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Rough solutions of the Einstein constraints on closed manifolds without nearCMC conditions. Submitted for publication. Available as arXiv:0712.0798v1 [grqc
"... ABSTRACT. We consider the conformal decomposition of Einstein’s constraint equations introduced by Lichnerowicz and York, on a closed manifold. We establish existence of nonCMC weak solutions using a combination of a priori estimates for the individual Hamiltonian and momentum constraints, barrier ..."
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Cited by 30 (14 self)
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ABSTRACT. We consider the conformal decomposition of Einstein’s constraint equations introduced by Lichnerowicz and York, on a closed manifold. We establish existence of nonCMC weak solutions using a combination of a priori estimates for the individual Hamiltonian and momentum constraints, barrier constructions and fixedpoint techniques for the Hamiltonian constraint, RieszSchauder theory for the momentum constraint, together with a topological fixedpoint argument for the coupled system. Although we present general existence results for nonCMC weak solutions when the rescaled background metric is in any of the three Yamabe classes, an important new feature of the results we present for the positive Yamabe class is the absence of the nearCMC assumption, if the freely specifiable part of the data given by the tracelesstransverse part of the rescaled extrinsic curvature and the matter fields are sufficiently small, and if the energy density of matter is not identically zero. In this case, the mean extrinsic curvature can be taken to be an arbitrary smooth function without restrictions on the size of its spatial derivatives, so that it can be arbitrarily far from constant, giving what is apparently the first existence results for nonCMC solutions without the
Murchadha, NonCMC conformal data sets which do not produce solutions of the Einstein constraint equations, Classical Quantum Gravity 21
, 2004
"... The conformal formulation provides a method for constructing and parametrizing solutions of the Einstein constraint equations by mapping freely chosen sets of conformal data to solutions, provided a certain set of coupled, elliptic determined PDEs (whose expression depends on the chosen conformal da ..."
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Cited by 19 (3 self)
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The conformal formulation provides a method for constructing and parametrizing solutions of the Einstein constraint equations by mapping freely chosen sets of conformal data to solutions, provided a certain set of coupled, elliptic determined PDEs (whose expression depends on the chosen conformal data) admit a unique solution. For constant mean curvature (CMC) data, it is known in almost all cases which sets of conformal data allow these PDEs to have solutions, and which do not. For non CMC data, much less is known. Here we exhibit the first class of non CMC data for which we can prove that no solutions exist. 1
2005 Generalized Korn’s inequality and conformal Killing vectors arXiv:grqc/0505022
"... Korn’s inequality plays an important role in linear elasticity theory. This inequality bounds the norm of the derivatives of the displacement vector by the norm of the linearized strain tensor. The kernel of the linearized strain tensor are the infinitesimal rigidbody translations and rotations (Ki ..."
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Cited by 17 (0 self)
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Korn’s inequality plays an important role in linear elasticity theory. This inequality bounds the norm of the derivatives of the displacement vector by the norm of the linearized strain tensor. The kernel of the linearized strain tensor are the infinitesimal rigidbody translations and rotations (Killing vectors). We generalize this inequality by replacing the linearized strain tensor by its trace free part. That is, we obtain a stronger inequality in which the kernel of the relevant operator are the conformal Killing vectors. The new inequality has applications in General Relativity. 1
Rough solutions of the Einstein constraint equations on closed manifolds without nearCMC conditions
 36 [ICBM92] [IM96] [IOM04] [Is79] [Is95
"... Abstract. We consider the conformal decomposition of Einstein’s constraint equations introduced by Lichnerowicz and York, on a compact manifold with boundary. We first develop some technical results for the momentum constraint operator under weak assumptions on the problem data, including generalize ..."
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Cited by 13 (2 self)
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Abstract. We consider the conformal decomposition of Einstein’s constraint equations introduced by Lichnerowicz and York, on a compact manifold with boundary. We first develop some technical results for the momentum constraint operator under weak assumptions on the problem data, including generalized Korn inequalities on manifolds with boundary not currently in the literature. We then consider the Hamiltonian constraint, and using order relations on appropriate Banach spaces we derive weak solution generalizations of known sub and supersolutions (barriers). We also establish some related a priori L ∞bounds on any W 1,2solution. The barriers are combined with variational methods to establish existence of solutions to the Hamiltonian constraint in L ∞ ∩ W 1,2. The result is established under weak assumptions on the problem data, and for scalar curvature R having any sign; nonnegative R requires additional positivity assumptions either on the matter energy density or on the tracefree divergencefree part of the extrinsic curvature. Although the formulation is different, the result can be viewed as extending the regularity of the recent result of Maxwell on “rough ” CMC solutions in W k,2 for k> 3/2 down to L ∞ ∩ W 1,2. The results for the individual constraints are then combined to establish existence of nonCMC solutions in W 1,p, p> 3 for the threemetric and in L q, q = 6p/(3 + p) for the extrinsic curvature. The result is obtained using fixedpoint iteration and compactness arguments directly, rather than by building a contraction map. The nonCMC result can be viewed as a type of extension of the regularity of the 1996 nonCMC result of Isenberg and Moncrief down to W 1,p for p> 3, and extending their result to R having any sign. Similarly, the result can also be viewed as type of extension of the recent work of Maxwell on rough solutions from the CMC case to the nonCMC case. Although our presentation is for 3manifolds, the results also hold in higher dimensions with minor adjustments. The results should also extend to other cases such as closed and (fully or partially) open manifolds without substantial difficulty.
BLACK HOLE INITIAL DATA WITH A HORIZON OF PRESCRIBED GEOMETRY
, 710
"... Abstract. The purpose of this work is to construct asymptotically flat, time symmetric initial data with an apparent horizon of prescribed intrinsic geometry. To do this, we use the parabolic partial differential equation for prescribing scalar curvature. In this equation the horizon geometry is con ..."
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Abstract. The purpose of this work is to construct asymptotically flat, time symmetric initial data with an apparent horizon of prescribed intrinsic geometry. To do this, we use the parabolic partial differential equation for prescribing scalar curvature. In this equation the horizon geometry is contained within the freely specifiable part of the metric. This contrasts with the conformal method in which the geometry of the horizon can only be specified up to a conformal factor. 1.
Generating initial data in general relativity using adaptive finite element methods
, 2008
"... The conformal formulation of the Einstein constraint equations is first reviewed, and we then consider the design, analysis, and implementation of adaptive multilevel finite elementtype numerical methods for the resulting coupled nonlinear elliptic system. We derive weak formulations of the coupl ..."
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Cited by 4 (0 self)
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The conformal formulation of the Einstein constraint equations is first reviewed, and we then consider the design, analysis, and implementation of adaptive multilevel finite elementtype numerical methods for the resulting coupled nonlinear elliptic system. We derive weak formulations of the coupled constraints, and review some new developments in the solution theory for the constraints in the cases of constant mean extrinsic curvature (CMC) data, nearCMC data, and arbitrarily prescribed mean extrinsic curvature data. We then outline some recent results on a priori and a posteriori error estimates for a broad class of Galerkintype approximation methods for this system which includes techniques such as finite element, wavelet, and spectral methods. We then use these estimates to construct an adaptive finite element method (AFEM) for solving this system numerically, and outline some new convergence and optimality results. We then describe in some detail an implementation of the methods using the FETK software package, which is an adaptive multilevel finite element code designed to solve nonlinear elliptic and parabolic systems on Riemannian
The Yamabe invariant for axially symmetric two Kerr black holes initial data
"... An explicit 3dimensional Riemannian metric is constructed which can be interpreted as the (conformal) sum of two Kerr black holes with aligned angular momentum. When the separation distance between them is large we prove that this metric has positive Ricci scalar and hence positive Yamabe invariant ..."
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An explicit 3dimensional Riemannian metric is constructed which can be interpreted as the (conformal) sum of two Kerr black holes with aligned angular momentum. When the separation distance between them is large we prove that this metric has positive Ricci scalar and hence positive Yamabe invariant. This metric can be used to construct axially symmetric initial data for two Kerr black holes with large angular momentum. 1
An Application Of A Generalized Korn Inequality To Energies Studied In General Relativity Giving The Smoothness Of Minimizers
, 2009
"... We combine a Korn type inequality with Widman’s hole filling technique to prove the interior regularity of minimizers for energies occurring in General Relativity. In addition we provide a new variant of this Korn type inequality valid for the nonquadratic case. ..."
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Cited by 1 (1 self)
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We combine a Korn type inequality with Widman’s hole filling technique to prove the interior regularity of minimizers for energies occurring in General Relativity. In addition we provide a new variant of this Korn type inequality valid for the nonquadratic case.
Elliptic systems
, 2004
"... Summary. In this article I will review some basic results on elliptic boundary value problems with applications to General Relativity. 1 ..."
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Summary. In this article I will review some basic results on elliptic boundary value problems with applications to General Relativity. 1