By Hans Ringstrom
ISBN-10: 3037190531
ISBN-13: 9783037190531
Advent define half I. historical past from the idea of partial differential equations: sensible research The Fourier rework Sobolev areas Sobolev embedding Symmetric hyperbolic platforms Linear wave equations neighborhood lifestyles, non-linear wave equations half II. history in geometry, international hyperbolicity and area of expertise: easy Lorentz geometry Characterizations of world hyperbolicity area of expertise of strategies to linear wave equations half III. basic relativity: The constraint equations neighborhood lifestyles Cauchy balance lifestyles of a maximal globally hyperbolic improvement half IV. Pathologies, powerful cosmic censorship: Preliminaries consistent suggest curvature preliminary information Einstein's vacuum equations Closed universe recollapse Asymptotic behaviour LRS Bianchi category A strategies life of extensions lifestyles of inequivalent extensions Appendices Bibliography Index
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Extra info for The Cauchy Problem in General Relativity (ESI Lectures in Mathematics and Physics)
Example text
Rn /, something we shall discuss in detail in the following chapter. 1) and to related equations, Sobolev spaces appear naturally. However, one is often interested in obtaining solutions that are k times continuously differentiable or even smooth. In that context, it is of interest to ask if it is possible to relate Sobolev space regularity to classical differentiability. 1. Rn /, then u is l times continuously differentiable assuming k > l C n=2, cf. 5. x/j D ˇˇ e u. 1 C j j / ju. 1 C j j / ju.
18) for correctly set up initial data, and this is a problem which essentially reduces to the type of non-linear wave equations we have already considered in these notes. It is of interest to note that this argument is essentially the same as the original one by Choquet-Bruhat [39]; the only difference is that she used the gauge source function F D 0. This choice is referred to as wave coordinates and is sometimes useful even for global problems, cf. [57], [58]. It is also of interest to note that the same kind of problem that arose above appears in the context of the Ricci flow and the context of Cheeger– Gromov convergence/collapse theory.
10) Proof. Rn /. Rn /, they then follow in general. We shall need the following lemma. 17. Rn ; C/ is such that all its derivatives are bounded. 18. Rn /. Proof. k/ . Rn / W kvk. k/ Ä 1g. Rn / to be such that v. 1 C j j2 /k u. k/1 ; 42 5. k/ ¤ 0. Then kvk. Z k/ D 1. k/ : n That this equality holds for u D 0 is quite clear as well. k/ k N vk. k/ kvk. 11) in order to obtain the first inequality and the fact that the result holds for k non-negative in order to obtain the second inequality. Taking the supremum over v 2 Ak , we obtain the statement of the lemma.