Skip to main content
SpringerLink
Log in

Elliptic equations in weighted Besov spaces on asymptotically flat Riemannian manifolds

  • Published:
Manuscripta Mathematica Aims and scope Submit manuscript

Abstract

This paper deals with the applications of weighted Besov spaces to elliptic equations on asymptotically flat Riemannian manifolds, and in particular to the solutions of Einstein’s constraints equations. We establish existence theorems for the Hamiltonian and the momentum constraints with constant mean curvature and with a background metric that satisfies very low regularity assumptions. These results extend the regularity results of Holst et al. (Commun Math Phys 288:547–613, 2009) about the constraint equations on compact manifolds in the Besov space \({B_{p,p}^{s}}\), to asymptotically flat manifolds. We also consider the Brill–Cantor criterion in the weighted Besov spaces. Our results improve the regularity assumptions on asymptotically flat manifolds (Choquet-Bruhat et al. in Chin Ann Math Ser B 27(1), 31–52, 2006; Maxwell in Commun Math Phys 253(3):561–583, 2005), as well as they enable us to construct the initial data for the Einstein–Euler system.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aubin T.: Some Nonlinear Problems in Riemannian Geometry. Springer, Berlin (1988)

    Google Scholar 

  2. Bahouri H., Chemin J.Y., Danchin R.: Fourier Analysis and Nonlinear Partial Differential Equations. Springer, Heidelberg (2011)

    Book  MATH  Google Scholar 

  3. Bartnik, R.: The mass of an asymptotically flat manifold. Commun. Pure Appl. Math. 39, 661–693 (1986)

  4. Bergh J., Löfström J.: Interpolation Spaces: An Introduction. Springer, Berlin (1976)

    Book  MATH  Google Scholar 

  5. Bourdaud G., Meyer Y.: Fonctions qui opèrent sur les espaces de sobolev. J. Funct. Anal. 97, 351–360 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  6. Brauer U., Karp L.: Well-posedness of the Einstein–Euler system in asymptotically flat spacetimes: the constraint equations. J. Differ. Equ. 251, 1428–1446 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Brauer U., Karp L.: Local existence of solutions of self gravitating relativistic perfect fluids. Commun. Math. Phys. 325, 105–141 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cantor M.: Spaces of functions with asymptotic conditions on \({{\mathbb{R}}^{n}}\). Indiana Univ. Math. J. 24(9), 897–902 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cantor M.: A necessary and sufficient condition for york data to specify an asymptotically flat spacetime. J. Math. Phys. 20(8), 1741–1744 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cantor M.: Some problems of global analysis on asymptotically simple manifolds. Compos. Math. 38(1), 3–35 (1979)

    MathSciNet  MATH  Google Scholar 

  11. Cantor M., Brill D.: The Laplacian on asymptotically flat manifolds and the specification of scalar curvature. Compos. Math. 43(3), 317–330 (1981)

    MathSciNet  MATH  Google Scholar 

  12. Choquet-Bruhat Y.: Einstein constraints on compact n-dimensional manifolds. Class. Quantum Grav. 21, 127–151 (2004)

    Article  MathSciNet  Google Scholar 

  13. Choquet-Bruhat Y.: General Relativity and Einstein Equations. Oxford Science Publications, Oxford (2009)

    MATH  Google Scholar 

  14. Choquet-Bruhat Y., Christodoulou D.: Elliptic systems in \({H_{s,\delta}}\) spaces on manifolds which are euclidian at infinity. Acta Math. 146, 129–150 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  15. Choquet-Bruhat Y., Isenberg J., Pollack D.: The Einstein-scalar field constraints on asymptotically euclidean manifolds. Chin. Ann. Math. Ser. B 27(1), 31–52 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Choquet-Bruhat Y., Isenberg J., York J.W.: Einstein constraints on asymptotically euclidean manifolds. Phys. Rev. D 61(8), 20 (2000)

    Article  MathSciNet  Google Scholar 

  17. Christodoulou D., O’Murchadha N.: The boost problem in general relativity. Commun. Math. Phys. 80(2), 271–300 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  18. Driver, B.K.: Analysis tools with applications and PDE notes. (2003) http://www.freebookcentre.net/maths-books-download/Analysis-Tools-with-Applications-and-PDE-Notes.html

  19. Edmunds D.E., Triebel H.: Entropy numbers and approximation numbers in function spaces. Proc. Lond. Math. Soc. 58(3), 137–152 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  20. Friedrich H.: Yamabe numbers and the Brill–Cantor criterion. Ann. Henri Poincaré 12, 1019–1025 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. Gilbarg D., Trudinger N.S.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer, Berlin (1983)

    Book  Google Scholar 

  22. Hebey E.: Sobolev Spaces on Riemannian Manifolds. Lecture Notes in Mathematics, No. 1635. Springer, Berlin (1996)

    Google Scholar 

  23. Holst G., Nagy M., Tsogtgerel G.: Rough solutions of the Einstein constraints on closed manifolds without near-CMC conditions. Commun. Math. Phys. 288, 547–613 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kateb D.: On the boundedness of the mapping \({f\mapsto \vert f \vert ^ \mu,\mu >1 }\) on Besov spaces. Math. Nachr. 248/249, 110–128 (2003)

    Article  MathSciNet  Google Scholar 

  25. Lee J.M., Parker T.H.: The yamabe problem. Bull. AMS 17(1), 37–91 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  26. Lockhart R.B., McOwen R.M.: On elliptic systems in \({{\mathbb{R}}^{n}}\). Acta Math. 150, 125–135 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  27. Makino, T.: On a local existence theorem for the evolution equation of gaseous stars. In: Nishida, T., Mimura, M., Fujii, H. (eds.) Patterns and Waves, pp. 459–479. North-Holland, Amsterdam (1986)

  28. Maxwell D.: Rough solutions of the Einstein constraint equations on compact manifolds. J. Hyperbolic Differ. Equ. 2(2), 521–546 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  29. Maxwell D.: Solutions of the Einstein constraint equations with apparent horizon boundaries. Commun. Math. Phys. 253(3), 561–583 (2005) MR MR2116728 (2006c:83008)

    Article  MathSciNet  MATH  Google Scholar 

  30. Maxwell D.: Rough solutions of the Einstein constraint equations. J. Reine Angew. Math. 590, 1–29 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  31. Nirenberg L., Walker H.: The null spaces of elliptic differential operators in \({{\mathbb{R}}^{n}}\). J. Math. Anal. Appl. 42, 271–301 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  32. Runst T., Sickel W.: Sobolev spaces of fractional order, nemytskij operators, and nonlinear partial differential equations. Walter de Gruyter, Berlin (1996)

    Book  MATH  Google Scholar 

  33. Schecter M, : Principles of Functional Analysis, Graduate Studies in Mathematics, vol. 136. American Mathematical Society, Providence, RI (2006)

    Google Scholar 

  34. Schoen R., Yau S.T.: On the proof of the positive mass conjecture in general relativity. Commun. Math. Phys. 65(1), 45–76 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  35. Stein E.M.: Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series No. 30. Princeton University Press, Princeton, NJ (1970)

    Google Scholar 

  36. Taibleson M.H.: On the theory of lipschitz spaces of distributions on euclidean n-space. J. Math. Mech. 13, 407–408 (1964)

    MathSciNet  Google Scholar 

  37. Tartar L.: An Introduction to Sobolev Spaces and Interpolation Spaces, Lecture Notes of the Unione Matematica Italiana. Springer, Berlin (2006)

    Google Scholar 

  38. Triebel H.: Spaces of Kudrjavcev type I. Interpolation, embedding, and structure. J. Math. Anal. Appl. 56(2), 253–277 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  39. Triebel H.: Spaces of Kudrjavcev type II. Spaces of distributions: duality, interpolation. J. Math. Anal. Appl. 56(2), 278–287 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  40. Triebel H.: Theory of Function Spaces II. Birkhäuser, Basel (1983)

    Book  Google Scholar 

  41. Zolesio J.L: Multiplication dans les espaces de besov. Proc. R. Soc. Edinb. 4(2), 113–117 (1977)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemática Aplicada, Universidad Complutense Madrid, 28040, Madrid, Spain

    Uwe Brauer

  2. Department of Mathematics, ORT Braude College, P.O. Box 78, 21982, Karmiel, Israel

    Lavi Karp

Authors
  1. Uwe Brauer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lavi Karp
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lavi Karp.

Additional information

Uwe Brauer’s research was partially supported by Grant MTM2012-31928.

Lavi Karp’s research was supported by ORT Braude College’s Research Authority.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Brauer, U., Karp, L. Elliptic equations in weighted Besov spaces on asymptotically flat Riemannian manifolds. manuscripta math. 148, 59–97 (2015). https://doi.org/10.1007/s00229-015-0728-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00229-015-0728-8

Mathematics Subject Classification

Search

Navigation