Advertisement

Existence and Regularity of Solutions of dω = f with Dirichlet Boundary Conditions

  • Bernard Dacorogna
Part of the International Mathematical Series book series (IMAT, volume 1)

Abstract

Given a bounded open set Ω ⊂ ℝ n and a (k + l)-form f satisfying some compatibility conditions, we solve the problem (in Hölder spaces)
$$ d\omega = f\;in\;\Omega, \quad \omega = 0\;on\;\partial \Omega $$
.

Keywords

Compatibility Condition Outward Unit Dual Version Bounded Convex Part Formula 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    C. Godbillon, Eléments de topologie algébrique, Hermann, Paris, 1971.MATHGoogle Scholar
  2. 2.
    M. E. Bogovski, Solution of the first boundary-value problem for the equation of continuity of an incompressible medium, Soviet Math. Dokl. 20 (1979), 1094–1098.Google Scholar
  3. 3.
    W. Borchers and H. Sohr, On the equations rot v = g and div u = f with zero boundary conditions, Hokkaido Math. J. 19 (1990), 67–87.MathSciNetMATHGoogle Scholar
  4. 4.
    B. Dacorogna and J. Moser, On a partial differential equation involving the Jacobian determinant, Ann. Inst. Henri Poincaré, Analyse non linéaire 7 (1990), 1–26.MathSciNetMATHGoogle Scholar
  5. 5.
    B. Dacorogna, Direct methods in the calculus of variations, Springer-Verlag, Berlin, 1989.CrossRefMATHGoogle Scholar
  6. 6.
    R. Dautray and J. L. Lions, Analyse mathématique et calcul numérique, 5, Masson, Paris, 1988.Google Scholar
  7. 7.
    G. P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations, Springer-Verlag, New York, 1994.CrossRefGoogle Scholar
  8. 8.
    V. Girault and P. A. Raviart, Finite element approximation of the Navier-Stokes equations, Lect. Notes Math. 749 (1979).Google Scholar
  9. 9.
    L. V. Kapitanskii and K. Pileckas, Certain problems of vector analysis, J. Soviet Math. 32 (1986), 469–483.CrossRefGoogle Scholar
  10. 10.
    O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, “Nauka”, Moscow, 1970; English transl. of 1st ed., Gordon and Breach, New York-London-Paris, 1969.MATHGoogle Scholar
  11. 11.
    O. A. Ladyzhenskaya and V. A. Solonnikov, Some problems of vector analysis and generalized formulations of boundary-value problems for the Navier-Stokes equations, J. Sov. Math. 10 (1978), 257–286.CrossRefGoogle Scholar
  12. 12.
    J. Necas, Les méthodes directes en théorie des équations elliptiques, Masson, Paris, 1967.MATHGoogle Scholar
  13. 13.
    L. Tartar, Topics in Nonlinear Analysis, Publ. Math. Orsay, 1978.Google Scholar
  14. 14.
    W. Von Wahl, On necessary and sufficient conditions for the solvability of the equations rot u = γ and div u = ε with u vanishing on the boundary, Lect. Notes Math. 1431 (1990), 152–157.CrossRefGoogle Scholar
  15. 15.
    W. Von Wahl, Vorlesung über das Aussenraumproblem für die instationären Gleichungen von Navier-Stokes, Rudolph-Lipschitz-Vorlesung. Sonderforschungsbereich 256 Nichtlineare Partielle Differentialgleichungen, Bonn, 1989.Google Scholar
  16. 16.
    R. Griesinger, On the boundary-value problem rot u = f, u vanishing at the boundary and related decomposition of L q and H 1,q 0 : existence, Ann. Univ. Ferrara Nuova Ser., Sez. VII 36 (1990), 15–43.MathSciNetMATHGoogle Scholar
  17. 17.
    G. F. Duff and D. C. Spencer, Harmonic tensors on Riemannian manifolds with boundary, Ann. Math. 56 (1952), 128–156.MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    C. B. Morrey, A variational method in the theory of harmonic integrals. II, Am. J. Math. 78 (1956), 137–170.MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    C. B. Morrey, Multiple integrals in the calculus of variations, Springer-Verlag, Berlin, 1966.MATHGoogle Scholar
  20. 20.
    R. Kress, Potentialtheoretische Randwertprobleme bei Tensorfeldern beliebiger Dimension und beliebigen Ranges, Arch. Ration. Mech. Anal. A47 (1972), 59–80.MathSciNetMATHGoogle Scholar
  21. 21.
    D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, Berlin, 1977.MATHGoogle Scholar
  22. 22.
    O. A. Ladyzhenskaya and N. N. Ural’tseva, Linear and Quasilinear Elliptic Equations, “Nauka”, Moscow, 1964; English transl., Academic Press, New York, 1968.Google Scholar

Copyright information

© Springer Science+Business Media New York 2002

Authors and Affiliations

  • Bernard Dacorogna

There are no affiliations available

Personalised recommendations