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

Abstract

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 Calderón 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-Hörmander 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.

Carleman estimates for stratified media

“…In [27,28] the authors show that this quadratic form is only nonnegative for low (tangential) frequencies. Here we shall recover this behavior where the tangential Fourier transform is replaced by Fourier series, built on a basis of eigenfunctions of the transverse part of the elliptic operator.…”

“…Assume that the coefficients associated with the transverse part of the operator are flat at the boundary. Then, by reflection at the boundary, the system under consideration can be turned into a problem with a smooth interface away from any boundary which permits to use the results of [27,28,25,26]. This situation is however not general.…”

“…For high (tangential) frequencies, the tangential derivative term (i.e., the action of a the first-order operator on u |S ) yields a negative contribution, unless a monotonicity assumption on the coefficient c is made as in [12]. In [5,27,28] the authors have used microlocal methods in the high frequency regime to solve this difficulty, and more recently in [26,25]. Here, because of the intersection of the interface with the boundary, and because of the little regularity required for the diffusion coefficients, such methods cannot be used directly.…”

“…The methods used in [5,27,28,25,26] focus on a neighborhood of a point at the interface where the interface can be given by {x n = 0} for an appropriate choice of coordinates x = (x , x n ), x ∈ R n−1 , x n ∈ R. Then, through microlocal techniques (Calderón projector or first-order factorization), a local Carleman estimate is proven. However, these methods require strong regularity for the coefficients and for the interface.…”

“…The case of an arbitrary dimension without any monotonicity condition in the elliptic case was solved in [5,27] and in the parabolic case in [28]. In [25,26] the case of a discontinuous anisotropic matrix coefficients is treated and a sharp condition on the weight function is provided for the Carleman estimate to hold.…”