman estimate is valid for any complex-valued function $u\in W^{2,2}$ 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.

23 pages

References in corpus (3)

• Scale-free quantitative unique continuation and equidistribution estimates for solutions of elliptic differential equations
• Multiscale unique continuation properties of eigenfunctions
• Equidistribution estimates for eigenfunctions and eigenvalue bounds for random operators

Cited by in corpus (5)

• Observability of the heat equation, geometric constants in control theory, and a conjecture of Luc Miller
• Scale-free quantitative unique continuation and equidistribution estimates for solutions of elliptic differential equations
• Spectral inequality with sensor sets of decaying density for Schrödinger operators with power growth potentials
• Quantitative unique continuation for spectral subspaces of Schrödinger operators with singular potentials
• Unique continuation for the gradient of eigenfunctions and Wegner estimates for random divergence-type operators