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An application of a new coercive inequality to variational problems studied in general relativity and in Cosserat elasticity giving the smoothness of minimizers

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Abstract

We combine a Korn-type inequality with Widman’s hole-filling technique to prove the interior regularity of minimizers of energies occurring in general relativity and in the theory of Cosserat elasticity. In addition, we provide a new variant of this Korn-type inequality valid for the nonquadratic case.

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References

  1. Adams R.A.: Sobolev spaces. Academic Press, New York-San Francisco-London (1975)

    MATH  Google Scholar 

  2. Bartnik R., Isenberg J.: The constraint equation. In: Chruściel, P.T., Friedrich, H.(eds) The Einstein equations and large scale behaviour of gravitational fields, pp. 1–38. Birkhäuser Verlag, Basel-Boston-Berlin (2004)

    Google Scholar 

  3. Bildhauer M., Fuchs M., Zhong X.: A lemma on the higher integrability of functions with applications to the regularity theory of two-dimensional generalized Newtonian fluids, Manus. Math. 116, 135–156 (2005)

    MATH  MathSciNet  Google Scholar 

  4. Dain S.: Generalized Korn’s inequality and conformal Killing vectors. Calc. Var. 25, 535–540 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. Giaquinta M.: Multiple integrals in the calculus of variations and nonlinear elliptic systems. Princeton University Press, Princeton (1983)

    MATH  Google Scholar 

  6. Gilbarg D., Trudinger N.S.: Elliptic partial differential equations of second order. Springer-Verlag, Berlin-Heidelberg-New York (1998)

    Google Scholar 

  7. J. Jeong and P. Neff, Existence, uniqueness and stability in linear Cosserat elasticity for weakest curvature conditions, Math. Mech. Solids (DOI:10.1177/1081286508093581, 2008).

  8. Jeong J. et al.: A numerical study for linear isotropic Cosserat elasticity with conformally invariant curvature. Z. Angew. Math. Mech. 89, 552–569 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  9. Meyers N.G.: An L p–estimate for the gradient of solutions of second order elliptic divergence equations. Ann. SNS Pisa III: 189(−206), 189–206 (1963)

    MathSciNet  Google Scholar 

  10. Neff P., Jeong J.: A new paradigm: the linear isotropic Cosserat model with conformally invariant curvature energy. Z. Angew. Math. Mech. 89, 107–122 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  11. P. Neff, J. Jeong, and A. Fischle, Stable identification of linear isotropic Cosserat parameters: bounded stiffness in bending and torsion implies conformal invariance of curvature, Acta Mechanica (in press 2009).

  12. P. Neff, J. Jeong and H. Ramezani, Subgrid interaction and micro-randomness-novel invariance requirements in infinitesimal gradient elasticity, Int. J. Solid Structures (in press 2009).

  13. O. Schirra, Regularity results for solutions to variational problems with applications in fluid mechanics and general relativity, PhD-thesis, Saarbrücken (2009).

  14. Widman K.O.: Hölder continuity of solutions of elliptic systems. Manus. Math. 5, 299–308 (1971)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Universität des Saarlandes, Fachbereich 6.1, Mathematik, P.O. Box 151150, 66041, Saarbrücken, Germany

    Martin Fuchs & Oliver Schirra

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  1. Martin Fuchs
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  2. Oliver Schirra
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Correspondence to Martin Fuchs.

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Fuchs, M., Schirra, O. An application of a new coercive inequality to variational problems studied in general relativity and in Cosserat elasticity giving the smoothness of minimizers. Arch. Math. 93, 587–596 (2009). https://doi.org/10.1007/s00013-009-0067-7

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  • DOI: https://doi.org/10.1007/s00013-009-0067-7

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