Solutions of the constraint equations in general relativity satisfying "hyperboloidal boundary conditions"

Andersson Lars ; Chruściel Piotr T.

GDML_Books, (1996), p.

Résumé

Abstract We prove existence of the solutions of the constraint equations satisfying "hyperboloidal boundary conditions" using the Choquet-Bruhat-Lichnerowicz-York conformal method and we analyze in detail their differentiability near the conformal boundary. We show that generic "hyperboloidal initial data" display asymptotic behaviour which is not compatible with Penrose's hypothesis of smoothness of ℐ. We also show that a large class of "non-generic" initial data satisfying Penrose smoothness conditions exists. The results are established by developing a theory of regularity up-to-boundary for a class of linear and semilinear elliptic systems of equations uniformly degenerating at the boundary.

CONTENTS 1. Introduction .............................................................................................5 2. The hyperboloidal initial value problem....................................................9  2.1. Conformal compactifications and Cauchy data....................................9  2.2. Some remarks on non-vacuum initial data sets..................................14 3. Definitions, preliminary results................................................................16  3.1. Function spaces.................................................................................16  3.2. Some embeddings..............................................................................22  3.3. Extensions of functions defined on ∂M...............................................24  3.4. Mapping properties of some integral operators..................................26 4. Regularity at the boundary: the linear problem......................................30  4.1. Tangential regularity below the threshold...........................................30  4.2. Boundary regularity for a class of second order systems...................39 5. Non-linear equations with polyhomogeneous coefficients......................52  5.1. Polyhomogeneity of solutions of some fully non-linear equations.......52 6. The vector constraint equation...............................................................57  6.1. Introductory remarks..........................................................................57  6.2. (Non-weighted) Hölder spaces on the compactified manifold.............58  6.3. Weighted Sobolev spaces..................................................................66 7. The Lichnerowicz equation.....................................................................74  7.1. Introductory remarks..........................................................................74  7.2. The linearized equation.....................................................................76  7.3. Existence of solutions of the non-linear problem................................79  7.4. Regularity at the boundary of the solutions........................................81 Appendix A. Genericity of log-terms...........................................................87  A.1. The vector constraint equation..........................................................88  A.2. The coupled system...........................................................................91 Appendix B.................................................................................................94  B.1. "Almost Gaussian" coordinates..........................................................94 References................................................................................................96 Index of symbols........................................................................................99 Index of terms..........................................................................................100

1991 Mathematics Subject Classification: Primary 83C05, 35Q75, 53C21; Secondary 58G20, 35J60, 35J65.

Détail

Zbl 0873.35101

EUDML-ID : urn:eudml:doc:275822