C. E. Kenig, J. Sjoestrand, G. Uhlmann
In this paper we improve an earlier result by Bukhgeim and Uhlmann, by showing that in dimension larger than or equal to three, the knowledge of the Cauchy data for the Schrödinger equation measured on possibly very small subsets of the boundary determines uniquely the potential. We follow the general strategy of Bukhgeim and Uhlmann but use a richer set of solutions to the Dirichlet problem.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We survey recent results on Calderon's inverse problem with partial data, focusing on three and higher dimensions.
In this paper we study the inverse conductivity problem with partial data. Moreover, we show that, in dimension $n\geq 3$ the uniqueness of the Calderón problem holds for the $C^{1}\bigcap H^{3/2, 2}$ conductivities.
We consider the Calderòn problem in an infinite cylindrical domain, whose cross section is a bounded domain of the plane. We prove log-log stability in the determination of the isotropic periodic conductivity coefficient from partial Dirichlet data and partial Neumann boundary observations of the solution.
We are concerned with the Calderón problem of determining an unknown conductivity of a body from the associated boundary measurement. We establish a logarithmic type stability estimate in terms of the Hausdorff distance in determining the support of a convex polygonal inclusion by a single partial boundary measurement. We also derive the uniqueness result in a more general scenario where the conductivities are piecewise constants supported in a nested polygonal geometry. Our methods in establishing the stability and uniqueness results have a significant technical initiative and a strong potent…
We solve the partial data Calderón problem on conformally transversallly anisotropic (CTA) manifolds with $L^{n/2}$ potentials - on par with sharp unique continuation result of \cite{JerKen}. A trivial consequence of this is the sharp regularity improvement to the result of Kenig-Sjöstrand-Uhlmann \cite{ksu}. This is done by constructing a "Green's function" which possesses both desirable boundary conditions {\em and} satisfies semiclassical type estimates in the suitable $L^{p}$ spaces. No Carleman estimates were used in the writing of this article which makes it starkly different from the tr…
This is a follow-up of a previous article where we proved local stability estimates for a potential in a Schrödinger equation on an open bounded set in dimension $n=3$ from the Dirichlet-to-Neumann map with partial data. The region under control was the penumbra delimited by a source of light outside of the convex hull of the open set. These local estimates provided stability of log-log type corresponding to the uniqueness results in Calderón's inverse problem with partial data proved by Kenig, Sjöstrand and Uhlmann. In this article, we prove the corresponding global estimates in all dimension…
In this article, we discuss quantitative Runge approximation properties for the acoustic Helmholtz equation and prove stability improvement results in the high frequency limit for an associated partial data inverse problem modelled on \cite{AU04, KU19}. The results rely on quantitative unique continuation estimates in suitable function spaces with explicit frequency dependence. We contrast the frequency dependence of interior Runge approximation results from non-convex and convex sets.
We show in two dimensions that measuring Dirichlet data for the conductivity equation on an open subset of the boundary and, roughly speaking, Neumann data in slightly larger set than the complement uniquely determines the conductivity on a simply connected domain. The proof is reduced to show a similar result for the Schrödinger equation. Using Carleman estimates with degenerate weights we construct appropriate complex geometrical optics solutions to prove the results.
Partial data Calderón problem for quasilinear conductivities in dimension 2
研究二维拟线性电导率的部分数据卡尔德隆问题，核心方法是基于高阶线性化及利用二维中部分边界消失的CGO解，通过改进余项估计、分析相位组合模式，最终证明唯一性结果。
Stability estimate for the discrete Calderon problem from partial data
针对 d≥3 维部分边界数据的离散 Calderón 问题，本文利用 CGO 解、新型离散 Carleman 估计与唯一延拓估计，建立了任意小边界部分上的对数稳定性估计。
We are concerned with the Calderón inverse inclusion problem, where one intends to recover the shape of an inhomogeneous conductive inclusion embedded in a homogeneous conductivity by the associated boundary measurements. We consider the highly challenging case with a single partial boundary measurement, which constitutes a long-standing open problem in the literature. It is shown in several existing works that corner singularities can help to resolve the uniqueness and stability issues for this inverse problem. In this paper, we show that the corner singularity can be relaxed to be a certain …
In this article we consider the simultaneous recovery of bulk and boundary potentials in (degenerate) elliptic equations modelling (degenerate) conducting media with inaccessible boundaries. This connects local and nonlocal Calderón type problems. We prove two main results on these type of problems: On the one hand, we derive simultaneous bulk and boundary Runge approximation results. Building on these, we deduce uniqueness for localized bulk and boundary potentials. On the other hand, we construct a family of CGO solutions associated with the corresponding equations. These allow us to deduce …
The partial data Calderón problem in dimension three
针对三维稳态Schrödinger方程，证明边界点附近的局部Dirichlet-to-Neumann映射唯一确定该点邻域内的势函数，通过加权X射线变换将问题归结为积分几何。
Linearized partial data Calderón problem for Biharmonic operators
本研究针对双调和算子的线性化部分数据Calderón问题，通过构造特殊解并借助Segal-Bargmann变换实现低阶扰动的恢复，推广了调和算子的相关结果。
We consider the problem of recovering the coefficient σ(x) of the elliptic equation \grad \cdot(σ\grad u)=0 in a body from measurements of the Cauchy data on possibly very small subsets of its surface. We give a constructive proof of a uniqueness result by Kenig, Sjöstrand, and Uhlmann. We construct a uniquely specified family of solutions such that their traces on the boundary can be calculated by solving an integral equation which involves only the given partial Cauchy data. The construction entails a new family of Green's functions for the Laplacian, and corresponding single layer potential…
In this Note, we present a Calderón-type uniqueness theorem on the Cauchy problem of stochastic partial differential equations. To this aim, we introduce the concept of stochastic pseudo-differential operators, and establish their boundedness and other fundamental properties. The proof of our uniqueness theorem is based on a new Carleman-type estimate.
We prove that uniqueness for the Calderón problem on a Riemannian manifold with boundary follows from a hypothetical unique continuation property for the elliptic operator $Δ+V+(Λ^{1}_{t}-q)\otimes (Λ^{2}_{t}-q)$ defined on $\partial\mathcal{M}^{2}\times [0,1]$ where $V$ and $q$ are potentials and $Λ^{i}_{t}$ is a Dirichlet-Neumann operator at depth $t$. This is done by showing that the difference of two Dirichlet-Neumann maps is equal to the Neumann boundary values of the solution to an inhomogeneous equation for said operator, where the source term is a measure supported on the diagonal of $…
We determine both the magnetic potential and the electric potential from the exterior partial measurements of the Dirichlet-to-Neumann map in the fractional linear magnetic Calderón problem by using an integral identity. We also determine both the magnetic potential and the nonlinearity in the fractional semilinear magnetic Calderón problem by using a first order linearization.
We introduce the fractional magnetic operator involving a magnetic potential and an electric potential. We formulate an inverse problem for the fractional magnetic operator. We determine the electric potential from the exterior partial measurements of the associated Dirichlet-to-Neumann map by using Runge approximation property.
We extend a global uniqueness result for the Calderón problem with partial data, due to Kenig-Sjöstrand-Uhlmann, to the case of less regular conductivities. Specifically, we show that in dimensions $n\ge 3$, the knowledge of the Diricihlet-to-Neumann map, measured on possibly very small subsets of the boundary, determines uniquely a conductivity having essentially $3/2$ derivatives in an $L^2$ sense.
In this paper we present a Calderón-Zygmund approach for a large class of parabolic equations with pseudo-differential operators $\mathcal{A}(t)$ of arbitrary order $γ\in(0,\infty)$. It is assumed that $\cA(t)$ is merely measurable with respect to the time variable. The unique solvability of the equation $$ \frac{\partial u}{\partial t}=\cA u-λu+f, \quad (t,x)\in \fR^{d+1} $$ and the $L_{q}(\fR,L_{p})$-estimate $$ \|u_{t}\|_{L_{q}(\fR,L_{p})}+\|(-Δ)^{γ/2}u\|_{L_{q}(\fR,L_{p})} +λ\|u\|_{L_{q}(\fR,L_{p})}\leq N\|f\|_{L_{q}(\fR,L_{p})} $$ are obtained for any $λ> 0$ and $p,q\in (1,\infty)$.
A generalized variant of the Calderón problem from electrical impedance tomography with partial data for anisotropic Lipschitz conductivities is considered in an arbitrary space dimension $n \geq 2$. The following two results are shown: (i) The selfadjoint Dirichlet operator associated with an elliptic differential expression on a bounded Lipschitz domain is determined uniquely up to unitary equivalence by the knowledge of the Dirichlet-to-Neumann map on an open subset of the boundary, and (ii) the Dirichlet operator can be reconstructed from the residuals of the Dirichlet-to-Neumann map on th…
We consider Calderon's inverse problem with partial data in dimensions $n \geq 3$. If the inaccessible part of the boundary satisfies a (conformal) flatness condition in one direction, we show that this problem reduces to the invertibility of a broken geodesic ray transform. In Euclidean space, sets satisfying the flatness condition include parts of cylindrical sets, conical sets, and surfaces of revolution. We prove local uniqueness in the Calderon problem with partial data in admissible geometries, and global uniqueness under an additional concavity assumption. This work unifies two earlier …
A Reduction of the Fractional Calderón Problem to the Local Calderón Problem by Means of the Caffarelli-Silvestre Extension
通过Caffarelli-Silvestre延拓，将分数阶Calderón问题的部分Dirichlet-to-Neumann数据转化为局部问题的完整数据，从而将非局部唯一性归结为局部唯一性，并给出构造性恢复方案及逆转障碍。
On a fixed smooth compact Riemann surface with boundary $(M_0,g)$, we show that for the Schrödinger operator $Δ+V$ with potential $V\in C^{1,α}(M_0)$ for some $α>0$, the Dirichlet-to-Neumann map $N|_Γ$ measured on an open set $Γ\subset \partial M_0$ determines uniquely the potential $V$. We also discuss briefly the corresponding consequences for potential scattering at 0 frequency on Riemann surfaces with asymptotically Euclidean or asymptotically hyperbolic ends.
We discuss the inverse problem of determining the, possibly anisotropic, conductivity of a body $Ω\subset\mathbb{R}^{n}$ when the so--called Dirichlet-to-Neumann map is locally given on a non empty portion $Γ$ of the boundary $\partialΩ$. We extend results of uniqueness and stability at the boundary, obtained by the same authors in SIAM J. Math. Anal. 33 (2001), no. 1, 153--171, where the Dirichlet-to-Neumann map was given on all of $\partialΩ$ instead. We also obtain a pointwise stability result at the boundary among the class of conductivities which are continuous at some point $y\inΓ$. Our …
In this article we consider a linearized Calderón problem for polyharmonic operators of order $2m\ (m\ge 2)$ in the spirit of Calderón's original work [Cal80]. We give a uniqueness result for determining coefficients of order $\leq 2m-1$ up to gauge, based on inverting momentum ray transforms.
Calderón's inverse problem via Vekua theory
本文通过维库亚理论在克利福德分析框架下证明了卡勒顿反问题的唯一性结果。
We prove a global logarithmic stability estimate for the Gel'fand-Calderon inverse problem on a two-dimensional domain.
We show, in a borderline case which was not covered before, the validity of nonlinear Calderón-Zygmund estimates for a class of non-uniformly elliptic problems driven by double phase energies.
We consider Calderón's inverse boundary value problems for a class of nonlinear Helmholtz Schrödinger equations and Maxwell's equations in a bounded domain in $\R^n$. The main method is the higher-order linearization of the Dirichlet-to-Neumann map of the corresponding equations. The local uniqueness of the linearized partial data Calderón's inverse problem is obtained following \cite{DKSU}. The Runge approximation properties and unique continuation principle allow us to extend to global situations. Simultaneous recovery of some unknown cavity$/$boundary and coefficients are given as some appl…
In this note we prove the exponential instability of the fractional Calderón problem and thus prove the optimality of the logarithmic stability estimate from \cite{RS17}. In order to infer this result, we follow the strategy introduced by Mandache in \cite{M01} for the standard Calderón problem. Here we exploit a close relation between the fractional Calderón problem and the classical Poisson operator. Moreover, using the construction of a suitable orthonormal basis, we also prove (almost) optimality of the Runge approximation result for the fractional Laplacian, which was derived in \cite{RS1…
We show global uniqueness in an inverse problem for the fractional Schrödinger equation: an unknown potential in a bounded domain is uniquely determined by exterior measurements of solutions. We also show global uniqueness in the partial data problem where the measurements are taken in arbitrary open, possibly disjoint, subsets of the exterior. The results apply in any dimension $\geq 2$ and are based on a strong approximation property of the fractional equation that extends earlier work. This special feature of the nonlocal equation renders the analysis of related inverse problems radically d…
In this paper, we use the theory of symmetric Dirichlet forms to give a probabilistic interpretation of Calderón's inverse conductivity problem in terms of reflecting diffusion processes and their corresponding boundary trace processes.
Remarks on the anisotropic Calderón problem
本文研究各向异性Calderón问题，证明了横截各向异性流形上该问题的唯一性，给出一般黎曼流形上Dirichlet-to-Neumann映射值域在零势附近的凸性，并得到半线性椭圆方程对应的Calderón型反问题的相关结果。
In this paper we prove log log type stability estimates for inverse boundary value problems on admissible Riemannian manifolds of dimension $n \geq 3$. The stability estimates correspond to a couple of uniqueness results by Dos Santos Ferrera, Kenig, Salo and Uhlmann. These inverse problems arise naturally when studying the anisotropic Calderón problem.
In these notes we prove log-type stability for the Calderón problem with conductivities in $ C^{1,\varepsilon}(\barΩ) $. We follow the lines of a recent work by Haberman and Tataru in which they prove uniqueness for $ C^1(\barΩ) $.
We review recent progress in the fractional Calderón problem, where one tries to determine an unknown coefficient in a fractional Schrödinger equation from exterior measurements of solutions. This equation enjoys remarkable uniqueness and approximation properties, which turn out to yield strong results in related inverse problems.
Anisotropic Calderón Problem for a Non-Local Second Order Elliptic Operator
本文研究了闭Riemann流形上二阶非局部椭圆算子的各向异性Calderón问题，证明了利用Cauchy数据集可在标准规范下恢复流形几何结构。
We show that Nachman's integral equations for the Calderón problem, derived for conductivities in $W^{2,p}(Ω)$, still hold for $L^\infty$ conductivities which are $1$ in a neighborhood of the boundary. We also prove convergence of scattering transforms for smooth approximations to the scattering transform of $L^\infty$ conductivities. We rely on Astala-Päivärinta's formulation of the Calderón problem for a framework in which these convergence results make sense.
The Calderón problem for near-Euclidean metrics
研究紧致带边黎曼流形上稳态薛定谔方程边界测量确定势的反问题，证明欧几里得度量小扰动下的全局唯一性。
Reconstruction in the Calderón problem on a fixed partition from finite and partial boundary data
本文改进了Calderón问题中从有限和部分边界数据重建分段常数电导率的方法，无需层状假设和电导率边界限制。
We consider the so called Calderón problem which corresponds to the determination of a conductivity appearing in an elliptic equation from boundary measurements. Using several known results we propose a simplified and self contained proof of this result.
We introduce a new approach to the anisotropic Calderón problem, based on a map called Poisson embedding that identifies the points of a Riemannian manifold with distributions on its boundary. We give a new uniqueness result for a large class of Calderón type inverse problems for quasilinear equations in the real analytic case. The approach also leads to a new proof of the result by Lassas and Uhlmann (2001) solving the Calderón problem on real analytic Riemannian manifolds. The proof uses the Poisson embedding to determine the harmonic functions in the manifold up to a harmonic morphism. The …
We study the inverse problem of determining the coefficients of the fractional power of a general second order elliptic operator given in the exterior of an open subset of the Euclidean space. We show the problem can be reduced into determining the coefficients from the boundary Cauchy data of the elliptic operator on the open set, the Calderón problem. As a corollary we establish several new results for nonlocal inverse problems by using the corresponding results for the local inverse problems. In particular the isotropic nonlocal Calderón problem can be resolved completely, assuming some reg…
Anisotropic Calderón problem of a nearly Laplace-Beltrami operator of order $2+$
本文研究闭黎曼流形上对数拉普拉斯算子的各向异性 Calderón 问题，证明柯西数据集可在标准规范等价意义下唯一恢复流形几何。
In this note, we study Calderón's problem for certain classes of conductivities in domains with circular symmetry in two and three dimensions. Explicit formulas are obtained for the reconstruction of the conductivity from the Dirichlet-to-Neumann map. As a consequence, we show that the reconstruction is Lipschitz stable.
We consider Schrödinger operators at a fixed high frequency on simply connected compact Riemannian manifolds with non-positive sectional curvatures and smooth strictly convex boundaries. We prove that the Dirichlet-to-Neumann map uniquely determines the potential.
The Dirichlet-to-Neumann map for Lorentzian Calderón problems with data on disjoint sets
针对带边界洛伦兹流形上波动方程的受限Dirichlet-to-Neumann映射，用滑行射线与微局部分析，证明可从该映射重建边界度量共形类与磁势，且已知部分区域度量时可重建更大区域度量。
The Calderón problem is an inverse problem with applications to electrical impedance tomography and geophysical prospection. We prove uniqueness in the Calderón problem in spatial dimension $n \geq 3$ for scalar conductivities in the Sobolev space $W^{1,p}$ with $p \geq n$. This generalizes a result of Haberman who considered the case $p \geq n$ and $n=3$ or $4$. Our method of proof combines a Fourier series approach with an analytic criterion for infinitesimal boundedness of potentials appearing in a Schrödinger equation with respect to the Laplacian.
Transfer of Stability from the Classical to the Fractional Anisotropic Calderón Problem
该研究针对两类带源-解测量的谱分数阶各向异性Calderón问题，通过定量唯一延拓方法建立其与经典Calderón问题的稳定性、唯一性传递关系，首次得到无Liouville约化的分数阶Calderón型问题主部稳定性结果，并证明经典问题两类测量的定量关联。
In this paper, we introduce an inverse problem of a Schrödinger type variable nonlocal elliptic operator $(-\nabla\cdot(A(x)\nabla))^{s}+q)$, for $0<s<1$. We determine the unknown bounded potential $q$ from the exterior partial measurements associated with the nonlocal Dirichlet-to-Neumann map for any dimension $n\geq2$. Our results generalize the recent initiative [16] of introducing and solving inverse problem for fractional Schrödinger operator $((-Δ)^{s}+q)$ for $0<s<1$. We also prove some regularity results of the direct problem corresponding to the variable coefficients fractional differ…
We deal with a global Calderón-Zygmund type estimate for elliptic obstacle problems of $p$-Laplacian type with measure data. For this paper, we focus on the singular case of growth exponent, i.e. $1<p \le 2-\frac{1}{n}$. In addition, the emphasis of this paper is in obtaining the Lorentz bounds for the gradient of solutions with the use of fractional maximal operators.
The unique determination of a measurable conductivity from the Dirichlet-to-Neumann map of the equation $\mathrm{div} (σ\nabla u) = 0$ is the subject of this note. A new strategy, based on Clifford algebras and a higher dimensional analogue of the Beltrami equation, is here proposed. This represents a possible first step for a proof of uniqueness for the Calderón problem in three and higher dimensions in the $L^\infty$ case.
Fractional anisotropic Calderón problem on closed Riemannian manifolds
本文解决了闭黎曼流形上的分数阶各向异性 Calderón 问题，证明由局部源到解映射可确定流形（等距意义下），为非局部类比提供了解答。
The purpose of this article is to extend the uniqueness results for the two dimensional Calderón problem to unbounded potentials on general geometric settings. We prove that the Cauchy data sets for Schrödinger equations uniquely determines potentials in $L^{p}$ for $p> 4/3$. In doing so, we first recover singularities of the potential, from which point a $L^2$-based method of stationary phase can be applied. Both steps are done via constructions of complex geometric optic solutions and Carleman estimates.
We show that there is non-uniqueness for the Calder{ó}n problem with partial data for Riemannian metrics with H{ö}lder continuous coefficients in dimension greater or equal than three. We provide simple counterexamples in the case of cylindrical Riemannian manifolds with boundary having two ends. The coefficients of these metrics are smooth in the interior of the manifold and are only H{ö}lder continuous of order $$ρ$\<1$ at the end where the measurements are made. More precisely, we construct a toroidal ring $(M, g)$ which is not a warped product manifold, and we show that there exist in the …
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, resp…
We study an inverse problem for the fractional Schrödinger equation (FSE) with a local perturbation by a linear partial differential operator (PDO) of order smaller than the order of the fractional Laplacian. We show that one can uniquely recover the coefficients of the PDO from the Dirichlet-to-Neumann (DN) map associated to the perturbed FSE. This is proved for two classes of coefficients: coefficients which belong to certain spaces of Sobolev multipliers and coefficients which belong to fractional Sobolev spaces with bounded derivatives. Our study generalizes recent results for the zeroth a…
Explicit diagonalization of the linearized Calderón problem on the disk: multiplicity normalization and exponential spectral attenuation
本文对单位圆盘上Calderón问题的线性化薛定谔形式，实现了正规算子的显式连续谱对角化，首次给出了完整角度依赖且重数归一化的线性化映射的广义奇异系统。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。