Carleman Estimates for Some Non-Smooth Anisotropic Media

Abstract

Let B be a n × n block diagonal matrix in which the first block C _τ is an hermitian matrix of order (n − 1) and the second block c is a positive function. Both are piecewise smooth in , a bounded domain of ℝ ⁿ . If S denotes the set where discontinuities of C _τ and c can occur, we suppose that Ω is stratified in a neighborhood of S in the sense that locally it takes the form Ω′ × (− δ, δ) with Ω′ ⊂ ℝ ⁿ⁻¹, δ > 0 and S = Ω′ × {0}. We prove a Carleman estimate for the elliptic operator A = − ∇·(B∇) with an arbitrary observation region. This Carleman estimate is obtained through the introduction of a suitable mesh of the neighborhood of S and an associated approximation of c involving the Carleman large parameters.

Keywords:

• Anisotropic elliptic operators
• Approximation
• Carleman estimate
• Non-smooth coefficients
• Observation location
• Stratified media

AMS 2010 Subject Classification:

• 35B37
• 35J15
• 35J60

Acknowledgments

The authors are grateful to F. Boyer (Aix Marseille Université) and J. Droniou (Monash University, Australia) for fruitful discussions on Appendix C.

The authors thank warmly the referees whose careful work has improved the presentation of this paper. L. Thevenet was partially supported by l'Agence Nationale de la Recherche under grant ANR-07-JCJC-0139-01.