scholarly journals The Cauchy Data and the Scattering Relation

2004 ◽  
pp. 263-287 ◽  
Author(s):  
Gunther Uhlmann
Keyword(s):  
Cauchy Data

scholarly journals Alternative Approach to Infinity

1974 ◽  
Vol 64 ◽  
pp. 99-99
Author(s):  
Peter G. Bergmann
Keyword(s):  
Large Class ◽  
Riemannian Manifolds ◽  
Quotient Space ◽  
Three Dimensional ◽  
Cauchy Data ◽  
Space Time ◽  
Equivalence Classes ◽  
Null Infinity ◽  
Alternative Approach ◽  
Dimensional Domain

Following Penrose's construction of space-time infinity by means of a conformal construction, in which null-infinity is a three-dimensional domain, whereas time- and space-infinities are points, Geroch has recently endowed space-infinity with a somewhat richer structure. An approach that might work with a large class of pseudo-Riemannian manifolds is to induce a topology on the set of all geodesics (whether complete or incomplete) by subjecting their Cauchy data to (small) displacements in space-time and Lorentz rotations, and to group the geodesics all of whose neighborhoods intersect into equivalence classes. The quotient space of geodesics over equivalence classes is to represent infinity. In the case of Minkowski, null-infinity has the usual structure, but I0, I+, and I- each become three-dimensional as well.

scholarly journals Imposition of Cauchy data to the Teukolsky equation. III. The rotating case

1998 ◽  
Vol 58 (8) ◽  
Author(s):  
Manuela Campanelli ◽  
Carlos O. Lousto ◽  
John Baker ◽  
Gaurav Khanna ◽  
Jorge Pullin
Keyword(s):  
Cauchy Data ◽  
Teukolsky Equation

scholarly journals Radial solutions to the Cauchy problem for □1+3U = 0 as limits of exterior solutions

2016 ◽  
Vol 13 (04) ◽  
pp. 833-860
Author(s):  
Helge Kristian Jenssen ◽  
Charis Tsikkou
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Three Dimensional ◽  
Initial Boundary ◽  
Cauchy Data ◽  
Linear Wave ◽  
Radial Solutions ◽  
Linear Wave Equation ◽  
Model Situation ◽  
Neumann Solutions

We consider the strategy of realizing the solution of a Cauchy problem (CP) with radial data as a limit of radial solutions to initial-boundary value problems posed on the exterior of vanishing balls centered at the origin. The goal is to gauge the effectiveness of this approach in a simple, concrete setting: the three-dimensional (3d), linear wave equation [Formula: see text] with radial Cauchy data [Formula: see text], [Formula: see text]. We are primarily interested in this as a model situation for other, possibly nonlinear, equations where neither formulae nor abstract existence results are available for the radial symmetric CP. In treating the 3d wave equation, we therefore insist on robust arguments based on energy methods and strong convergence. (In particular, this work does not address what can be established via solution formulae.) Our findings for the 3d wave equation show that while one can obtain existence of radial Cauchy solutions via exterior solutions, one should not expect such results to be optimal. The standard existence result for the linear wave equation guarantees a unique solution in [Formula: see text] whenever [Formula: see text]. However, within the constrained framework outlined above, we obtain strictly lower regularity for solutions obtained as limits of exterior solutions. We also show that external Neumann solutions yield better regularity than external Dirichlet solutions. Specifically, for Cauchy data in [Formula: see text], we obtain [Formula: see text]-solutions via exterior Neumann solutions, and only [Formula: see text]-solutions via exterior Dirichlet solutions.

scholarly journals BOUNDARY CONTROL PROBLEM FOR ONE DEGENERATE HIBERBOLIC EQUATION

2019 ◽  
pp. 19-27
Author(s):  
А.Х. Аттаев
Keyword(s):  
Control Problem ◽  
Boundary Control ◽  
Sufficient Conditions ◽  
Analytical Form ◽  
Cauchy Data ◽  
Boundary Control Problem ◽  
Order Hyperbolic Equation ◽  
Boundary Controls ◽  
Necessary And Sufficient ◽  
Second Order Hyperbolic Equation

В работе изучается задача граничного управления для вырождающегося гиперболического уравнения второго порядка. Установлены необходимые и достаточные условия управляемости данными Коши за минимальный промежуток времени. Граничные управления предъявлены в явном аналитическом виде. The paper studies the boundary control problem for a degenerate second-order hyperbolic equation. Necessary and sufficient conditions are established for minimal time controllability over Cauchy data. Boundary controls are presented in an explicit analytical form.

HYPERBOLIC DOMAINS OF DETERMINACY AND HAMILTON–JACOBI EQUATIONS

2005 ◽  
Vol 02 (03) ◽  
pp. 713-744 ◽  
Author(s):  
JEAN-LUC JOLY ◽  
GUY MÉTIVIER ◽  
JEFFREY RAUCH
Keyword(s):  
Differential Operators ◽  
Cauchy Data ◽  
Influence Curves ◽  
Partial Differential Operators ◽  
Spacelike Hypersurfaces ◽  
The Cauchy Problem ◽  
Local Convex ◽  
Jacobi Equations ◽  
Set Of Points ◽  
Linear Hyperbolic System

If L(t, x, ∂t, ∂x) is a linear hyperbolic system of partial differential operators for which local uniqueness in the Cauchy problem at spacelike hypersurfaces is known, we find nearly optimal domains of determinacy of open sets Ω0⊂ {t = 0}. The frozen constant coefficient operators [Formula: see text] determine local convex propagation cones, [Formula: see text]. Influence curves are curves whose tangent always lies in these cones. We prove that the set of points Ω which cannot be reached by influence curves beginning in the exterior of Ω0is a domain of determinacy in the sense that solutions of Lu = 0 whose Cauchy data vanish in Ω0must vanish in Ω. We prove that Ω is swept out by continuous spacelike deformations of Ω0and is also the set described by maximal solutions of a natural Hamilton–Jacobi equation (HJE). The HJE provides a method for computing approximate domains and is also the bridge from the raylike description using influence curves to that depending on spacelike deformations. The deformations are obtained from level surfaces of mollified solutions of HJEs.

Dynamical black holes

2020 ◽  
pp. 312-336
Author(s):  
Piotr T. Chruściel
Keyword(s):  
Black Holes ◽  
Black Hole ◽  
Cauchy Data ◽  
Einstein Equations ◽  
Black Hole Spacetimes ◽  
Alternative Approaches ◽  
Definition Of ◽  
Meaningful Definition ◽  
Stability Results ◽  
Black Hole Solutions

In this chapter we review what is known about dynamical black hole-solutions of Einstein equations. We discuss the Robinson–Trautman black holes, with or without a cosmological constant. We review the Cauchy-data approach to the construction of black-hole spacetimes. We propose some alternative approaches to a meaningful definition of black hole in a dynamical spacetime, and we review the nonlinear stability results for black-hole solutions of vacuum Einstein equations.

A Moving Pseudo-Boundary MFS for Three-Dimensional Void Detection

2013 ◽  
Vol 5 (04) ◽  
pp. 510-527 ◽  
Author(s):  
Andreas Karageorghis ◽  
Daniel Lesnic ◽  
Liviu Marin
Keyword(s):  
Three Dimensional ◽  
Nonlinear Least Squares ◽  
Fundamental Solutions ◽  
Cauchy Data ◽  
Three Dimensions ◽  
Method Of Fundamental Solutions ◽  
Void Detection ◽  
Spherical Parametrization ◽  
Least Squares Minimization

AbstractWe propose a new moving pseudo-boundary method of fundamental solutions (MFS) for the determination of the boundary of a three-dimensional void (rigid inclusion or cavity) within a conducting homogeneous host medium from overdetermined Cauchy data on the accessible exterior boundary. The algorithm for imaging the interior of the medium also makes use of radial spherical parametrization of the unknown star-shaped void and its centre in three dimensions. We also include the contraction and dilation factors in selecting the fictitious surfaces where the MFS sources are to be positioned in the set of unknowns in the resulting regularized nonlinear least-squares minimization. The feasibility of this new method is illustrated in several numerical examples.

Hyperbolic Mixed Problems for Harmonic Tensors

1957 ◽  
Vol 9 ◽  
pp. 161-179 ◽  
Author(s):  
G. F. D. Duff
Keyword(s):  
Boundary Condition ◽  
Cauchy Problem ◽  
Principal Part ◽  
Wave Equations ◽  
Initial Conditions ◽  
Boundary Surface ◽  
Cauchy Data ◽  
Riemann Space ◽  
Initial Value ◽  
Mixed Problems

This paper may be regarded as a sequel to (1), where the initial value or Cauchy problem for harmonic tensors on a normal hyperbolic Riemann space was treated. The mixed problems to be studied here involve boundary conditions on a timelike boundary surface in addition to the Cauchy data on a spacelike initial manifold. The components of a harmonic tensor satisfy a system of wave equations with similar principal part, and we assign two initial conditions and one boundary condition for each component.

Inverse problem for one-dimensional wave equation with matrix potential

2019 ◽  
Vol 27 (2) ◽  
pp. 217-223 ◽  
Author(s):  
Ammar Khanfer ◽  
Alexander Bukhgeim
Keyword(s):  
Inverse Problem ◽  
Wave Equation ◽  
Uniqueness Theorem ◽  
Cauchy Data ◽  
Global Uniqueness ◽  
One Dimensional ◽  
Matrix Potential ◽  
The Matrix ◽  
Real Matrices ◽  
Dimensional Wave

AbstractWe prove a global uniqueness theorem of reconstruction of a matrix-potential {a(x,t)} of one-dimensional wave equation {\square u+au=0}, {x>0,t>0}, {\square=\partial_{t}^{2}-\partial_{x}^{2}} with zero Cauchy data for {t=0} and given Cauchy data for {x=0}, {u(0,t)=0}, {u_{x}(0,t)=g(t)}. Here {u,a,f}, and g are {n\times n} smooth real matrices, {\det(f(0))\neq 0}, and the matrix {\partial_{t}a} is known.

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