Local and global Carleman estimates for parabolic operators with coefficients with jumps at interfaces

Abstract

In (0,T)×Ω, Ω open subset of ℝ n , n≥2, we consider a parabolic operator P=∂ t −∇ x δ(t,x)∇ x , where the (scalar) coefficient δ(t,x) is piecewise smooth in space yet discontinuous across a smooth interface S. We prove a global in time, local in space Carleman estimate for P in the neighborhood of any point of the interface. The “observation” region can be chosen independently of the sign of the jump of the coefficient δ at the considered point. The derivation of this estimate relies on the separation of the problem into three microlocal regions related to high and low tangential frequencies at the interface. In the high-frequency regime we use Calderon projectors. In the low-frequency regime we follow a more classical approach. Because of the parabolic nature of the problem we need to introduce Weyl-Hormander anisotropic metrics, symbol classes and pseudo-differential operators. Each frequency regime and the associated technique require a different calculus. A global in time and space Carleman estimate on (0,T)×M, M a manifold, is also derived from the local result.