31 research outputs found
Carleman Estimates for Some Non-Smooth Anisotropic Media
International audienceLet 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
Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications
International audienceWe study the observability and some of its consequences for the one-dimensional heat equation with a discontinuous coefficient (piecewise \Con^1). The observability, for a {\em linear} equation, is obtained by a Carleman-type estimate. This kind of observability inequality yields results of controllability to the trajectories for {\em semilinear} equations. It also yields a stability result for the inverse problem of the identification of the diffusion coefficient
Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications to controllability and an inverse problem
International audienceWe study the observability and some of its consequences (controllability, identification of diffusion coefficients) for one-dimensional heat equations with discontinuous coefficients (piecewise \Con^1). The observability, for a linear equation, is obtained by a Carleman-type estimate. This kind of observability inequality yields controllability results for a semi-linear equation as well as a stability result for the identification of the diffusion coefficient
Concentration and non-concentration of eigenfunctions of second-order elliptic operators in layered media
This work is concerned with operators of the type A = --c acting in
domains := ' x (0, H) R^d x R ^+. The diffusion
coefficient c > 0 depends on one coordinate y (0, H) and is bounded but
may be discontinuous. This corresponds to the physical model of ''layered
media'', appearing in acoustics, elasticity, optical fibers... Dirichlet
boundary conditions are assumed. In general, for each > 0, the set
of eigenfunctions is divided into a disjoint union of three subsets : Fng
(non-guided), Fg (guided) and Fres (residual). The residual set shrinks as
0. The customary physical terminology of
guided/non-guided is often replaced in the mathematical literature by
concentrating/non-concentrating solutions, respectively. For guided waves, the
assumption of ''layered media'' enables us to obtain rigorous estimates of
their exponential decay away from concentration zones. The case of non-guided
waves has attracted less attention in the literature. While it is not so
closely connected to physical models, it leads to some very interesting
questions concerning oscillatory solutions and their asymptotic properties.
Classical asymptotic methods are available for c(y) C 2 but a lesser
degree of regularity excludes such methods. The associated eigenfunctions (in
Fng) are oscillatory. However, this fact by itself does not exclude the
possibility of ''flattening out'' of the solution between two consecutive
zeros, leading to concentration in the complementary segment. Here we show it
cannot happen when c(y) is of bounded variation, by proving a ''minimal
amplitude hypothesis''. However the validity of such results when c(y) is not
of bounded variation (even if it is continuous) remains an open problem
On the controllability of linear parabolic equations with an arbitrary control location for stratified media
We prove a null controllability result with an arbitrary control location in dimension greater than or equal to two for a class of linear parabolic operators with non-smooth coefficients. The coefficients are assumed to be smooth in all but one directions
On the guided states of 3D biperiodic Schrödinger operators
International audienceWe consider the Laplacian operator H_0 perturbed by a non-positive potential , which is periodic in two directions, and decays in the remaining one. We are interested in the characterization and decay properties of the guided states, defined as the eigenfunctions of the reduced operators in the Bloch-Floquet-Gelfand transform of H_0+V in the periodic variables. If V is sufficiently small and decreases fast enough in the infinite direction, we prove that, generically, these guided states are characterized by quasi-momenta belonging to some one-dimensional compact real analytic submanifold of the Brillouin zone. Moreover they decay faster than any polynomial function in the infinite direction