An identity involving exterior derivatives and applications to Gaffney inequality

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June 2012, 5(3): 531-544. doi: 10.3934/dcdss.2012.5.531

An identity involving exterior derivatives and applications to Gaffney inequality

Gyula Csató ^1, and Bernard Dacorogna ^2,

1.	Section de Mathématiques, EPFL, 1015 Lausanne, Switzerland
2.	Section de Mathématiques, Station 8, EPFL, 1015 Lausanne

Received July 2010 Published October 2011

Abstract
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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.

Keywords: Gaffney inequality., exterior derivative, Differential forms.

Mathematics Subject Classification: Primary: 58A1.

Citation: Gyula Csató, Bernard Dacorogna. An identity involving exterior derivatives and applications to Gaffney inequality. Discrete and Continuous Dynamical Systems - S, 2012, 5 (3) : 531-544. doi: 10.3934/dcdss.2012.5.531

References:

[1]	, G. Csató,, Ph.D thesis, ().
[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.

show all references

References:

[1]	, G. Csató,, Ph.D thesis, ().
[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.

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