arXiv subjects
Publications and source records attributed to Gunther Uhlmann.
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.
The aim of this work is to derive sharp quantitative estimates of the qualitative convergence results developed in [28] for regularized full- and partial-cloaks via the transformation-optics approach. Let $Γ_0$ be a compact set in $\mathbb{R}^3$ and $Γ_δ$ be a $δ$-neighborhood of $Γ_0$ for $δ\in\mathbb{R}_+$. $Γ_δ$ represents the virtual domain used for the blow-up construction. By incorporating suitably designed lossy layers, it is shown that if the generating set $Γ_0$ is a generic curve, then one would have an approximate full-cloak within $δ^2$ to the perfect full-cloak; whereas if $Γ_0$ is the closure of an open subset on a flat surface, then one would have an approximate partial-cloak within $δ$ to its perfect counterpart. The estimates derived are independent of the contents being cloaked; that is, the cloaking devices are capable of nearly cloaking an arbitrary content. Furthermore, as a significant byproduct, our argument allows the relaxation of the convexity requirement on $Γ_0$ in [28], which is critical for the Mosco convergence argument therein.
We show that the spectrum of a Schrödinger operator on $\mathbb{R}^n$, $n\ge 3$, with a periodic smooth Riemannian metric, whose conformal multiple has a product structure with one Euclidean direction, and with a periodic electric potential in $L^{n/2}_{\text{loc}}(\mathbb{R}^n)$, is purely absolutely continuous. Previously known results in the case of a general metric are obtained in [12], see also [8], under the assumption that the metric, as well as the potential, are reflection symmetric.
We show that the knowledge of the Dirichlet-to-Neumann map on the boundary of a bounded open set in $R^n$ for the perturbed polyharmonic operator $(-Δ)^m +q$ with $q\in L^{n/2m}$, $n>2m$, determines the potential $q$ in the set uniquely. In the course of the proof, we construct a special Green function for the polyharmonic operator and establish its mapping properties in suitable weighted $L^2$ and $L^p$ spaces. The $L^p$ estimates for the special Green function are derived from $L^p$ Carleman estimates with linear weights for the polyharmonic operator.
We present explicit reconstruction algorithms for fully anisotropic unknown elasticity tensors from knowledge of a finite number of internal displacement fields, with applications to transient elastography. Under certain rank-maximality assumptions satified by the strain fields, explicit algebraic reconstruction formulas are provided. A discussion ensues on how to fulfill these assumptions, describing the range of validity of the approach. We also show how the general method can be applied to more specific cases such as the transversely isotropic one.
We show that given two hyperbolic Dirichlet to Neumann maps associated to two Riemannian metrics of a Riemannian manifold with boundary which coincide near the boundary are close then the lens data of the two metrics is the same. As a consequence, we prove uniqueness of recovery a conformal factor (sound speed) locally under some conditions on the latter.
We consider how microlocal methods developed for tomographic problems can be used to detect singularities of the Lorentzian metric of the Universe using measurements of the Cosmic Microwave Background radiation. The physical model we study is mathematically rigorous but highly idealized.
We consider the linearization of the Dirichlet-to-Neumann (DN) map as a function of the potential. We show that it is injective at a real analytic potential for measurements made at an open subset of analyticity of the boundary. More generally, we relate the analyticity up to the boundary of the variations of the potential to the analyticity of the symbols of the corresponding variations of the DN-map.
We study the microlocal properties of the geodesic X-ray transform $\mathcal{X}$ on a manifold with boundary allowing the presence of conjugate points. Assuming that there are no self-intersecting geodesics and all conjugate pairs are nonsingular we show that the normal operator $\mathcal{N} = \mathcal{X}^t \circ \mathcal{X}$ can be decomposed as the sum of a pseudodifferential operator of order $-1$ and a sum of Fourier integral operators. We also apply this decomposition to prove inversion of $\mathcal{X}$ is only mildly ill-posed in dimension three or higher.
This paper concerns thermoacoustic tomography and photoacoustic tomography, two couple-physics imaging modalities that attempt to combine the high resolution of ultrasound and the high contrast capabilities of electromagnetic waves. We give sufficient conditions to recover both the sound speed of the medium being probed and the source.
In the recent articles \cite{PSU1,PSU3}, a number of tensor tomography results were proved on two-dimensional manifolds. The purpose of this paper is to extend some of these methods to manifolds of any dimension. A central concept is the surjectivity of the adjoint of the geodesic ray transform, or equivalently the existence of certain distributions that are invariant under geodesic flow. We prove that on any Anosov manifold, one can find invariant distributions with controlled first Fourier coefficients. The proof is based on subelliptic type estimates and a Pestov identity. We present an alternative construction valid on manifolds with nonpositive curvature, based on the fact that a natural Beurling transform on such manifolds turns out to be essentially a contraction. Finally, we obtain uniqueness results in tensor tomography both on simple and Anosov manifolds that improve earlier results by assuming a condition on the terminator value for a modified Jacobi equation.
We prove the local invertibility, up to potential fields, and stability of the geodesic X-ray transform on tensor fields of order 1 and 2 near a strictly convex boundary point, on manifolds with boundary of dimension n>=3. We also present an inversion formula. Under the condition that the manifold can be foliated with a continuous family of strictly convex surfaces, we prove a global result which also implies a lens rigidity result near such a metric. The class of manifolds satisfying the foliation condition includes manifolds with no focal points, and does not exclude existence of conjugate points.
Given a bounded domain $M$ in $\mathbb{R}^n$ with a conformally Euclidean metric $g=ρ\,dx^2$, in this paper we consider the inverse problem of recovering a semigeodesic neighborhood of a domain $Γ\subset \partial M$ and the conformal factor $ρ$ in the neighborhood from the travel time data (defined below) and the Cartesian coordinates of $Γ$. We develop an explicit reconstruction procedure for this problem. The key ingredient is the relation between the reconstruction and a Cauchy problem of the conformal Killing equation.
We propose a fundamentally new method for the design of metamaterial arrays, valid for any waves modeled by the Helmholtz equation, including scalar optics and acoustics. The design and analysis of these devices is based on eigenvalue and eigenfunction asymptotics of solutions to Schrödinger wave equations with harmonic and degenerate potentials. These resonators behave superdimensionally, with a higher local density of eigenvalues and greater concentration of waves than expected from the physical dimension, e.g., planar resonators function as 3- or higher-dimensional media, and bulk material as effectively of dimension 4 or higher. Applications include antennas with a high density of resonant frequencies and giant focussing, and are potentially broadband.
We consider transformation-optics based cloaking in acoustic and electromagnetic scattering. The blueprints for an ideal cloak use singular acoustic and electromagnetic materials, posing server difficulties to both theoretical analysis and practical fabrication. In order to avoid the singular structures, various regularized approximate cloaking schemes have been developed. We survey these developments in this paper. We also propose some challenging issues for further investigation.
We consider inverse problems for the coupled Einstein equations and the matter field equations on a 4-dimensional globally hyperbolic Lorentzian manifold $(M,g)$. We give a positive answer to the question: Do the active measurements, done in a neighborhood $U\subset M$ of a freely falling observed $μ=μ([s_-,s_+])$, determine the conformal structure of the spacetime in the minimal causal diamond-type set $V_g=J_g^+(μ(s_-))\cap J_g^-(μ(s_+))\subset M$ containing $μ$? More precisely, we consider the Einstein equations coupled with the scalar field equations and study the system $Ein(g)=T$, $T=T(g,ϕ)+F_1$, and $\square_gϕ-\mathcal V^\prime(ϕ)=F_2$, where the sources $F=(F_1,F_2)$ correspond to perturbations of the physical fields which we control. The sources $F$ need to be such that the fields $(g,ϕ,F)$ are solutions of this system and satisfy the conservation law $\nabla_jT^{jk}=0$. Let $(\hat g,\hat ϕ)$ be the background fields corresponding to the vanishing source $F$. We prove that the observation of the solutions $(g,ϕ)$ in the set $U$ corresponding to sufficiently small sources $F$ supported in $U$ determine $V_{\hat g}$ as a differentiable manifold and the conformal structure of the metric $\hat g$ in the domain $V_{\hat g}$. The methods developed here have potential to be applied to a large class of inverse problems for non-linear hyperbolic equations encountered e.g. in various practical imaging problems.
We study the Einstein equations coupled with the scalar field equations, $\hbox{Ein}(g)=T$, $T=T(g,ϕ)+F^1$, and $\square_gϕ^\ell-m^2ϕ^\ell= F^2$, where the sources $F=(F^1, F^2)$ correspond to perturbations of the physical fields which we control. Here $ϕ=(ϕ^\ell)_{\ell=1}^L$ and $(M,g)$ is a 4-dimensional globally hyperbolic Lorentzian manifold. The sources $F$ need to be such that the fields $(g,ϕ,F)$ satisfy the conservation law $\hbox{div}_g(T)=0$. If $(g_ε,ϕ_ε)$ solves the above equations, $\dot g=\partial_εg_ε|_{ε=0}$, $\dotϕ=ϕ_ε|_{ε=0}$, and $f=(f^1,f^2)= \partial_εF_ε|_{ε=0}$ solve the linearized Einstein equations and the linearized conservation law $$ \frac 12 \hat g^{pk}\hat \nabla_p f^1_{kj}+ \sum_{\ell=1}^L f^2_\ell \, \partial_j\hatϕ_\ell=0, $$ where $\hat g= g_ε|_{ε=0}$ and $\hat ϕ= ϕ_ε|_{ε=0}$. Then $(\hat g,\hat ϕ)$ and $f$ have the linearization stability property. Here ask the converse: If $\dot g$, $\dot ϕ$, and $f$ solve the linearized Einstein equations and the linearized conservation law, are there $F_ε=(F^1_ε,F^2_ε)$ and $(g_ε,ϕ_ε)$ depending on $ε\in [0,ε_0)$, $ε_0>0$, such that $(g_ε,ϕ_ε)$ solves the Einstein-scalar field equations and the conservation law. When $\hat g$ and $\hat ϕ$ vary enough and $L\geq 5$, we prove a microlocal version of this: When $Y\subset M$ is a 2-surface and $(y,η)\in N^*Y$, there is $f$ that is a conormal distibutions wrt. the surface $Y$ with a given principal symbol at $(y,η)$ such that $(\hat g,\hat ϕ)$ and $f$ have the linearization stability property.
This article considers inverse problems on closed Riemannian surfaces whose geodesic flow is Anosov. We prove spectral rigidity for any Anosov surface and injectivity of the geodesic ray transform on solenoidal 2-tensors. We also establish surjectivity results for the adjoint of the geodesic ray transform on solenoidal tensors. The surjectivity results are of independent interest and imply the existence of many geometric invariant distributions on the unit sphere bundle. In particular, we show that on any Anosov surface $(M,g)$, given a smooth function $f$ on $M$ there is a distribution in the Sobolev space $H^{-1}(SM)$ that is invariant under the geodesic flow and whose projection to $M$ is the given function $f$.