Results 1  10
of
12
The wave equation on asymptotically de Sitterlike spaces
, 2007
"... Abstract. In this paper we obtain the asymptotic behavior of solutions of the KleinGordon equation on Lorentzian manifolds (X ◦ , g) which are de Sitterlike at infinity. Such manifolds are Lorentzian analogues of the socalled Riemannian conformally compact (or asymptotically hyperbolic) spaces. Un ..."
Abstract

Cited by 10 (6 self)
 Add to MetaCart
Abstract. In this paper we obtain the asymptotic behavior of solutions of the KleinGordon equation on Lorentzian manifolds (X ◦ , g) which are de Sitterlike at infinity. Such manifolds are Lorentzian analogues of the socalled Riemannian conformally compact (or asymptotically hyperbolic) spaces. Under global assumptions on the (null)bicharacteristic flow, namely that the boundary of the compactification X is a union of two disjoint manifolds, Y±, and each bicharacteristic converges to one of these two manifolds as the parameter along the bicharacteristic goes to +∞, and to the other manifold as the parameter goes to −∞, we also define the scattering operator, and show that it is a Fourier integral operator associated to the bicharacteristic flow from Y+ to Y−. 1.
Radiation fields, scattering and inverse scattering on asymptotically hyperbolic manifolds
, 2004
"... ..."
ASYMPTOTICS OF SOLUTIONS OF THE WAVE EQUATION ON DE SITTERSCHWARZSCHILD SPACE
"... Abstract. Solutions to the wave equation on de SitterSchwarzschild space with smooth initial data on a Cauchy surface are shown to decay exponentially to a constant at temporal in nity, with corresponding uniform decay on the appropriately compacti ed space. 1. ..."
Abstract

Cited by 5 (3 self)
 Add to MetaCart
Abstract. Solutions to the wave equation on de SitterSchwarzschild space with smooth initial data on a Cauchy surface are shown to decay exponentially to a constant at temporal in nity, with corresponding uniform decay on the appropriately compacti ed space. 1.
A SUPPORT THEOREM FOR THE RADIATION FIELDS ON ASYMPTOTICALLY EUCLIDEAN MANIFOLDS
, 709
"... We prove a support theorem for the radiation fields on asymptotically Euclidean manifolds with metrics which are warped products near infinity. It generalizes to this setting the well known support theorem for the Radon transform in R n. The main reason we are interested in proving such a theorem is ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
We prove a support theorem for the radiation fields on asymptotically Euclidean manifolds with metrics which are warped products near infinity. It generalizes to this setting the well known support theorem for the Radon transform in R n. The main reason we are interested in proving such a theorem is the possible application to the problem of reconstructing an asymptotically Euclidean manifold from the scattering
Radiation fields for semilinear wave equations
 In preparation
, 2012
"... Abstract. We define the radiation fields of solutions to critical semilinear wave equations in R 3 and use them to define the scattering operator. We also prove a support theorem for the radiation fields with radial initial data. This extends the well known support theorem for the Radon transform to ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Abstract. We define the radiation fields of solutions to critical semilinear wave equations in R 3 and use them to define the scattering operator. We also prove a support theorem for the radiation fields with radial initial data. This extends the well known support theorem for the Radon transform to this setting and can also be interpreted as a PaleyWiener theorem for the distorted nonlinear Fourier transform of radial functions. 1.
MORAWETZ ESTIMATES FOR THE WAVE EQUATION AT LOW FREQUENCY
"... Abstract. We consider Morawetz estimates for weighted energy decay of solutions to the wave equation on scattering manifolds (i.e., those with large conic ends). We show that a Morawetz estimate persists for solutions that are localized at low frequencies, independent of the geometry of the compact ..."
Abstract
 Add to MetaCart
Abstract. We consider Morawetz estimates for weighted energy decay of solutions to the wave equation on scattering manifolds (i.e., those with large conic ends). We show that a Morawetz estimate persists for solutions that are localized at low frequencies, independent of the geometry of the compact part of the manifold. We further prove a new type of Morawetz estimate in this context, with both hypotheses and conclusion localized inside the forward light cone. This result allows us to gain a 1/2 power of t decay relative to what would be dictated by energy estimates, in a small part of spacetime. 1.
On Lars Hörmander’s remark on the characteristic Cauchy problem
, 2005
"... We extend the results of a work by L. Hörmander [9] concerning the resolution of the characteristic Cauchy problem for second order wave equations with regular first order potentials. The geometrical background of this work was a spatially compact spacetime with smooth metric. The initial data surfa ..."
Abstract
 Add to MetaCart
We extend the results of a work by L. Hörmander [9] concerning the resolution of the characteristic Cauchy problem for second order wave equations with regular first order potentials. The geometrical background of this work was a spatially compact spacetime with smooth metric. The initial data surface was spacelike or null at each point and merely Lipschitz. We lower the regularity hypotheses on the metric and potential and obtain similar results. The Cauchy problem for a spacelike initial data surface is solved for a Lipschitz metric and coefficients of the first order potential that are L ∞ loc, with the same finite energy solution space as in the smooth case. We also solve the fully characteristic Cauchy problem with very slightly more regular metric and potential, namely a C1 metric and a potential with continuous first order terms and locally L ∞ coefficients for the terms of order 0. Résumé Nous étendons des résultats dus à L. Hörmander [9] concernant la résolution du problème de Cauchy caractéristique pour des équations d’onde du second ordre avec un potentiel régulier du premier ordre. Le cadre géométrique de [9] était un espacetemps spatialement compact avec une métrique régulière. L’hypersurface sur laquelle les données initiales sont fixées était spatiale ou caractéristique en chaque point et simplement de régularité Lipschitz. Nous affaiblissons les hypothèses de régularité sur la métrique et le potentiel et nous obtenons des résultats analogues. Le problème de Cauchy pour une hypersurface spatiale est résolu dans le cas d’une métrique Lipschitz et pour un potentiel dont les coefficients sont localement L ∞ , avec le même espace de solutions que dans le cas régulier. Nous résolvons également le problème de Cauchy totalement caractéristique dans un cadre très légèrement plus régulier: une métrique C 1 et un potentiel dont les coefficients des termes du premier ordre sont continus et ceux des termes d’ordre 0 sont localement L ∞.
On Lars Hörmander’s remark on the characteristic Cauchy problem
, 2005
"... We extend the results of a work by L. Hörmander [9] concerning the resolution of the characteristic Cauchy problem for second order wave equations with regular first order potentials. The geometrical background of this work was a spatially compact spacetime with smooth metric. The initial data surfa ..."
Abstract
 Add to MetaCart
We extend the results of a work by L. Hörmander [9] concerning the resolution of the characteristic Cauchy problem for second order wave equations with regular first order potentials. The geometrical background of this work was a spatially compact spacetime with smooth metric. The initial data surface was spacelike or null at each point and merely Lipschitz. We lower the regularity hypotheses on the metric and potential and obtain similar results. The Cauchy problem for a spacelike initial data surface is solved for a Lipschitz metric and coefficients of the first order potential that are L ∞ loc, with the same finite energy solution space as in the smooth case. We also solve the fully characteristic Cauchy problem with very slightly more regular metric and potential: essentially, a C1 metric and a potential with continuous coefficients of the first order terms and locally L ∞ coefficients for the terms of order 0. Résumé Nous étendons des résultats dus à L. Hörmander [9] concernant la résolution du problème de Cauchy caractéristique pour des équations d’onde du second ordre avec un potentiel régulier du premier ordre. Le cadre géométrique de [9] était un espacetemps spatialement compact avec
Some global aspects of linear wave equations
"... This paper surveys a few aspects of the global theory of wave equations. This material is structured around the contents of a minicourse given by the second author during the CMI/ETH Summer School on evolution equations during the Summer of 2008. ..."
Abstract
 Add to MetaCart
This paper surveys a few aspects of the global theory of wave equations. This material is structured around the contents of a minicourse given by the second author during the CMI/ETH Summer School on evolution equations during the Summer of 2008.
EQUIPARTITION OF ENERGY IN GEOMETRIC SCATTERING THEORY
"... Abstract. In this note, we use an elementary argument to show that the existence and unitarity of radiation fields implies asymptotic partition of energy for the corresponding wave equation. This argument establishes the equipartition of energy for the wave equation on scattering manifolds, asymptot ..."
Abstract
 Add to MetaCart
Abstract. In this note, we use an elementary argument to show that the existence and unitarity of radiation fields implies asymptotic partition of energy for the corresponding wave equation. This argument establishes the equipartition of energy for the wave equation on scattering manifolds, asymptotically hyperbolic manifolds, asymptotically complex hyperbolic manifolds, and the Schwarzschild spacetime. It also establishes equipartition of energy for the energycritical semilinear wave equation on R3. 1.