Displaying similar documents to “Regularity and geometric properties of solutions of the Einstein-Vacuum equations”

Around the bounded L 2 curvature conjecture in general relativity

Sergiu Klainerman, Igor Rodnianski, Jeremie Szeftel (2008)

Journées Équations aux dérivées partielles

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We report on recent progress obtained on the construction and control of a parametrix to the homogeneous wave equation □ g φ = 0 , where ≫ is a rough metric satisfying the Einstein vacuum equations. Controlling such a parametrix as well as its error term when one only assumes L 2 bounds on the curvature tensor R of ≫ is a major step towards the proof of the bounded L 2 curvature conjecture.

Global existence for a quasilinear wave equation outside of star-shaped domains

Hart F. Smith (2001)

Journées équations aux dérivées partielles

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This talk describes joint work with Chris Sogge and Markus Keel, in which we establish a global existence theorem for null-type quasilinear wave equations in three space dimensions, where we impose Dirichlet conditions on a smooth, compact star-shaped obstacle 𝒦 ⊂ ℝ 3 . The key tool, following Christodoulou [1], is to use the Penrose compactification of Minkowski space. In the case under consideration, this reduces matters to a local existence theorem for a singular obstacle problem. Full details...

On Lars Hörmander’s remark on the characteristic Cauchy problem

Jean-Philippe Nicolas (2006)

Annales de l’institut Fourier

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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...

Dispersive and Strichartz estimates for the wave equation in domains with boundary

Oana Ivanovici (2010)

Journées Équations aux dérivées partielles

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In this note we consider a strictly convex domain Ω ⊂ ℝ d of dimension d ≥ 2 with smooth boundary ∂ Ω ≠ ∅ and we describe the dispersive and Strichartz estimates for the wave equation with the Dirichlet boundary condition. We obtain counterexamples to the optimal Strichartz estimates of the flat case; we also discuss the some results concerning the dispersive estimates.