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 ...
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 ...
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) .
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 ...
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 ...
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) .
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