Local Carleman Estimate Research Articles

Carleman estimate for complex second order elliptic operators with discontinuous Lipschitz coefficients

In this paper, we derive a local Carleman estimate for the complex second order elliptic operator with Lipschitz coefficients having jump discontinuities. Combing the result by M. Bellassoued and J. Le Rousseau (2018) and the arguments by M. Di Cristo, E. Francini, C.-L. Lin, S. Vessella, and J.-N. Wang (2017), we present an elementary method to derive the Carleman estimate under the optimal regularity assumption on the coefficients.

Conditional stability for a Cauchy problem for the ultrahyperbolic Schrödinger equation

ABSTRACT In this work, we first establish a local Carleman estimate for the ultrahyperbolic Schrödinger equation which arises in some theories of modern physics such as quantum mechanics and string theory. Next, we prove a Hölder stability estimate for the Cauchy Problem. In the proof, we introduce a suitable cut-off function and extend Cauchy data in a Sobolev space to reduce the problem to another problem for functions with compact supports and then we apply the Carleman estimate.

Inverse coefficient problems for a transport equation by local Carleman estimate

We consider the transport equation in where is a bounded domain, and discuss two inverse problems which consist of determining a vector-valued function or a real-valued function by initial values and data on a subboundary of . Our results are conditional stability of Hölder type in a subdomain D provided that the outward normal component of is positive on . The proofs are based on a Carleman estimate where the weight function depends on H.

Boundary feedback stabilization of the Boussinesq system with mixed boundary conditions

We study the feedback stabilization of the Boussinesq system in a two dimensional domain, with mixed boundary conditions. After ascertaining the precise loss of regularity of the solution in such models, we prove first Green's formulas for functions belonging to weighted Sobolev spaces and then correctly define the underlying control system. This provides a rigorous mathematical framework for models studied in the engineering literature. We prove the stabilizability by extending to the linearized Boussinesq system a local Carleman estimate already established for the Oseen system. Then we determine a feedback control law able to stabilize the linearized system around the stationary solution, with any prescribed exponential decay rate, and able to stabilize locally the nonlinear system.

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⁡(τφ) where τ 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 φ=exp⁡(γψ), with γ acting as a second large parameter, and we derive estimates where the dependency upon the two parameters, τ and γ, is made explicit. Applications to unique continuation properties are given.

Carleman estimates for the parabolic transmission problem and Hölder propagation of smallness across an interface

In this paper we prove a Hölder propagation of smallness for solutions to second order parabolic equations whose general anisotropic leading coefficient has a jump at an interface. We assume that the leading coefficient is Lipschitz continuous with respect to the parabolic distance on both sides of the interface. The main effort consists in proving a local Carleman estimate for this parabolic operator.

Determining the conductivity for a nonautonomous hyperbolic operator in a cylindrical domain

This paper is devoted to the reconstruction of the conductivity coefficient for a nonautonomous hyperbolic operator an infinite cylindrical domain. Applying a local Carleman estimate, we prove the uniqueness and a Hölder stability in the determination of the conductivity using a single measurement data on the lateral boundary. Our numerical examples show good reconstruction of the location and contrast of the conductivity function in 3 dimensions.

Carleman estimate for second order elliptic equations with Lipschitz leading coefficients and jumps at an interface

In this paper we prove a local Carleman estimate for second order elliptic equations with a general anisotropic Lipschitz coefficients having a jump at an interface. The argument we use is of microlocal nature. Yet, not relying on pseudodifferential calculus, our approach allows one to achieve almost optimal assumptions on the regularity of the coefficients and, consequently, of the interface.

Carleman estimate for Biot consolidation system in poro‐elasticity and application to inverse problems

In this paper, we consider a coupled system of mixed hyperbolic–parabolic type, which describes the Biot consolidation model in poro‐elasticity. We establish a local Carleman estimate for Biot consolidation system. Using this estimate, we prove the uniqueness and a Hölder stability in determining on the one hand a physical parameter arising in connection with secondary consolidation effects λ∗ and on the other hand the two spatially varying densities by a single measurement of solution over ω × (0,T), where T > 0 is a sufficiently large time and a suitable subdomain ω satisfying ∂ω⊃∂Ω. Copyright © 2016 John Wiley & Sons, Ltd.

Carleman estimates for elliptic operators with complex coefficients. Part I: Boundary value problems

We consider elliptic operators with complex coefficients and we derive microlocal and local Carleman estimates near a boundary, under sub-ellipticity and strong Lopatinskii condition. Carleman estimates are weighted a priori estimates for the solutions of the associated elliptic boundary problem. The weight is of exponential form, exp⁡(τφ) where τ is meant to be taken as large as desired. Such estimates have numerous applications in unique continuation, inverse problems, and control theory. Based on inequalities for interior and boundary differential quadratic forms, the proof relies on the microlocal factorization of the symbol of the conjugated operator in connection with the sign of the imaginary part of its roots. We further consider weight functions of the previous form with moreover φ=exp⁡(γψ), where γ meant to be taken as large as desired, and we derive Carleman estimates where the dependency upon the two large parameters, τ and γ, is made explicit. Applications on unique continuation properties are given.

On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations

Local and global Carleman estimates play a central role in the study of some partial differential equations regarding questions such as unique continuation and controllability. We survey and prove such estimates in the case of elliptic and parabolic operators by means of semi-classical microlocal techniques. Optimality results for these estimates and some of their consequences are presented. We point out the connexion of these optimality results to the local phase-space geometry after conjugation with the weight function. Firstly, we introduce local Carleman estimates for elliptic operators and deduce unique continuation properties as well as interpolation inequalities. These latter inequalities yield a remarkable spectral inequality and the null controllability of the heat equation. Secondly, we prove Carleman estimates for parabolic operators. We state them locally in space at first, and patch them together to obtain a global estimate. This second approach also yields the null controllability of the heat equation.

Carleman Estimate for Elliptic Operators with Coefficients with Jumps at an Interface in Arbitrary Dimension and Application to the Null Controllability of Linear Parabolic Equations

In a bounded domain of R n+1, n ≧ 2, we consider a second-order elliptic operator, $${A=-{\partial_{x_0}^2} - \nabla_x \cdot (c(x) \nabla_x)}$$ , where the (scalar) coefficient c(x) is piecewise smooth yet discontinuous across a smooth interface S. We prove a local Carleman estimate for A in the neighborhood of any point of the interface. The “observation” region can be chosen independently of the sign of the jump of the coefficient c at the considered point. The derivation of this estimate relies on the separation of the problem into three microlocal regions and the Calderón projector technique. Following the method of Lebeau and Robbiano (Comm Partial Differ Equ 20:335–356, 1995) we then prove the null controllability for the linear parabolic initial problem with Dirichlet boundary conditions associated with the operator $${{\partial_t - \nabla_x \cdot (c(x) \nabla_x)}}$$ .

Uniqueness and partial identification in a geometric inverse problem for the Boussinesq system

We analyze the inverse problem of the identification of a rigid body immersed in a fluid governed by the stationary Boussinesq system. First, we establish a uniqueness result. Then, we present a new method for the partial identification of the body. The proofs use local Carleman estimates, differentiation with respect to domains, data assimilation techniques and controllability results for PDEs. To cite this article: A. Doubova et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).

THE UNIQUENESS AND CONTINUOUS DEPENDENCE FOR CHARACTERISTIC CAUCHY PROBLEM OF PARTIAL DIFFERENTIAL EQUATIONS

In this paper, we investigate the connection between the uniqueness and the continuous dependence for the Cauchy problem of linear partial differential equations with everywhere characteristic initial hypersurface. The local Carleman's estimate used by S. Alinhac and M. S. Baouendi to prove the uniqueness for the characteristic Cauchy problem is improved to the global estimate with a specific peaking weight factor. Using this estimate, we can prove similarly the uniqueness of the characteristic Cauchy problem. In particular, we derive also the continuous dependence for the solution of this problem in the function class that a certain norm is bounded by the aid of the same inference of proving the global estimate.