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-separable, anisotropic conductivities. The motivation for our study stems from its relation to the classical anisotropic Calderón problem, which at the time of writing is one of the main open problems in the field.
We study the existence and uniqueness for weak solutions to some classes of anisotropic elliptic Dirichlet problems with data belonging to the natural dual space.
We prove uniqueness for minimizers of the weighted least gradient problem \[\inf \left\lbrace \int_Ω a|Du|: \ \ u\in BV(Ω), \ \ u|_{\partial Ω}=f \right\rbrace.\] The weight function $a$ is assumed to be continuous and it is allowed to vanish in certain subsets of $Ω$. Existence is assumed a priori. Our approach is motivated by the hybrid inverse problem of imaging electric conductivity from interior knowledge (obtainable by MRI) of the magnitude of one current density vector field.
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.
We consider the inverse problem of determining, the possibly anisotropic, conductivity of a body by means of the so called local Neumann to Dirichlet map on a curved portion $Σ$ of the boundary. Motivated by the uniqueness result for piecewise constant anisotropic conductivities proved in \cite{Al-dH-G}, we provide a Hölder stability estimate on $Σ$ when the conductivity is a priori known to be a constant matrix near $Σ$.
We study the perfect conductivity problem when two perfectly conducting inclusions are closely located to each other in an anisotropic background medium. We establish optimal upper and lower gradient bounds for the solution in any dimension which characterize the singular behavior of the electric field as the distance between the inclusions goes to zero.
The aim of the paper is twofold. Firstly, we would like to derive quantitative uniqueness estimates for solutions of the general complex conductivity equation. It is still unknown whether the \emph{strong} unique continuation property holds for such equations. Nonetheless, in this paper, we show that the unique continuation property in the form of three-ball inequalities is satisfied for the complex conductivity equation under only Lipschitz assumption on the leading coefficients. The derivation of such estimates relies on a delicate Carleman estimate. Secondly, we study the problem of estimat…
The purpose of this paper is to derive anisotropic mean curvature flow as the limit of the anisotropic Allen-Cahn equation. We rely on distributional solution concepts for both the diffuse and sharp interface models, and prove convergence using relative entropy methods, which have recently proven to be a powerful tool in interface evolution problems. With the same relative entropy, we prove a weak-strong uniqueness result, which relies on the construction of gradient flow calibrations for our anisotropic energy functionals.
We consider the stability issue of the inverse conductivity problem for a conformal class of anisotropic conductivities in terms of the local Dirichlet-to-Neumann map. We extend here the stability result obtained by Alessandrini and Vessella in Advances in Applied Mathematics 35:207-241, where the authors considered the piecewise constant isotropic case.
We consider an inverse boundary value problem for the heat equation with a nonsmooth coefficient of conductivity which models the displacement of a moving body inside a nonhomogeneous background. We prove the uniqueness of the moving inclusion from the knowledge of the Dirichlet-to-Neumann operator by using a dynamical probe method.
The unique determination of electrical conductivity is extensively studied for isotropic conductivity ever since Calderon's suggestion of the EIT (Electrical Impedance Tomography) problem. However, it is known that there are many anisotropic conductivities producing the same Dirichlet-to-Neumann map; moreover the anisotropic conductivities giving the same Dirichlet-to-Neumann map are classified using the equivalence relation with respect to the change of variables. The change of variable argument is applied to the theory of near-cloaking: We are under an illusion that the domain of anomaly is …
We study in this paper the axisymmetric 3-D Navier-Stokes system where the horizontal viscosity is zero. We prove the existence of a unique global solution to the system with initial data of Yudovitch type
Studying elliptic measure data problem with strongly nonlinear operator whose growth is described by the means of fully anisotropic $N$-function, we prove the uniqueness for a broad class of measures. In order to provide it, the framework of capacities in fully anisotropic Orlicz-Sobolev spaces is developed and the~capacitary characterization of a~bounded measure is given. Moreover, we give an example of an anisotropic Young function $Φ$, such that $|ξ|^p \lesssimΦ(ξ)\lesssim |ξ|^p\log^α(1+|ξ|)$, with arbitrary $p\geq 1$, $α>0$, but so irregularly growing that % we call it essentially fully an…
We study a modified three-dimensional incompressible anisotropic Navier-Stokes equations. The modification consists in the addition of a power term to the nonlinear convective one. This modification appears naturally in porous media when a fluid obeys the Darcy-Forchheimer law instead of the classical Darcy law. We prove global in time existence and uniqueness of solutions without assuming the smallness condition on the initial data. This improves the result obtained for the classical 3D incompressible anisotropic Navier-Stokes equations.
We discuss the inverse problem of determining the, possibly anisotropic, conductivity of a body $Ω\subset\mathbb{R}^{n}$ when the so-called Neumann-to-Dirichlet map is locally given on a non empty curved portion $Σ$ of the boundary $\partialΩ$. We prove that anisotropic conductivities that are \textit{a-priori} known to be piecewise constant matrices on a given partition of $Ω$ with curved interfaces can be uniquely determined in the interior from the knowledge of the local Neumann-to-Dirichlet map.
We consider the inverse boundary value problem of recovering piecewise homogeneous elastic tensor and piecewise homogeneous mass density from a localized lateral Dirichlet-to-Neumann or Neumann-to-Dirichlet map for the elasticity equation in the space-time domain. We derive uniqueness for identifying these tensor and density on all domains of homogeneity that may be reached from the part of the boundary where the measurements are taken by a chain of subdomains whose successive interfaces contain a curved portion.
We prove uniqueness of the inverse conductivity problem in three dimensions for complex conductivities in $W^{1,\infty}$. We apply quaternionic analysis to transform the inverse problem into an inverse Dirac scattering problem, as established in two dimensions by Brown and Uhlmann. This is a novel methodology that allows to extend the uniqueness result from once-differentiable real conductivities to complex ones.
We consider the issues of stability and reconstruction of the electrical anisotropic conductivity of biological tissues in a domain $Ω\subset\mathbb{R}^3$ by means of the hybrid inverse problem of magneto-acoustic tomography with magnetic induction (MAT-MI). The class of anisotropic conductivities considered here is of type $σ(\cdot)=A(\cdot,γ(\cdot))$ in $Ω$, where $[λ^{-1}, λ]\ni t\mapsto A(\cdot, t)$ is a one-parameter family of matrix-valued functions which are \textit{a-priori} known to be $C^{1,β}$, allowing us to stably reconstruct $γ$ in $Ω$ in terms of an internal functional $F(σ)$. O…
Consider a three dimensional piecewise homogeneous anisotropic elastic medium $Ω$ which is a bounded domain consisting of a finite number of bounded subdomains $D_α$, with each $D_α$ a homogeneous elastic medium. One typical example is a finite element model with elements with curvilinear interfaces for an ansiotropic elastic medium. Assuming the $D_α$ are known and Lipschitz, we are concerned with the uniqueness in the inverse boundary value problem of identifying the anisotropic elasticity tensor on $Ω$ from a localized Dirichlet to Neumann map given on a part of the boundary $\partial D_{α_…
We consider the impedance tomography problem in the plane. Using Bukhgeim's scattering data for the Dirac problem, we prove that the conductivity is uniquely determined by the Dirichlet-to-Neuman map
This paper concerns the reconstruction of an anisotropic conductivity tensor $γ$ from internal current densities of the form $J = γ\nabla u$, where $u$ solves a second-order elliptic equation $\nabla\cdot(γ\nabla u) = 0$ on a bounded domain $X$ with prescribed boundary conditions. A minimum number of such functionals equal to $n + 2$, where $n$ is the spatial dimension, is sufficient to guarantee a local reconstruction. We show that $γ$ can be uniquely reconstructed with a loss of one derivative compared to errors in the measurement of $J$. In the special case where $γ$ is scalar, it can be re…
In this paper, we focus on the two-dimensional surface quasi-geostrophic equation with fractional horizontal dissipation and vertical thermal diffusion which represents a general case of the classical surface quasi-geostrophic equation. On the one hand, we will show the local existence and uniqueness of the solution in Sobolev space $H^{2-2α}(\mathbb{R}^2)\cap H^{2-2β}(\mathbb{R}^2)$, which is the critical space in the classical case. Furthermore, we will demonstrate that the solution is global even when the initial data is very small. Finally, we will study the asymptotic representation of ou…
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.
Let $(Ω^3,g)$ be a compact smooth Riemannian manifold with smooth boundary and suppose that $U$ is a an open set in $Ω$ such that $g|_U$ is the Euclidean metric. Let $Γ= \overline{U} \cap \partial Ω$ be connected and suppose that $U$ is the convex hull of $Γ$. We will study the uniqueness of an unknown potential for the Schrödinger operator $ -\triangle_g + q $ from the associated Dirichlet to Neumann map, $Λ_q$. We will prove that if the potential $q$ is a priori explicitly known in $U^c$, then one can uniquely reconstruct $q$ over the convex hull of $Γ$ from $Λ_q$. We will also outline a rec…
We consider the inverse problem of determining the unknown function $α: \mathbb{R} \rightarrow \mathbb{R}$ from the DN map associated to the operator $\mbox{div}(A(x',α(x\_3))\nabla \cdot)$ acting in the infinite straight cylindrical waveguide $Ω=ω\times \mathbb{R}$, where $ω$ is a bounded domain of $\mathbb{R}^2$. Here $A=(A\_{ij}(x))$, $x=(x',x\_3) \in Ω$, is a matrix-valued metric on $Ω$ obtained by straightening a twisted waveguide. This inverse anisotropic conductivity problem remains generally open, unless the unknown function $α$ is assumed to be constant. In this case we prove Lipschit…
We study the local recovery of an unknown piecewise constant anisotropic conductivity in EIT (electric impedance tomography) on certain bounded Lipschitz domains $Ω$ in $\mathbb{R}^2$ with corners. The measurement is conducted on a connected open subset of the boundary $\partialΩ$ 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 t…
We present a procedure for recovering the conformal factor of an anisotropic conductivity matrix in a known conformal class in a domain in Euclidean space of dimension greater than or equal to 2. The method requires one internal measurement, together with a priori knowledge of the conformal class (local orientation) of the conductivity matrix. This problem arises in the coupled-physics medical imaging modality of Current Density Impedance Imaging (CDII) and the assumptions on the data are suitable for measurements determinable from cross-property based couplings of the two imaging modalities C…
We prove for a two dimensional bounded domain that the Cauchy data for the Schroedinger equation measured on an arbitrary open subset of the boundary determines uniquely the potential. This implies, for the conductivity equation, that if we measure the current fluxes at the boundary on an arbitrary open subset of the boundary produced by voltage potentials supported in the same subset, we can determine uniquely the conductivity. We use Carleman estimates with degenerate weight functions to construct appropriate complex geometrical optics solutions to prove the results.
Given an elliptic integrand of class $ \mathscr{C}^{3} $, we prove that finite unions of disjoint open Wulff shapes with equal radii are the only volume-constrained critical points of the anisotropic surface energy among all sets with finite perimeter and reduced boundary almost equal to its closure.
We are interested in the uniqueness of solutions to Maxwell's equations when the magnetic permeability $μ$ and the permittivity $\varepsilon$ are symmetric positive definite matrix-valued functions in $\mathbb{R}^{3}$. We show that a unique continuation result for globally $W^{1,\infty}$ coefficients in a smooth, bounded domain, allows one to prove that the solution is unique in the case of coefficients which are piecewise $W^{1,\infty}$ with respect to a suitable countable collection of sub-domains with $C^{0}$ boundaries. Such suitable collections include any bounded finite collection. The p…
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 discuss the Cauchy problem for anisotropic wave equations. Precisely, we address the question to know which kind of Cauchy data on the lateral boundary are necessary to guarantee uniqueness of solutions of an anisotropic wave equation. In the case where the uniqueness holds, the natural problem that arise in this context is to estimate the solutions, in some appropriate space, in terms of the Cauchy data. We aim in this paper to transfer, via a reduced Fourrier-Bros-Iagolnitzer transform, the known stability estimates for the Cauchy problem for elliptic equations to that for waves equations…
The multifrequency electrical impedance tomography is considered in order to image a conductivity inclusion inside a homogeneous background medium by injecting one current. An original spectral decomposition of the solution of the forward conductivity problem is used to retrieve the Cauchy data corresponding to the extreme case of perfect conductor. Using results based on the unique continuation we then prove the uniqueness of multifrequency electrical impedance tomography and obtain rigorous stability estimates. Our results in this paper are quite surprising in inverse conductivity problem si…
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.
We study an inverse problem involving the unique recovery of several lower order anisotropic tensor perturbations of a polyharmonic operator in a bounded domain from the knowledge of the Dirichlet to Neumann map on a part of boundary. The uniqueness proof relies on the inversion of generalized momentum ray transforms (MRT) for symmetric tensor fields, which we introduce for the first time to study Calderón-type inverse problems. We construct suitable complex geometric optics (CGO) solutions for the polyharmonic operators that reduces the inverse problem to uniqueness results for a generalized …
Uniqueness and stability of time and space-dependent conductivity in a hyperbolic cylindrical domain
Chalmers University of Technology and Gothenburg University（查尔姆斯理工大学和哥德堡大学）；Institut de Mathématiques de Marseille, CNRS, UMR 7373, École Centrale, Aix-Marseille Université（马赛数学研究所，法国国家科学研究中心，联合研究实验室7373，中央理工学院，艾克斯-马赛大学）；Key Laboratory of Wu Wen-Tsun Mathematics, Chinese Academy of Sciences, School of Mathematical Sciences, University of Science and Technology of China（中国科学院吴文俊数学重点实验室，中国科学技术大学数学科学学院）
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…
Let $\hat Ω\subset \mathbb R^2$ be a bounded domain with smooth boundary and $\hat σ$ a smooth anisotropic conductivity on $\hat Ω$. Starting from the Dirichlet-to-Neumann operator $Λ_{\hat σ}$ on $\partial \hat Ω$, we give an explicit procedure to find a unique domain $Ω$, an isotropic conductivity $σ$ on $Ω$ and the boundary values of a quasiconformal diffeomorphism $F:\hat Ω\to Ω$ which transforms $\hat σ$ into $σ$.
We study the linear Vlasov equation with a given electric field $E \in \mathcal{S}$, where $\mathcal{S}$ is the space of Schwartz functions. The associated damped partial differential equation has a unique tempered solution, which fixes the needed Cauchy datum. This tempered solution then converges to the causal solution of the linear Vlasov equation when the damping parameter goes to zero. This result allows us to define the plasma conductivity operator $σ$, which gives the current density $j = σ(E)$ induced by the electric field $E$. We prove that $σ$ is continuous from $\mathcal{S}$ to its …
We investigate the problem of reconstructing a fully anisotropic conductivity tensor $γ$ from internal functionals of the form $\nabla u\cdotγ\nabla u$ where $u$ solves $\nabla\cdot(γ\nabla u) = 0$ over a given bounded domain $X$ with prescribed Dirichlet boundary condition. This work motivated by hybrid medical imaging methods covers the case $n\ge 3$, following the previously published case $n=2$ \cite{Monard2011}. Under knowledge of enough such functionals, and writing $γ= τ\tilde γ$ ($\det \tildeγ= 1$) with $τ$ a positive scalar function, we show that all of $γ$ can be explicitely and loca…
We prove that the electromagnetic material parameters are uniquely determined by boundary measurements for the time-harmonic Maxwell equations in certain anisotropic settings. We give a uniqueness result in the inverse problem for Maxwell equations on an admissible Riemannian manifold, and a uniqueness result for Maxwell equations in Euclidean space with admissible matrix coefficients. The proofs are based on a new Fourier analytic construction of complex geometrical optics solutions on admissible manifolds, and involve a proper notion of uniqueness for such solutions.
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 $\…
A note on uniqueness in the identification of a spacewise dependent source and diffusion coefficient for the heat equation
Institute of Mathematics Statistics and Physics, Federal University of Rio Grande（里奥格兰德联邦大学数学、统计与物理研究所）；School of Mathematics, University of Birmingham（伯明翰大学数学学院）
In this paper we prove uniqueness for an inverse boundary value problem (IBVP) arising in electrodynamics. We assume that the electromagnetic properties of the medium, namely the magnetic permeability, the electric permittivity and the conductivity, are described by continuously differentiable functions.
