HAL (Le Centre pour la Communication Scientifique Directe), Mar 14, 2007
In this paper we prove the unique continuation property of the solution for the elastic transvers... more In this paper we prove the unique continuation property of the solution for the elastic transversally isotropic dynamical systems with smooth coefficients satisfying some conditions and apply it to extending the Dirichlet to Neumann map. The proof is based on the localized Fourier-Gauss transformation and Carleman type estimate.
We study the local recovery of an unknown piecewise constant anisotropic conductivity in electric... more We study the local recovery of an unknown piecewise constant anisotropic conductivity in electric impedance tomography on certain bounded Lipschitz domains Ω in R 2 with corners. The measurement is conducted on a connected open subset of the boundary ∂ Ω of Ω containing corners and is given as a localized Neumann-to-Dirichlet map. The above unknown conductivity is defined via a decomposition of Ω into polygonal cells. Specifically, we consider a parallelogram-based decomposition and a trapezoid-based decomposition. We assume that the decomposition is known, but the conductivity on each cell is unknown. We prove that the local recovery is almost surely true near a known piecewise constant anisotropic conductivity γ 0. We do so by proving that the injectivity of the Fréchet derivative F ′ ( γ 0 ) of the forward map F, say, at γ 0 is almost surely true. The proof presented, here, involves defining different classes of decompositions for γ 0 and a perturbation or contrast H in a proper ...
In this paper we prove a quantitative form of the strong unique continuation property for the Lam... more In this paper we prove a quantitative form of the strong unique continuation property for the Lamé system when the Lamé coefficients µ is Lipschitz and λ is essentially bounded in dimension n ≥ 2. This result is an improvement of our earlier result [5] in which both µ and λ were assumed to be Lipschitz.
As for the unique continuation property (UCP) of solutions in (0,T)×Ω with a domain Ω⊂ℝ^n, n∈ℕ fo... more As for the unique continuation property (UCP) of solutions in (0,T)×Ω with a domain Ω⊂ℝ^n, n∈ℕ for a multi-terms time fractional diffusion equation, we have already shown it by assuming that the solutions are zero for t≤0 (see <cit.>). Here the strongly elliptic operator for this diffusion equation can depend on time and the orders of its time fractional derivatives are in (0,2). This paper is a continuation of the previous study. The aim of this paper is to drop the assumption that the solutions are zero for t≤0. We have achieved this aim by first using the usual Holmgren transformation together with the argument in <cit.> to derive the UCP in (T_1,T_2)× B_1 for some 0<T_1<T_2<T and a ball B_1⊂Ω. Then if u is the solution of the equation with u=0 in (T_1,T_2)× B_1, we show u=0 also in ((0,T_1]∪[T_2,T))× B_r for some r<1 by using the argument in <cit.> which uses two Holmgren type transformations different from the usual one. This together with spatial coordinates transformation, we can obtain the usual UCP which we call it the classical UCP given in the title of this paper for our time fractional diffusion equation.
This paper concerns the weak unique continuation property of solutions of a general system of dif... more This paper concerns the weak unique continuation property of solutions of a general system of differential equation/inequality with a second order strongly elliptic system as its leading part. We put not only some natural assumptions which we call basic assumptions, but also some technical assumptions which we call further assumptions. It is shown as usual by first applying the Holmgren transform to this equation/inequality and then establishing a Carleman estimate for the leading part of the transformed inequality. The Carleman estimate is given via a partition of unity and the Carleman estimate for the operator with constant coefficients obtained by freezing the coefficients of the transformed leading part at a point. A little more details about this are as follows. Factorize this operator with constant coefficients into two first order differential operators. Conjugate each factor by a Carleman weight, and derive an estimate which is uniform with respect to the point at which we ...
Proceedings of the American Mathematical Society, 2019
In this paper, we study a Liouville-type theorem for the Lamé system with rough coefficients in t... more In this paper, we study a Liouville-type theorem for the Lamé system with rough coefficients in the plane. Let u be a real-valued twovector in R 2 satisfying ∇u ∈ L p (R 2) for some p > 2 and the equation div µ ∇u + (∇u) T + ∇(λdivu) = 0 in R 2. When ∇µ L 2 (R 2) is not large, we show that u ≡ constant in R 2. As by-products, we prove the weak unique continuation property and the uniqueness of the Cauchy problem for the Lamé system with small µ W 1,2 .
In this paper, we would like to derive a quantitative uniqueness estimate, the three-region inequ... more In this paper, we would like to derive a quantitative uniqueness estimate, the three-region inequality, for the second order elliptic equation with jump discontinuous coefficients. The derivation of the inequality relies on the Carleman estimate proved in our previous work [5]. We then apply the three-region inequality to study the size estimate problem with one boundary measurement.
In this paper we prove three spheres inequalities for a two-dimensional strongly elliptic system.... more In this paper we prove three spheres inequalities for a two-dimensional strongly elliptic system. We then give an application of these three spheres inequalities to the inverse problem of identifying cavities by partial boundary measurements.
We study the asymptotic behavior of an incompressible fluid around a bounded obstacle. The proble... more We study the asymptotic behavior of an incompressible fluid around a bounded obstacle. The problem is modeled by the stationary Navier-Stokes equations in an exterior domain in R n with n ≥ 2. We will show that, under some assumptions, any nontrivial velocity field obeys a minimal decaying rate exp(−Ct 2 log t) at infinity. Our proof is based on appropriate Carleman estimates.
We investigate inverse problems in the determination of leading coefficients for nonlocal parabol... more We investigate inverse problems in the determination of leading coefficients for nonlocal parabolic operators, by knowing the corresponding Cauchy data in the exterior space-time domain. The key contribution is that we reduce nonlocal parabolic inverse problems to the corresponding local inverse problems with the lateral boundary Cauchy data. In addition, we derive a new equation and offer a novel proof of the unique continuation property for this new equation. We also build both uniqueness and non-uniqueness results for both nonlocal isotropic and anisotropic parabolic Calderón problems, respectively.
In this paper we study the local behavior of a solution to second order elliptic operators with s... more In this paper we study the local behavior of a solution to second order elliptic operators with sharp singular coefficients inlower order terms. One of the main results is the bound on the vanishing order of the solution, which is a quantitative estimate of the strong unique continuation property. Our proof relies on Carleman estimates with carefully chosen phases. A key strategy in the proof is to derive doubling inequalities via three-sphere inequalities. Our method can also be applied to certain elliptic systems with similar singular coefficients.
In this paper we prove the unique continuation property of the solution for the transversally iso... more In this paper we prove the unique continuation property of the solution for the transversally isotropic dynamical systems with smooth coefficients satisfying some conditions and apply it to extending the Dirichlet to Neumann map. The proof is based on the localized Fourier-Gauss transformation and Carleman type estimate.
We study the local behavior of a solution to a generalized non-stationary Stokes system with sing... more We study the local behavior of a solution to a generalized non-stationary Stokes system with singular coefficients in Rn with n≥ 2. One of our main results is a bound on the vanishing order of a no...
In this paper we prove strong unique continuation for u u satisfying an inequality of the form | ... more In this paper we prove strong unique continuation for u u satisfying an inequality of the form | △ m u | ≤ f ( x , u , D u , ⋯ , D k u ) |\triangle ^m u| \leq f(x,u,Du,\cdots ,D^ku) , where k k is up to [ 3 m / 2 ] [3m/2] . This result gives an improvement of a work by Colombini and Grammatico (1999) in some sense. The proof of the main theorem is based on Carleman estimates with three-parameter weights | x | 2 σ 1 ( log | x | ) 2 σ 2 exp ( β 2 ( log | x | ) 2 ) |x|^{2\sigma _1}(\log |x|)^{2\sigma _2}\!\exp (\frac {\beta }{2}(\log |x|)^2) .
A Carleman estimate and the unique continuation property of solutions for a multi-terms time frac... more A Carleman estimate and the unique continuation property of solutions for a multi-terms time fractional diffusion equation up to order α (0 < α < 2) and general time dependent second order strongly elliptic time elliptic operator for the diffusion. The estimate is derived via some subelliptic estimate for an operator associated to this equation using calculus of pseudo-differential operators. A special Holmgren type transformation which is linear with respect to time is used to show the local unique continuation of solutions. We developed a new argument to derive the global unique continuation of solutions. Here the global unique continuation means as follows. If u is a solution of the multiterms time fractional diffusion equation in a domain over the time interval (0, T), then a zero set of solution over a subdomain of Ω can be continued to (0, T) × Ω.
In this paper we study the local behavior of a solution to the Lamé system when the Lamé coeffici... more In this paper we study the local behavior of a solution to the Lamé system when the Lamé coefficients λ and µ satisfy that µ is Lipschitz and λ is essentially bounded in dimension n ≥ 2. One of the main results is the local doubling inequality for the solution of the Lamé system. This is a quantitative estimate of the strong unique continuation property. Our proof relies on Carleman estimates with carefully chosen weights. Furthermore, we also prove the global doubling inequality, which is useful in some inverse problems. 1991 Mathematics Subject Classification. 35J47. Key words and phrases. Elliptic systems, doubling, Carleman inequalities, quantitative uniqueness. Koch is partially supported by the DFG through SFB 1060. Lin is partially supported by the Ministry of Science and Technology of Taiwan. Wang is partially supported by MOST102-2115-M-002-009-MY3.
HAL (Le Centre pour la Communication Scientifique Directe), Mar 14, 2007
In this paper we prove the unique continuation property of the solution for the elastic transvers... more In this paper we prove the unique continuation property of the solution for the elastic transversally isotropic dynamical systems with smooth coefficients satisfying some conditions and apply it to extending the Dirichlet to Neumann map. The proof is based on the localized Fourier-Gauss transformation and Carleman type estimate.
We study the local recovery of an unknown piecewise constant anisotropic conductivity in electric... more We study the local recovery of an unknown piecewise constant anisotropic conductivity in electric impedance tomography on certain bounded Lipschitz domains Ω in R 2 with corners. The measurement is conducted on a connected open subset of the boundary ∂ Ω of Ω containing corners and is given as a localized Neumann-to-Dirichlet map. The above unknown conductivity is defined via a decomposition of Ω into polygonal cells. Specifically, we consider a parallelogram-based decomposition and a trapezoid-based decomposition. We assume that the decomposition is known, but the conductivity on each cell is unknown. We prove that the local recovery is almost surely true near a known piecewise constant anisotropic conductivity γ 0. We do so by proving that the injectivity of the Fréchet derivative F ′ ( γ 0 ) of the forward map F, say, at γ 0 is almost surely true. The proof presented, here, involves defining different classes of decompositions for γ 0 and a perturbation or contrast H in a proper ...
In this paper we prove a quantitative form of the strong unique continuation property for the Lam... more In this paper we prove a quantitative form of the strong unique continuation property for the Lamé system when the Lamé coefficients µ is Lipschitz and λ is essentially bounded in dimension n ≥ 2. This result is an improvement of our earlier result [5] in which both µ and λ were assumed to be Lipschitz.
As for the unique continuation property (UCP) of solutions in (0,T)×Ω with a domain Ω⊂ℝ^n, n∈ℕ fo... more As for the unique continuation property (UCP) of solutions in (0,T)×Ω with a domain Ω⊂ℝ^n, n∈ℕ for a multi-terms time fractional diffusion equation, we have already shown it by assuming that the solutions are zero for t≤0 (see <cit.>). Here the strongly elliptic operator for this diffusion equation can depend on time and the orders of its time fractional derivatives are in (0,2). This paper is a continuation of the previous study. The aim of this paper is to drop the assumption that the solutions are zero for t≤0. We have achieved this aim by first using the usual Holmgren transformation together with the argument in <cit.> to derive the UCP in (T_1,T_2)× B_1 for some 0<T_1<T_2<T and a ball B_1⊂Ω. Then if u is the solution of the equation with u=0 in (T_1,T_2)× B_1, we show u=0 also in ((0,T_1]∪[T_2,T))× B_r for some r<1 by using the argument in <cit.> which uses two Holmgren type transformations different from the usual one. This together with spatial coordinates transformation, we can obtain the usual UCP which we call it the classical UCP given in the title of this paper for our time fractional diffusion equation.
This paper concerns the weak unique continuation property of solutions of a general system of dif... more This paper concerns the weak unique continuation property of solutions of a general system of differential equation/inequality with a second order strongly elliptic system as its leading part. We put not only some natural assumptions which we call basic assumptions, but also some technical assumptions which we call further assumptions. It is shown as usual by first applying the Holmgren transform to this equation/inequality and then establishing a Carleman estimate for the leading part of the transformed inequality. The Carleman estimate is given via a partition of unity and the Carleman estimate for the operator with constant coefficients obtained by freezing the coefficients of the transformed leading part at a point. A little more details about this are as follows. Factorize this operator with constant coefficients into two first order differential operators. Conjugate each factor by a Carleman weight, and derive an estimate which is uniform with respect to the point at which we ...
Proceedings of the American Mathematical Society, 2019
In this paper, we study a Liouville-type theorem for the Lamé system with rough coefficients in t... more In this paper, we study a Liouville-type theorem for the Lamé system with rough coefficients in the plane. Let u be a real-valued twovector in R 2 satisfying ∇u ∈ L p (R 2) for some p > 2 and the equation div µ ∇u + (∇u) T + ∇(λdivu) = 0 in R 2. When ∇µ L 2 (R 2) is not large, we show that u ≡ constant in R 2. As by-products, we prove the weak unique continuation property and the uniqueness of the Cauchy problem for the Lamé system with small µ W 1,2 .
In this paper, we would like to derive a quantitative uniqueness estimate, the three-region inequ... more In this paper, we would like to derive a quantitative uniqueness estimate, the three-region inequality, for the second order elliptic equation with jump discontinuous coefficients. The derivation of the inequality relies on the Carleman estimate proved in our previous work [5]. We then apply the three-region inequality to study the size estimate problem with one boundary measurement.
In this paper we prove three spheres inequalities for a two-dimensional strongly elliptic system.... more In this paper we prove three spheres inequalities for a two-dimensional strongly elliptic system. We then give an application of these three spheres inequalities to the inverse problem of identifying cavities by partial boundary measurements.
We study the asymptotic behavior of an incompressible fluid around a bounded obstacle. The proble... more We study the asymptotic behavior of an incompressible fluid around a bounded obstacle. The problem is modeled by the stationary Navier-Stokes equations in an exterior domain in R n with n ≥ 2. We will show that, under some assumptions, any nontrivial velocity field obeys a minimal decaying rate exp(−Ct 2 log t) at infinity. Our proof is based on appropriate Carleman estimates.
We investigate inverse problems in the determination of leading coefficients for nonlocal parabol... more We investigate inverse problems in the determination of leading coefficients for nonlocal parabolic operators, by knowing the corresponding Cauchy data in the exterior space-time domain. The key contribution is that we reduce nonlocal parabolic inverse problems to the corresponding local inverse problems with the lateral boundary Cauchy data. In addition, we derive a new equation and offer a novel proof of the unique continuation property for this new equation. We also build both uniqueness and non-uniqueness results for both nonlocal isotropic and anisotropic parabolic Calderón problems, respectively.
In this paper we study the local behavior of a solution to second order elliptic operators with s... more In this paper we study the local behavior of a solution to second order elliptic operators with sharp singular coefficients inlower order terms. One of the main results is the bound on the vanishing order of the solution, which is a quantitative estimate of the strong unique continuation property. Our proof relies on Carleman estimates with carefully chosen phases. A key strategy in the proof is to derive doubling inequalities via three-sphere inequalities. Our method can also be applied to certain elliptic systems with similar singular coefficients.
In this paper we prove the unique continuation property of the solution for the transversally iso... more In this paper we prove the unique continuation property of the solution for the transversally isotropic dynamical systems with smooth coefficients satisfying some conditions and apply it to extending the Dirichlet to Neumann map. The proof is based on the localized Fourier-Gauss transformation and Carleman type estimate.
We study the local behavior of a solution to a generalized non-stationary Stokes system with sing... more We study the local behavior of a solution to a generalized non-stationary Stokes system with singular coefficients in Rn with n≥ 2. One of our main results is a bound on the vanishing order of a no...
In this paper we prove strong unique continuation for u u satisfying an inequality of the form | ... more In this paper we prove strong unique continuation for u u satisfying an inequality of the form | △ m u | ≤ f ( x , u , D u , ⋯ , D k u ) |\triangle ^m u| \leq f(x,u,Du,\cdots ,D^ku) , where k k is up to [ 3 m / 2 ] [3m/2] . This result gives an improvement of a work by Colombini and Grammatico (1999) in some sense. The proof of the main theorem is based on Carleman estimates with three-parameter weights | x | 2 σ 1 ( log | x | ) 2 σ 2 exp ( β 2 ( log | x | ) 2 ) |x|^{2\sigma _1}(\log |x|)^{2\sigma _2}\!\exp (\frac {\beta }{2}(\log |x|)^2) .
A Carleman estimate and the unique continuation property of solutions for a multi-terms time frac... more A Carleman estimate and the unique continuation property of solutions for a multi-terms time fractional diffusion equation up to order α (0 < α < 2) and general time dependent second order strongly elliptic time elliptic operator for the diffusion. The estimate is derived via some subelliptic estimate for an operator associated to this equation using calculus of pseudo-differential operators. A special Holmgren type transformation which is linear with respect to time is used to show the local unique continuation of solutions. We developed a new argument to derive the global unique continuation of solutions. Here the global unique continuation means as follows. If u is a solution of the multiterms time fractional diffusion equation in a domain over the time interval (0, T), then a zero set of solution over a subdomain of Ω can be continued to (0, T) × Ω.
In this paper we study the local behavior of a solution to the Lamé system when the Lamé coeffici... more In this paper we study the local behavior of a solution to the Lamé system when the Lamé coefficients λ and µ satisfy that µ is Lipschitz and λ is essentially bounded in dimension n ≥ 2. One of the main results is the local doubling inequality for the solution of the Lamé system. This is a quantitative estimate of the strong unique continuation property. Our proof relies on Carleman estimates with carefully chosen weights. Furthermore, we also prove the global doubling inequality, which is useful in some inverse problems. 1991 Mathematics Subject Classification. 35J47. Key words and phrases. Elliptic systems, doubling, Carleman inequalities, quantitative uniqueness. Koch is partially supported by the DFG through SFB 1060. Lin is partially supported by the Ministry of Science and Technology of Taiwan. Wang is partially supported by MOST102-2115-M-002-009-MY3.
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