Solutions of the constraint equations in general relativity satisfying “hyperboloidal boundary conditions”. (English) Zbl 0873.35101
Authors’ summary: 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 \(\mathcal S\). 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.
Reviewer: A.D.Osborne (Keele)
MSC:
35Q75 | PDEs in connection with relativity and gravitational theory |
83C05 | Einstein’s equations (general structure, canonical formalism, Cauchy problems) |
53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |
35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |
58J32 | Boundary value problems on manifolds |
35J60 | Nonlinear elliptic equations |
35J65 | Nonlinear boundary value problems for linear elliptic equations |