Carleman estimates for elliptic operators with complex coefficients Part II: transmission problems

We consider elliptic transmission problems with complex coefficients across an interface. Under proper transmission conditions, that extend known conditions for well-posedness, and sub-ellipticity we derive microlocal and local Carleman estimates near the interface. Carleman estimates are weighted a priori estimates of the solutions of the elliptic transmission problem. The weight is of exponential form, exp($tau$ {\phi}) where $tau$ can be taken as large as desired. Such estimates have numerous applications in unique continuation, inverse problems, and control theory. The proof relies on microlocal factorizations of the symbols of the conjugated operators in connection with the sign of the imaginary part of their roots. We further consider weight functions where {\phi} = exp($\gamma$$\psi$), with $\gamma$ acting as a second large paremeter, and we derive estimates where the dependency upon the two parameters, $tau$ and $\gamma$, is made explicit. Applications to unique continuation properties are given.Comment: 58 page

Carleman estimates and controllability results for the one-dimensional heat equation with {\em BV} coefficients

International audienceWe derive global Carleman estimates for one-dimensional linear parabolic operators \d_t \pm \d_x(c \d_x) with a coefficient $c$ with bounded variations. These estimates are obtained by approximating $c$ by piecewise regular coefficients, c_\eps, and passing to the limit in the Carleman estimates associated to the operators defined with c_\eps. Such estimates yield results of controllability to the trajectories for a classe of {\em semilinear} parabolic equations

On the convergence of some products of Fourier integral operators

An approximation Ansatz for the operator solution, $U(z',z)$, of a hyperbolic first-order pseudodifferential equation, \d_z + a(z,x,D_x) with $\Re (a) \geq 0$, is constructed as the composition of global Fourier integral operators with complex phases. We prove a convergence result for the Ansatz to $U(z',z)$ in some Sobolev space as the number of operators in the composition goes to $\infty$, with a convergence of order $\alpha$, if the symbol $a(z,.)$ is in \Con^{0,\alpha} with respect to the evolution parameter $z$. We also study the consequences of some truncation approximations of the symbol $a(z,.)$ in the construction of the Ansatz

Geometric control condition for the wave equation with a time-dependent observation domain

We characterize the observability property (and, by duality, the controllability and the stabilization) of the wave equation on a Riemannian manifold $\Omega,$ with or without boundary, where the observation (or control) domain is time-varying. We provide a condition ensuring observability, in terms of propagating bicharacteristics. This condition extends the well-known geometric control condition established for fixed observation domains. As one of the consequences, we prove that it is always possible to find a time-dependent observation domain of arbitrarily small measure for which the observability property holds. From a practical point of view, this means that it is possible to reconstruct the solutions of the wave equation with only few sensors (in the Lebesgue measure sense), at the price of moving the sensors in the domain in an adequate way.We provide several illustrating examples, in which the observationdomain is the rigid displacement in $\Omega$ of a fixed domain, withspeed $v,$ showing that the observability property depends both on $v$and on the wave speed. Despite the apparent simplicity of some of ourexamples, the observability property can depend on nontrivial arithmeticconsiderations

Fourier-integral operator approximation of solutions to first-order hyperbolic pseudodifferential equations II: microlocal analysis

redaction 2004/2005An approximation Ansatz for the operator solution, $U(z',z)$, of a hyperbolic first-order pseudodifferential equation, \d_z + a(z,x,D_x) with $\Re (a) \geq 0$, is constructed as the composition of global Fourier integral operators with complex phases. We investigate the propagation of singularities for this Ansatz and prove microlocal convergence: the wavefront set of the approximated solution is shown to converge to that of the exact solution away from the region where the phase is complex

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

Stability of discontinuous diffusion coefficients and initial conditions in an inverse problem for the heat equation

International audienceWe consider the heat equation with a discontinuous diffusion coefficient and give uniqueness and stability results for both the diffusion coefficient and the initial condition from a measurement of the solution on an arbitrary part of the boundary and at some arbitrary positive time. The key ingredient is the derivation of a Carleman-type estimate. The diffusion coefficient is assumed to be discontinuous across interfaces with a monotonicity condition

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