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 Ω, 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.

Uniform null controllability for parabolic equations with discontinuous diffusion coefficients

“…It would be very natural to try to develop suitable Carleman estimates in the spirit of the works [5][6][7][20][21][22] to derive uniform observability estimates (1.12) for the solutions of (1.11) for conductivities σ of the form (1.1) when σ 1 → ∞ under the only condition that the control set ω is non-empty and ω ⊂ Ω 2 . This is so far an open problem, even when considering the restrictive geometric setting of [10].…”

“…Later on, the case of BV coefficients in 1d was dealt with using the Fursikov-Imanuvilov approach in [19]. The case of piecewise smooth coefficients with a smooth surface of discontinuity was then studied under no geometric assumptions in the works [5][6][7][20][21][22], and null-controllability of (1.2) was proved in those cases.…”