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.