Gunther Uhlmann, Yiran Wang
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.
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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。
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。
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。
We study inverse conductivity problem for an anisotropic conductivity in $L^\infty$ in bounded and unbounded domains. Also, we give applications of the results in the case when Dirichlet-to-Neumann and Neumann-to-Dirichlet maps are given only on a part of the boundary.
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。
Direct reconstruction of anisotropic self-adjoint inclusions in the Calderón problem
本文提出了一种在Calderón问题中直接重建各向异性自 adjoint包含物的方法，通过局部Neumann到Dirichlet映射和扰动条件，解决了非唯一性问题。
Revisiting the Anisotropic Fractional Calderón Problem Using the Caffarelli-Silvestre Extension
本文利用Caffarelli-Silvestre延拓方法重访各向异性分数阶Calderón问题，证明热核与波核可互相恢复，并建立Poisson核与波核的直接联系，从而简化恢复过程并联系局部与非局部问题。
Boundary Reconstruction for the Anisotropic Fractional Calderón Problem
针对各向异性分数 Calderón 问题，论文利用退化椭圆扩张进行局部化，并基于 Brown 方法证明可由源到解数据重构测量集上的度量。
Prescribed nonlinearity helps in an anisotropic Calderón-type problem
研究拟线性各向异性椭圆方程的逆边值问题，证明三维及以上维度中其Dirichlet-to-Neumann映射可唯一确定系数矩阵，揭示预设非线性可突破经典线性Calderón问题的唯一性障碍。
On the transfer of stability from the local to the fractional anisotropic Calderón problem with exterior measurements
研究经典到分数Calderón问题的唯一性定量转移，基于外部数据推导出无Liouville变换下的主部稳定性估计，利用定量唯一延续和Runge逼近估计解决无界几何和维度不匹配的挑战。
Stability for the Calderón's problem for a class of anisotropic conductivities via an ad-hoc misfit functional
本文针对形式为σ=γA的各向异性电导率，通过构造特设失配泛函，建立了Calderón问题在该类电导率下的稳定性估计。
Local rigidity of conformally Euclidean metrics for the anisotropic Calderón problem
本文证明在光滑紧致域上，若一个黎曼度量与欧几里得度量具有相同的边界度量和Dirichlet-to-Neumann映射且足够接近，则它们通过固定边界的微分同胚等距，即欧几里得度量具有局部刚性。
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。
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…
The anisotropic Calderón problem: rigidity near the Euclidean metric
该文证明三维及以上区域中各向异性Calderón问题在欧几里得度量附近的刚性：若光滑度量充分接近欧几里得度量且具有相同Dirichlet-to-Neumann映射，则必相差一个固定边界的微分同胚，方法结合调和坐标与Carleman估计。
Fractional anisotropic Calderón problem on complete Riemannian manifolds
本文证明了完备黎曼流形的度量张量可由局部源到解算子唯一确定（等距意义下），该算子与分数阶Laplace-Beltrami算子相关，并推广至闭流形情形。
The Calderón Problem for Quasilinear Conductivities of Conformally Transversally Anisotropic Media
本文研究维数n≥3的共形横向各向异性流形上含电势和电场依赖的拟线性电导率Calderón问题，证明其在(s,p)=(t,0)处的导数由边界测量唯一确定，且解析电导率可唯一恢复。
Fractional anisotropic Calderón problem with external data
本文在欧几里得空间中解决了二维及以上的分数阶各向异性Calderón问题，证明了部分外部Dirichlet--to--Neumann映射可确定黎曼度量至微分同胚，并给出了两种证明方法。
The local Calderón problem and the determination at the boundary of a complex anisotropic admittivity
研究通过局部Dirichlet到Neumann映射确定各向异性复电导率，建立Lipschitz和Hölder稳定性估计。
The anisotropic Calderón problem at large fixed frequency on manifolds with invertible ray transform
本文研究大固定频率下由 Dirichlet 到 Neumann 映射恢复势的逆问题，将已有结果推广至简单流形及射线变换稳定可逆的流形，采用不变的高斯波束拟模构造并保证常数一致有界。
The Calderón problem for piecewise polynomial anisotropic conductivities on partitions with many flat faces
针对n≥3维分段多项式各向异性电导率，通过平面界面逐步恢复和Runge逼近，证明局部Dirichlet-to-Neumann映射唯一确定所有多项式块，并具有Hölder稳定性。
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…
The fractional anisotropic Calderón problem for a nonlocal parabolic equation on closed Riemannian manifolds
本文研究闭黎曼流形上非局部抛物型方程的分数阶各向异性Calderón问题，证明通过任意小开圆柱体内的局部源到解映射可在等距意义下确定流形，并借助半群理论分析了非局部抛物型算子的谱性质。
Quasianalyticity and geometric rigidity in anisotropic Calderón's problem
本文针对n≥3维未解决的各向异性Calderón问题，在两种互补情形下建立唯一性结果，揭示了正则性、几何结构与边界访问间的权衡关系。
In this article we study the linearized anisotropic Calderon problem. In a compact manifold with boundary, this problem amounts to showing that products of harmonic functions form a complete set. Assuming that the manifold is transversally anisotropic, we show that the boundary measurements determine an FBI type transform at certain points in the transversal manifold. This leads to recovery of transversal singularities in the linearized problem. The method requires a geometric condition on the transversal manifold related to pairs of intersecting geodesics, but it does not involve the geodesic…
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。
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.
On the anisotropic Calderón's problem
该研究证明了维数 n≥3 的紧流形上，Laplace–Beltrami 算子的 Dirichlet-to-Neumann 映射可在边界固定等距意义下唯一确定 Riemann 度量。
The conformal Poisson embedding method for the anisotropic Calderón problem with a potential
该研究针对带位势的各向异性Calderón问题，证明了Δ_g+q的局部Dirichlet-to-Neumann映射可确定对应黎曼流形与位势，发展共形泊松嵌入方法并获新结果。
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…
Anisotropic Calderón problem for a logarithmic Schrödinger operator of order $2+$ on closed Riemannian manifolds
本文研究了闭黎曼流形上非局部对数薛定谔算子的各向异性Calderón问题，证明了在底层流形相同或集合互异情形下，可通过Cauchy数据唯一恢复黎曼度量与势函数。
The local complex Calderón problem. Stability in a layered medium for a special type of anisotropic admittivity
本文研究分层各向异性介质中的局部复Calderón问题，在导纳为已知矩阵与未知分层仿射复值函数乘积的假设下，基于失配泛函和局部Dirichlet-to-Neumann映射给出了相应的Hölder和Lipschitz稳定性估计。
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 …
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。
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 $\…
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…
In this article we study the linearized anisotropic Calderón problem on a compact Riemannian manifold with boundary. This problem amounts to showing that products of pairs of harmonic functions of the manifold form a complete set. We assume that the manifold is transversally anisotropic and that the transversal manifold is real analytic and satisfies a geometric condition related to the geometry of pairs of intersecting geodesics. In this case, we solve the linearized anisotropic Calderón problem. The geometric condition does not involve the injectivity of the geodesic X-ray transform. Crucial…
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…
Conformally St{ä}ckel manifolds can be characterized as the class of n-dimensional pseudo-Riemannian manifolds (M, G) on which the Hamilton-Jacobi equation G($\nabla$u, $\nabla$u) = 0 for null geodesics and the Laplace equation --$Δ$ G $ψ$ = 0 are solvable by R-separation of variables. In the particular case in which the metric has Riemannian signature, they provide explicit examples of metrics admitting a set of n--1 commuting conformal symmetry operators for the Laplace-Beltrami operator $Δ$ G. In this paper, we solve the anisotropic Calder{ó}n problem on compact 3-dimensional Riemannian man…
We construct anisotropic conductivities with the same Dirichlet-to-Neumann map as a homogeneous isotropic conductivity. These conductivities are singular close to a surface inside the body.
The Calderón inverse problem for isotropic quasilinear conductivities
该研究证明了高维有界区域上一般拟线性各向同性电导率方程的Calderón逆问题全局唯一性，通过高阶线性化和复几何光学解实现。
After giving a general introduction to the main known results on the anisotropic Calder{ó}n problem on n-dimensional compact Riemannian manifolds with boundary, we give a motivated review of some recent non-uniqueness results obtained in [5, 6] for the anisotropic Calder{ó}n problem at fixed frequency, in dimension n $\ge$ 3, when the Dirichlet and Neumann data are measured on disjoint subsets of the boundary. These non-uniqueness results are of the following nature: given a smooth compact connected Riemannian manifold with boundary (M, g) of dimension n $\ge$ 3, we first show that there exist…
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…
Let $X$ be a smooth bordered surface in $\real^3$ with smooth boundary and $\hat σ$ a smooth anisotropic conductivity on $X$. If the genus of $X$ is given, then starting from the Dirichlet-to-Neumann operator $Λ_{\hat σ}$ on $\partial X$, we give an explicit procedure to find a unique Riemann surface $Y$ (up to a biholomorphism), an isotropic conductivity $σ$ on $Y$ and the boundary values of a quasiconformal diffeomorphism $F: X \to Y$ which transforms $\hat σ$ into $σ$. As a corollary we obtain the following uniqueness result: if $σ_1, σ_2$ are two smooth anisotropic conductivities on $X$ wi…
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 $$ρ$\<1$ at the end where the measurements are made. More precisely, we construct a toroidal ring $(M, g)$ which is not a warped product manifold, and we show that there exist in the …
We outline an approach to the inverse problem of Calderón that highlights the role of microlocal normal forms and propagation of singularities and extends a number of earlier results also in the anisotropic case. The main result states that from the boundary measurements it is possible to recover integrals of the unknown coefficient over certain two-dimensional manifolds called good bicharacteristic leaves. This reduces the Calderón problem into solving a linear integral geometry problem (inversion of a bicharacteristic leaf transform).
Stability in the anisotropic Calderón problem for Painlevé-Liouville Riemannian manifolds
研究Painlevé-Liouville黎曼流形中各向异性Calderón反问题的稳定性，证明了对数稳定性结果及部分问题的类似结论。
Let $\vec{a}:=(a_1,\ldots,a_n)\in[1,\infty)^n$, $\vec{p}:=(p_1,\ldots,p_n)\in(0,\infty)^n$ and $H_{\vec{a}}^{\vec{p}}(\mathbb{R}^n)$ be the anisotropic mixed-norm Hardy space associated with $\vec{a}$ defined via the non-tangential grand maximal function. In this article, via first establishing a Calderón-Zygmund decomposition and a discrete Calderón reproducing formula, the authors then characterize $H_{\vec{a}}^{\vec{p}}(\mathbb{R}^n)$, respectively, by means of atoms, the Lusin area function, the Littlewood-Paley $g$-function or $g_λ^\ast$-function. The obtained Littlewood-Paley $g$-functio…
Given a general symmetric elliptic operator $$ L\_{a} := \sum\_{k,,j=1}^d \p\_k (a\_{kj} \p\_j) + \sum\_{k=1}^d a\_k \p\_k - \p\_k(\overline{a\_k} .) + a\_0$$we define the associated Dirichlet-to-Neumann (D-t-N) operator with partial data, i.e., data supported in a part of the boundary. We prove positivity, $L^p$-estimates and domination properties for the semigroup associated with this D-t-N operator. Given $L\_a $ and $L\_b$ of the previous type with bounded measurable coefficients $a = \{a\_{kj}, \ a\_k, a\_0 \}$ and $b = \{b\_{kj}, \ b\_k, b\_0 \}$, we prove that if their partial D-t-N ope…
In this paper, we investigate the anisotropic Calder{ó}n problem on cylindrical Riemannian manifolds with boundary having two ends and equipped with singular metrics of (simple or double) warped product type, that is whose warping factors only depend on the horizontal direction of the cylinder. By singular, we mean that these factors are only assumed to be positive almost everywhere and to belong to some L p space with 0 < p $\le$ $\infty$. Using recent developments on the Weyl-Titchmarsh theory for singular Sturm-Liouville operators, we prove that the local Dirichlet-to-Neumann maps at each e…
The anisotropic Calderón problem for Riemannian metrics at high fixed frequency
该论文证明了在固定高频下，黎曼流形上 Helmholtz 方程的 Dirichlet-to-Neumann 映射相等可推出透镜数据相等，从而建立 Calderón 问题与透镜刚性之间的高频桥梁，并给出定量版本。
Reconstruction in the Calderón problem on conformally transversally anisotropic manifolds
本文在共形横各向异性流形上，利用高斯波束拟模构造复几何光学解，从Dirichlet-to-Neumann映射构造性重构连续势，给出Calderón问题的构造性解法。
On Instability Properties of the Fractional Calderón Problem
该研究针对分数阶幂次s∈(0,1)的分数阶Calderón问题及其电导率公式，证明其指数不稳定性性质，推广了已有结果至一般几何与变系数情形，验证了对数稳定性估计的最优性。
The Calderón problem for the Schrödinger equation in transversally anisotropic geometries with partial data
该研究针对横向各向异性几何中带部分数据的薛定谔方程Calderón问题，通过三步法（边界确定、关联非局部椭圆反问题、约化为波动方程反问题）实现了区域上度量与势函数的恢复。
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…
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…
The Born approximation for the fixed energy Calderón problem
本文研究了固定能量下Calderón问题的Born近似，证明其作为径向分布的良定义性，并展示了其在恢复势中的应用及高能极限下的收敛性。
Applications of the Stone-Weierstrass theorem in the Calderón problem
本文利用 Stone-Weierstrass 定理研究 Calderón 反问题，证明了全纯可分 Kähler 流形上线性化问题及共形横截各向异性流形上非线性问题的唯一性，并推广简化了已有结果。
In this note we show that on any compact subdomain of a Kähler manifold that admits sufficiently many global holomorphic functions, the products of harmonic functions form a complete set. This gives a positive answer to the linearized anisotropic Calderón problem on a class of complex manifolds that includes compact subdomains of Stein manifolds and sufficiently small subdomains of Kähler manifolds. Some of these manifolds do not admit limiting Carleman weights, and thus cannot by treated by standard methods for the Calderón problem in higher dimensions. The argument is based on constructing M…
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。
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