In dimensions greater than or equal to three, we establish global uniqueness and obtain reconstruction in the Calderon problem for the Schrodinger equation with certain singular potentials. The potentials considered are conormal of order less than 1-k with respect to submanifolds (of arbitrary codimension k). This gives positive results for (conormal) conductivities which are Holder of any order > 1. A related problem for highly singular potentials is shown to exhibit nonuniqueness.
We study the multi-channel Gel'fand-Calderón inverse problem in two dimensions, i.e. the inverse boundary value problem for the equation $-Δψ+ v(x) ψ= 0$, $x\in D$, where $v$ is a smooth matrix-valued potential defined on a bounded planar domain $D$. We give an exact global reconstruction method for finding $v$ from the associated Dirichlet-to-Neumann operator. This also yields a global uniqueness results: if two smooth matrix-valued potentials defined on a bounded planar domain have the same Dirichlet-to-Neumann operator then they coincide.
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 use X^{s,b}-inspired spaces to prove a uniqueness result for Calderon's problem in a Lipschitz domain under the assumption that the conductivity is Lipschitz. For Lipschitz conductivities, we obtain uniqueness for conductivities close to the identity in a suitable sense. We also prove uniqueness for arbitrary C^1 conductivities.
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 uniqueness for Calderón's problem with Lipschitz conductivities in higher dimensions. Combined with the recent work of Haberman, who treated the three and four dimensional cases, this confirms a conjecture of Uhlmann. Our proof builds on the work of Sylvester and Uhlmann, Brown, and Haberman and Tataru who proved uniqueness for $C^1$ conductivities and Lipschitz conductivities sufficiently close to the identity.
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.
We study the fractional Schrödinger equation with quasilocal perturbations. These are a family of nonlocal perturbations vanishing at infinity, which include e.g. convolutions against Schwartz functions. We show that the qualitative unique continuation and Runge approximation properties hold in the assumption of sufficient decay. Quantitative versions of both results are also obtained via a propagation of smallness analysis for the Caffarelli-Silvestre extension. The results are then used to show uniqueness in the inverse problem of retrieving a quasilocal perturbation from DN data under suita…
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.
We prove uniqueness in the inverse conductivity problem for uniformly elliptic conductivities in $W^{s,p}(Ω)$, where $Ω\subset \mathbb R^n$ is Lipschitz, $3\leq n \leq 6$, and $s$ and $p$ are such that $ W^{s,p}(Ω)\not \subset W^{1,\infty}(Ω)$. In particular, we obtain uniqueness for conductivities in $W^{1,n}(Ω)$ ($n=3,4$). This improves on the result of the author and Tataru, who assumed that the conductivity is Lipschitz.
We study Green functions for stationary Stokes systems satisfying the conormal derivative boundary condition. We establish existence, uniqueness, and various estimates for the Green function under the assumption that weak solutions of the Stokes system are continuous in the interior of the domain. Also, we establish the global pointwise bound for the Green function under the additional assumption that weak solutions of the conormal derivative problem for the Stokes system are locally bounded up to the boundary. We provide some examples satisfying such continuity and boundedness properties.
We show global uniqueness in the fractional Calderón problem with a single measurement and with data on arbitrary, possibly disjoint subsets of the exterior. The previous work \cite{GhoshSaloUhlmann} considered the case of infinitely many measurements. The method is again based on the strong uniqueness properties for the fractional equation, this time combined with a unique continuation principle from sets of measure zero. We also give a constructive procedure for determining an unknown potential from a single exterior measurement, based on constructive versions of the unique continuation resu…
We consider the determination of a conductivity function in a two-dimensional domain from the Cauchy data of the solutions of the conductivity equation on the boundary. We prove uniqueness results for this inverse problem, posed by Calderon, for conductivities that are degenerate, that is, they may not be bounded from above or below. In particular, for scalar conductivities we solve the inverse problem in a class which is larger than $L^\infty$. Also, we give new counterexamples for the uniqueness of the inverse conductivity problem. We say that a conductivity is visible if the inverse problem…
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…
Uniqueness in the Calderón problem in dimension bigger than two was usually studied under the assumption that conductivity has bounded gradient. For conductivities with unbounded gradients uniqueness results have not been known until recent years. The latest result due to Haberman basically relies on the optimal $L^2$ restriction estimate for hypersurface which is known as the Tomas-Stein restriction theorem. In the course of developments of the Fourier restriction problem bilinear and multilinear generalizations of the (adjoint) restriction estimates under suitable transversality condition be…
We show that there is generically non-uniqueness for the anisotropic Calderón problem at fixed frequency when the Dirichlet and Neumann data are measured on disjoint sets of the boundary of a given domain. More precisely, we first show that given a smooth compact connected Riemannian manifold with boundary $(M,g)$ of dimension $n\geq 3$, there exist in the conformal class of $g$ an infinite number of Riemannian metrics $\tilde{g}$ such that their corresponding DN maps at a fixed frequency coincide when the Dirichlet data $Γ_D$ and Neumann data $Γ_N$ are measured on disjoint sets and satisfy $\…
In this paper we define a Dirichlet-to-Neumann map for a twisted Dirac Laplacian acting on bundle-valued spinors over a spin manifold. We show that this map is a pseudodifferential operator of order 1 whose symbol determines the Taylor series of the metric and connection at the boundary. We go on to show that if two real-analytic connections couple to a spinor via the Yang--Mills--Dirac equations with appropriate boundary conditions, and have equal Dirichlet-to-Neumann maps, then the two connections are locally gauge equivalent. In the abelian case, the connections are globally gauge equivalen…
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.
A Sharp Regularity Threshold for Uniqueness in Riemannian Calderón-type Problems
Université Marie et Louis Pasteur（玛丽和路易·巴斯德大学）；Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas（高等科学研究委员会数学科学研究所）；Nantes Université（南特大学）；McGill University（麦吉尔大学）
In this short note we prove the logarithmic stability of the single measurement uniqueness result for the fractional Calderón problem which had been derived in \cite{GRSU18}. To this end, we use the quantitative uniqueness results established in \cite{RS20a} and complement these bounds with a boundary doubling estimate. The latter yields control of the order of vanishing of solutions to the fractional Schrödinger equation. Then, following a scheme introduced in \cite{S10,ASV13} in the context of the determination of a surface impedance from far field measurements, this allows us to deduce loga…
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 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…
Via Carleman estimates we prove uniqueness and continuous dependence results for lateral Cauchy problems for linear integro-differential parabolic equations without initial conditions. The additional information supplied prescribes the conormal derivative of the temperature on a relatively open subset of the lateral boundary of the space-time domain.
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 …
We investigate global uniqueness for an inverse problem for a nonlocal diffusion equation on domains that are bounded in one direction. The coefficients are assumed to be unknown and isotropic on the entire space. We first show that the partial exterior Dirichlet-to-Neumann map locally determines the diffusion coefficient in the exterior domain. In addition, we introduce a novel analysis of nonlocal Neumann derivatives to prove an interior determination result. Interior and exterior determination yield the desired global uniqueness theorem for the Calderón problem of nonlocal diffusion equatio…
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 …
The Calderón problem for the fractional Schrödinger equation was introduced in the work \cite{GSU}, which gave a global uniqueness result also in the partial data case. This article improves this result in two ways. First, we prove a quantitative uniqueness result showing that this inverse problem enjoys logarithmic stability under suitable a priori bounds. Second, we show that the results are valid for potentials in scale-invariant $L^p$ or negative order Sobolev spaces. A key point is a quantitative approximation property for solutions of fractional equations, obtained by combining a careful…
We investigate the Calderón problem for the fractional Schrödinger equation with drift, proving that the unknown drift and potential in a bounded domain can be determined simultaneously and uniquely by an infinite number of exterior measurements. In particular, in contrast to its local analogue, this nonlocal problem does \emph{not} enjoy a gauge invariance. The uniqueness result is complemented by an associated logarithmic stability estimate under suitable apriori assumptions. Also uniqueness under finitely many \emph{generic} measurements is discussed. Here the genericity is obtained through…
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.
In this paper we show uniqueness of the conductivity for the quasilinear Calderón's inverse problem. The nonlinear conductivity depends, in a nonlinear fashion, of the potential itself and its gradient. Under some structural assumptions on the direct problem, a real-valued conductivity allowing a small analytic continuation to the complex plane induces a unique Dirichlet-to-Neumann (DN) map. The method of proof considers some complex-valued, linear test functions based on a point of the boundary of the domain, and a linearization of the DN map placed at these particular set of solutions.
We consider one-dimensional Calderón's problem for the variable exponent $p(\cdot)$-Laplace equation and find out that more can be seen than in the constant exponent case. The problem is to recover an unknown weight (conductivity) in the weighted $p(\cdot)$-Laplace equation from Dirichlet and Neumann data of solutions. We give a constructive and local uniqueness proof for conductivities in $L^\infty$ restricted to the coarsest sigma-algebra that makes the exponent $p(\cdot)$ measurable.
A direct reconstruction algorithm based on Calderón's linearization method for the reconstruction of isotropic conductivities is proposed for anisotropic conductivities in two-dimensions. To overcome the non-uniqueness of the anisotropic inverse conductivity problem, the entries of the unperturbed anisotropic tensors are assumed known \emph{a priori}, and it remains to reconstruct the multiplicative scalar field. The quasi-conformal map in the plane facilitates the Calderón-based approach for anisotropic conductivities. The method is demonstrated on discontinuous radially symmetric conductivit…
Determining $W^{1,n}$ conductivities in Calderón's problem for $n\geq4$
Key Laboratory of Algebraic Lie Theory and Analysis, Ministry of Education, School of Mathematics and Statistics, Beijing Institute of Technology（北京理工大学数学与统计学院代数李理论与分析教育部重点实验室）；Department of Mathematics, City University of Hong Kong（香港城市大学数学系）
Given any elliptic system with $t$-independent coefficients in the upper-half space, we obtain representation and trace for the conormal gradient of solutions in the natural classes for the boundary value problems of Dirichlet and Neumann types with area integral control or non-tangential maximal control. The trace spaces are obtained in a natural range of boundary spaces which is parametrized by properties of some Hardy spaces. This implies a complete picture of uniqueness vs solvability and well-posedness.
We prove a Plancherel theorem for a nonlinear Fourier transform in two dimensions arising in the Inverse Scattering method for the defocusing Davey-Stewartson II equation. We then use it to prove global well-posedness and scattering in $L^2$ for defocusing DSII. This Plancherel theorem also implies global uniqueness in the inverse boundary value problem of Calderón in dimension $2$, for conductivities $σ>0$ with $\log σ\in \dot H^1$. The proof of the nonlinear Plancherel theorem includes new estimates on classical fractional integrals, as well as a new result on $L^2$-boundedness of pseudo-dif…
We prove that an $L^\infty$ potential in the Schrödinger equation in three and higher dimensions can be uniquely determined from a finite number of boundary measurements, provided it belongs to a known finite dimensional subspace $\mathcal W$. As a corollary, we obtain a similar result for Calderón's inverse conductivity problem. Lipschitz stability estimates and a globally convergent nonlinear reconstruction algorithm for both inverse problems are also presented. These are the first results on global uniqueness, stability and reconstruction for nonlinear inverse boundary value problems with f…
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…
Electrical Impedance Tomography gives rise to the severely ill-posed Calderón problem of determining the electrical conductivity distribution in a bounded domain from knowledge of the associated Dirichlet-to-Neumann map for the governing equation. The uniqueness and stability questions for the three-dimensional problem were largely answered in the affirmative in the 1980's using complex geometrical optics solutions, and this led further to a direct reconstruction method relying on a non-physical scattering transform. In this paper, the reconstruction problem is taken one step further towards p…
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 …
We consider the anisotropic Calderon problem of recovering a conductivity matrix or a Riemannian metric from electrical boundary measurements in three and higher dimensions. In the earlier work \cite{DKSaU}, it was shown that a metric in a fixed conformal class is uniquely determined by boundary measurements under two conditions: (1) the metric is conformally transversally anisotropic (CTA), and (2) the transversal manifold is simple. In this paper we will consider geometries satisfying (1) but not (2). The first main result states that the boundary measurements uniquely determine a mixed Four…
In this paper we study an inverse problem for fractional anisotropic conductivity. Our nonlocal operator is based on the well-developed theory of nonlocal vector calculus, and differs substantially from other generalizations of the classical anisotropic conductivity operator obtained spectrally. We show that the anisotropic conductivity matrix can be recovered uniquely from fractional Dirichlet-to-Neumann data up to a natural gauge. Our analysis makes use of techniques recently developed for the study of the isotropic fractional elasticity equation, and generalizes them to the case of non-sepa…
