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

• 成果类型: Article
• 署名作者: Le Rousseau, Jerome; Robbiano, Luc
• 署名单位: Universite de Orleans; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite Paris Saclay; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI)
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-010-0278-3
• 发表日期: 2011
• 页码: 245-336
• 关键词: dimensional heat-equation; exact controllability

摘要

In (0, T) x Omega, Omega open subset of R-n, n >= 2, we consider a parabolic operator P = partial derivative(t) - del(x)delta(t, x)del(x), where the ( scalar) coefficient delta(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 d 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) x M, M a manifold, is also derived from the local result.