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On existence and behaviour of asymptotically flat solutions to the stationary Einstein equations

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Abstract

We show existence and uniqueness of asymptotically flat solutions to the stationary Einstein equations inS=ℝ3−B r , whereB r is a ball of radiousr>0, when a small enough continuous complex function û on ∂S is given. Regularity and decay estimates imply that these solutions are analytic in the interior ofS and also at infinity, when suitably conformally rescaled.

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References

  1. Geroch, R.: J. Math. Phys.12, 918 (1971)

    Google Scholar 

  2. Hansen, R.O.: J. Math. Phys.15, 46 (1974)

    Google Scholar 

  3. Kundu, P.: J. Math. Phys.22, 2006 (1981)

    Google Scholar 

  4. H. Müller zum Hagen: Proc. Camb. Phil. Soc.67, 415 (1970); Proc. Camb. Phil. Soc.68, 199 (1970)

    Google Scholar 

  5. Morrey, C.B.: Contributions to theory of partial differential equations. Annals of Mathematics Studies, No. 33. Princeton, NJ: Princeton U.P., 1954; Am. J. Math.80, 198 (1958)

    Google Scholar 

  6. Beig, R.: Gen. Rel. Grav.12, 439 (1980)

    Google Scholar 

  7. Beig, R., Simon, W.: Gen. Rel. Grav.12, 1003 (1980)

    Google Scholar 

  8. Geroch, R.: J. Math. Phys.11, 2580 (1970)

    Google Scholar 

  9. Beig, R., Simon, W.: J. Math. Phys.24, 1163–1171 (1983)

    Google Scholar 

  10. Beig, R., Simon, W.: Proc. R. Soc. Lond. A376, 333 (1981)

    Google Scholar 

  11. Beig, R., Simon, W.: Commun. Math. Phys.78, 75–82 (1980)

    Google Scholar 

  12. Choquet-Bruhat, Y., York, J.W.: The Cauchy problem, general relativity and gravitation. Held, A. (ed.). London: Plenum Press 1980

    Google Scholar 

  13. Wald, R.: General relativity. Chicago, IL: The Univ. of Chicago Press 1984

    Google Scholar 

  14. Reula, O.: A configuration space for quantum gravity and solutions to the Euclidean Einstein Equations in a slab region. Class. Quant. Grav. (to appear)

  15. Choquet-Bruhat, Y., Christodoulou, D.: Acta Math.146, 129 (1981)

    Google Scholar 

  16. Choquet-Bruhat, Y., Marsden, J.E.: Commun. Math. Phys.51, 283 (1976)

    Google Scholar 

  17. Bartnik, R.: Note the different convention for the functional spaces indices. Commun. Pure Appl. Math.39, 661 (1986)

    Google Scholar 

  18. Abraham, R., Marsden, J., Ratiu, T.: Manifolds, tensor analysis, and applications. Reading, MA: Addison-Wesley 1983

    Google Scholar 

  19. Cartan, H.: Differential calculus. London: Kershaw Publishing 1971

    Google Scholar 

  20. Hörmander, L.: The analysis of linear partial operators III. Chap. XX. Berlin, Heidelberg, New York: Springer 1985

    Google Scholar 

  21. Hörmander, L.: Linear partial differential operators. Fourth Printing. Berlin, Heidelberg, New York: Springer 1976

    Google Scholar 

  22. Lopatinskii, Ya.B.: Ukrain. Mat. Ž.5, 123–151 (1953)

    Google Scholar 

  23. Sapiro, Ja.Z.: Isv. AN, ser. Matem.17, 539–562 (1953)

    Google Scholar 

  24. McOwen, R.C.: Commun. Pure Appl. Math.32, 783 (1979)

    Google Scholar 

  25. Lockhart, R.: Trans. Am. Math. Soc.301, 1–35 (1987)

    Google Scholar 

  26. Lockhart, R., McOwen, R.C.: Ann. Scuola Norm. Sup. Pisa22 (1985)

  27. Cantor, M.: J. Diff. Eqns.34, 102–113 (1979)

    Google Scholar 

  28. Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Berlin, Heidelberg, New York: Springer 1977

    Google Scholar 

  29. Meyers, N.: J. Math. Mechs.12, 247 (1963)

    Google Scholar 

  30. Simon, W.: Thesis, Univ. Wien (1980)

  31. Martin, A.: Private communication

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  1. Max-Planck-Institut für Physik und Astrophysik, Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Strasse 1, D-8046, Garching bei München, Federal Republic of Germany

    Oscar Reula

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  1. Oscar Reula
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Communicated by S.-T. Yau

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Reula, O. On existence and behaviour of asymptotically flat solutions to the stationary Einstein equations. Commun.Math. Phys. 122, 615–624 (1989). https://doi.org/10.1007/BF01256497

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  • DOI: https://doi.org/10.1007/BF01256497

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