Oleg Yu. Imanuvilov

Global Lipschitz stability in an inverse hyperbolic problem by interior observations

For the solution u(p) = u(p)(x,t) to ∂t2u(x,t)-Δu(x,t)-p(x)u(x,t) = 0 in Ω×(0,T) and (∂u/∂ν)|∂Ω×(0,T) = 0 with given u(,0) and ∂tu(,0), we consider an inverse problem of determining p(x), xΩ, from data u|ω×(0,T). Here Ω⊂n, n = 1,2,3, is a bounded domain, ω is a sub-domain of Ω and T>0. For suitable ω⊂Ω and T>0, we prove an upper and lower estimate of Lipschitz type between ||p-q||L2(Ω) and ||∂t(u(p)-u(q))||L2(ω×(0,T)) + ||∂t2(u(p)-u(q))||L2(ω×(0,T)).

GLOBAL UNIQUENESS AND STABILITY IN DETERMINING COEFFICIENTS OF WAVE EQUATIONS

We consider an inverse problem of determining p(x), x ∈ Ω in (∂2 u/∂t 2)(x, t) − Δu(x, t) − p(x)u(x, t) = 0 in Ω × (0, T) and (∂u/∂ν)|∂Ω×(0, T) = 0 with given u(·, 0) and (∂u/∂t)(·, 0). Here Ω ⊂ R n , n = 1, 2, 3, is a bounded domain. We prove the Lipschitz stability in determining p from u|∂Ω×(0,T), under the assumption that T is greater than the diameter of Ω and u(·,0) > 0 on . Our stability holds over Ω without any knowledge of boundary values of p. The proof is based on the Carleman estimate.

Determination of a coefficient in an acoustic equation with a single measurement

For the solution u(p) = u(p)(t, x) to ∂t2 u(t, x) − div(p(x)∇u(t, x)) = 0 in (0, T) × Ω with given u|(0,T)×∂Ω, u(0, ·) and ∂t u(0, ·), we consider an inverse problem concerning the determination of the coefficient p(x), x Ω from data u|(0,T)×ω. Here Ω ⊂ n is a bounded domain, and ω is some subdomain of Ω and 0

Carleman estimate for a stationary isotropic Lamé system and the applications

>T > 0. For suitable ω ⊂ Ω and 0

Global Carleman estimates for weak solutions of elliptic nonhomogeneous Dirichlet problems

>T > 0, we prove an estimate of the Holder type: |p − q|L2 (Ω) ≤ C( ∑ j = 23 |∂ tj (u(p) − u(q))| L2 ((0,T)×ω)) κwith some κ (0, 1), provided that p, q satisfy a priori uniform boundedness conditions, compatible conditions and some positivity conditions. The keys are Carleman estimates for a hyperbolic operator in an H−1-space.

Inverse Source Problem for the Stokes System

For the isotropic stationary Lamé system with variable coefficients equipped with the Dirichlet or surface stress boundary condition, we obtain a Carleman estimate such that (i) the right hand side is estimated in a weighted L 2-space and (ii) the estimate includes nonhomogeneous surface displacement or surface stress. Using this estimate we establish the conditional stability in Sobolevs norm of the displacement by means of measurements in an arbitrary subdomain or measurements of surface displacement and stress on an arbitrary subboundary. Finally by the Carleman estimate, we prove the uniqueness and conditional stability for an inverse problem of determining a source term by a single interior measurement.

Existence and nonexistence in Chern–Simons–Higgs theory with a constant electric charge density

We consider a general second order elliptic equation with right-hand side f+∑j=0N∂fj∂xj∈H−1(Ω) where f,fj∈L2(Ω) and Dirichlet boundary condition g∈H1/2(Γ). We prove a global Carleman estimate for the solution y of this equation in terms of the weighted L2 norms of f and fj and the H1/2 norm of g. This estimate depends on two real parameters s and λ which are supposed to be large enough and is sharp with respect to the exponents of these parameters. This allows us to obtain, for example, sharper estimates on the pressure term in the linearized Navier–Stokes equations and it turns out to be very useful in the context of controllability problems. To cite this article: O.Y. Imanuvilov, J.-P. Puel, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 33–38.

The Existence of Non-Topological Multivortex Solutions in the Relativistic Self-Dual Chern–Simons Theory

We consider a Stokes system with external force term:

Carleman Inequalities for Parabolic Equations in Sobolev Spaces of Negative Order and Exact Controllability for Semilinear Parabolic Equations

\frac{{\partial y}}{{\partial t}} = \Delta y - \nabla p + r\left( t \right)f\left( x \right),\nabla \cdot y = 0in\left( {0,T} \right) \times \Omega