manuscripta mathematica

, Volume 43, Issue 1, pp 45–72 | Cite as

An existence theorem for a non-regular variational problem

  • L. M. Sibner


A “Hodge” theorem is proved for a non-linear system of equations which are not uniformly elliptic. The solutions are p-forms which minimize a non-regular energy functional over cohomology classes. The theorem is proved by regularizing the functional and proving weak L2 existence. To obtain regularity, we first show that a scalar function of the solution is a subsolution of an elliptic equation from which it follows that the solution is bounded. Hölder continuity is then proved by comparison with a solution of a simpler system known to be Hölder continuous. The solution of the non-regular problem is then obtained from the Shiffman regularization.


Number Theory Elliptic Equation Variational Problem Scalar Function Algebraic Geometry 
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  1. [1]
    BERS, L., Mathematical aspects of subsonic and transonic gas dynamics, Wiley (1958).Google Scholar
  2. [2]
    DUFF, G.F.D., and SPENCER, D.C., Harmonic tensors on Riemannian manifolds with boundary, Ann. of Math. 56, (1952), 128–156.Google Scholar
  3. [3]
    GIAQUINTA, M., Multiple integrals in the calculus of variations, Lecture Notes, SFB 72, University of Bonn.Google Scholar
  4. [4]
    MORREY, C.B., Multiple integrals in the calculus of variations, Springer-Verlag, Berlin, 1966.Google Scholar
  5. [5]
    SHIFFMAN, M., On the existence of subsonic flows of a compressible fluid. J. Rat. Mech. Anal.1 (1952) 605–652.Google Scholar
  6. [6]
    SIBNER, L.M. and SIBNER, R.J., A non-linear Hodge-de Rham theorem, Acta Math.125 (1970) 57–73.Google Scholar
  7. [7]
    —, Non-linear Hodge theory: Applications, Advances in Math.31 (1979) 1–15.Google Scholar
  8. [8]
    —, A sub-elliptic estimate for a class of invariantly defined elliptic systems, Pacific J. Math.92 (1981) 417–421.Google Scholar
  9. [9]
    SMITH, P.D., Non-linear Hodge theory on punctured Riemannian manifolds, Indiana J. Math., Vol. 31, 4 (1982) 553–577.Google Scholar
  10. [10]
    UHLENBECK, K., Regularity for a class of non-linear elliptic systems, Acta Math.138 (1977) 219–240.Google Scholar

Copyright information

© Springer-Verlag 1983

Authors and Affiliations

  • L. M. Sibner
    • 1
  1. 1.Polytechnic Institute of New YorkBrooklynUSA

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