Carleman estimates for stratified media

Abstract

International audienceWe consider anisotropic elliptic and parabolic operators in a bounded stratified media in $\R^n$ characterized by discontinuties of the coefficients in one direction. The surfaces of discontinuities cross the boundary of the domain. We prove Carleman estimates for these operators with an arbitrary observation region

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.…”

Stability of inverse source problem for a transmission wave equation with multiple interfaces of discontinuity

“…For further details on the Carleman estimate for hyperbolic equations, see for example, see Bellassoued and Yamamoto [8] and Imanuvilov [17]. On the other hand, for Carleman estimates for elliptic or parabolic equations with discontinuous coefficients , we refer to Benabdallah, Gaitan and Le Rousseau [11,12], and Le Rousseau and Lerner [28].…”