Carleman estimates for some non smooth anisotropic media
Résumé
Let B be a n × n block diagonal matrix in which the first block Cτ is an hermitian matrix of order (n − 1) and the second block c is a positive function. Both are piecewise smooth in Ω, a bounded domain of Rn. If S denotes the set where discontinuities of Cτ 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′ ⊂ Rn−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.
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