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 an...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.展开更多
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 ...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.展开更多
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 Carlem...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.展开更多
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 ...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.展开更多
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...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.展开更多
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 domai...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.展开更多
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 p...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.展开更多
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 mediu...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.展开更多
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 lo...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.展开更多
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 s...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.展开更多
基金supported by NSF grant DMS (No. 0808130) ANR Project (No. C-QUID 06-BLAN-0052).
文摘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.
基金supported by the Japanese Government Scholarship,the National Natural Science Foundation ofChina(No.10801030)the Science Foundation for Young Teachers of Northeast Normal University(No.20080103)+1 种基金the Japan Society for the Promotion of Science(No.15340027)the Grant from the Ministryof Education,Cultures,Sports and Technology of Japan(No.17654019)
文摘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.
文摘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.
基金The project supported by National Natural Science Foundation of China, Grant No. 10371099.
文摘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.
文摘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.
文摘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.
基金Project supported by the Rotary Yoneyama Doctor Course Scholarship (Japan) the Fujyu-kai (Tokyo, Japan)+1 种基金the 21st Century Center of Excellence Program at Graduate School of Mathematical Sciences, the University of Tokyo, the Japan Society for the Promotion of Science (No. 15340027)the Ministry of Education, Cultures, Sports and Technology (No. 17654019).
文摘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.
基金supported by the National Natural Science Foundation of China(No.11101093)Shanghai Science and Technology Commission(Nos.11ZR1402800,11PJ1400800)
文摘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.
基金supported by the Council of Higher Education of Turkey(No.16.01.2012:558–2233)
文摘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.
基金supported by the National Science Foundation (No. DMS-0104305)the Air Force Office ofScientific Research under Grant FA 9550-09-1-0459
文摘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.
