Carleman Estimates for Parabolic Equations with Nonhomogeneous Boundary Conditions

The authors prove a new Carleman estimate for general linear second order parabolic equation with nonhomogeneous boundary conditions. On the basis of this estimate, improved Carleman estimates for the Stokes system and for a system of parabolic equations with a penalty term are obtained. This system can be viewed as an approximation of the Stokes system.

Carleman estimates and unique continuation property for the anisotropic differential-operator equations

The unique continuation theorems for the anisotropic partial differential-operator equations with variable coefficients in Banach-valued L p -spaces are studied. To obtain the uniform maximal regularity and the Carleman type estimates for parameter depended differential-operator equations, the sufficient conditions are founded. By using these facts, the unique continuation properties are established. In the application part, the unique continuation properties and Carleman estimates for finite or infinite systems of quasielliptic partial differential equations are studied.

Carleman Estimates for the Schrdinger Equation and Applications to an Inverse Problem and an Observability Inequality

The authors prove Carleman estimates for spaces of negative orders, and use these estimates to problem of determining L^p-potentials. An L^2-1evel continuation results for the SchrSdinger equation are the Schrodinger equation in Sobolev prove the uniqueness in the inverse observability inequality and unique also obtained.

Global Lipschitz Stability in an Inverse Problem for the Non-stationary Transport Equation

An inverse problem of determining the source term of the non-stationary transport equation was considered.The extra lateral boundary condition was chosen as the observation data.Based on a Carleman estimate on the non-stationary transport equation,a global Lipschitz stability for the inverse problem was proved.

A CARLEMAN ESTIMATE ON GROUPS OF HEISENBERG TYPE

A Pohozaev-Rellich type identity for the p-sub-Laplacian on groups of Heisenberg type, G, is given. A Carleman estimate for the sub-Laplacian on G is established and, as a consequence, a unique continuation result is proved.

An Inverse Problem for Maxwell's Equations in Anisotropic Media

The authors consider Maxwell＇s equations for an isomagnetic anisotropic and inhomogeneous medium in two dimensions, and discuss an inverse problem of determining the permittivity tensor （ε1,ε2,ε2,ε3 ） and the permeability μ in the constitutive relations from a finite number of lateral boundary measurements. Applying a Carleman estimate, the authors prove an estimate of the Lipschitz type for stability, provided that ε1,ε2,ε3,μ satisfy some a priori conditions.

Local Stability for an Inverse Coefficient Problem of a Fractional Diffusion Equation

Time-fractional diffusion equations are of great interest and importance on describing the power law decay for diffusion in porous media. In this paper, to identify the diffusion rate, i.e., the heterogeneity of medium, the authors consider an inverse coefficient problem utilizing finite measurements and obtain a local HSlder type conditional stability based upon two Carleman estimates for the corresponding differential equations of integer orders.

Stability of Inverse Problems for Ultrahyperbolic Equations

In this paper, the authors consider inverse problems of determining a coemcient or a source term in an ultrahyperbolic equation by some lateral boundary data. The authors prove HSlder estimates which are global and local and the key tool is Carleman estimate.

A Stability Estimate for an Inverse Problem of Determining a Coefficient in a Hyperbolic Equation with a Point Source

For the solution toαt^(2)u(x,t)-△u(x,t)+q(x)u(x,t)=δ(x,t)and u|t<0=0,consider an inverse problem of determining q(x),x∈Ωfrom data f=u|sT and g=(σu/σn)|sT.HereΩ■{(X1,x2,x3)∈R^(3)|X1>0}is a bounded domain,ST={(x,t)|x∈a2,|x|<t<T+|x|},n=n(x)is the outward unit normal n to aΩ,and T>0.For suitable T>0,prove a Lipschitz stability estimation:||q1-q2||L^(2)(Ω)≤C{||f1-f2||H^(1)(ST)+||g1-g2||L^(2)(ST)},provided that q1 satisfies a priori uniform boundedness conditions and q2 satisfies apriori uniform smallness conditions,where ux is the solution to problem(1.1)withq=qk,k=1,2.

Recovering of Damping Coefficients for a System of Coupled Wave Equations with Neumann Boundary Conditions:Uniqueness and Stability

The authors study the inverse problem of recovering damping coefficients for two coupled hyperbolic PDEs with Neumann boundary conditions by means of an additional measurement of Dirichlet boundary traces of the two solutions on a suitable, explicit subportion F1 of the boundary F, and over a computable time interval T 〉 0. Under sharp conditions on Г0= Г／Г1, T 〉 0, the uniqueness and stability of the damping coefficients are established. The proof uses critically the Carleman estimate due to Lasiecka et al. in 2000, together with a convenient tactical route ＂post-Carleman estimates＂ suggested by Isakov in 2006.