\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

An identity involving exterior derivatives and applications to Gaffney inequality

Abstract Related Papers Cited by
  • Given two $k-$forms $\alpha$ and $\beta$ we derive an identity relating $$ %TCIMACRO{\dint _{\Omega}} %BeginExpansion {\displaystyle\int_{\Omega}} %EndExpansion \left( \langle d\alpha;d\beta\rangle+\langle\delta\alpha;\delta\beta \rangle-\langle\nabla\alpha;\nabla\beta\rangle\right) $$ to an integral on the boundary of the domain and involving only the tangential and the normal components of $\alpha$ and $\beta.$ We use this identity to deduce in a very simple way the classical Gaffney inequality and a generalization of it.
    Mathematics Subject Classification: Primary: 58A10.

    Citation:

    \begin{equation} \\ \end{equation}
  • [1]
    [2]

    G. Duvaut and J.-L. Lions, "Inequalities in Mechanics and Physics," Grundlehren der Mathematischen Wissenschaften, 219, Springer-Verlag, Berlin-New York, 1976.

    [3]

    M. P. Gaffney, The harmonic operator for exterior differential forms, Proc. Nat. Acad. of Sci. U. S. A., 37 (1951), 48-50.

    [4]

    M. P. Gaffney, Hilbert space methods in the theory of harmonic integrals, Trans. Amer. Math. Soc., 78 (1955), 426-444.doi: 10.1090/S0002-9947-1955-0068888-1.

    [5]

    T. Iwaniec, C. Scott and B. Stroffolini, Nonlinear Hodge theory on manifolds with boundary, Ann. Mat. Pura Appl. (4), 177 (1999), 37-115.doi: 10.1007/BF02505905.

    [6]

    S. G. Krantz and H. R. Parks, "The Geometry of Domains in Space," Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Boston, Inc., Boston, 1999.

    [7]

    C. B. Morrey, Jr., A variational method in the theory of harmonic integrals II, Amer. J. Math., 78 (1956), 137-170.doi: 10.2307/2372488.

    [8]

    C. B. Morrey, Jr., "Multiple Integrals in the Calculus of Variations," Die Grundlehren der mathematischen Wissenschaften, Band 130, Springer-Verlag, New-York, 1966.

    [9]

    G. Schwarz, "Hodge Decomposition-A Method for Solving Boundary Value Problems," Lecture Notes in Math., 1607, Springer-Verlag, Berlin, 1995.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(93) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return