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

An identity involving exterior derivatives and applications to Gaffney inequality

1. 

Section de Mathématiques, EPFL, 1015 Lausanne, Switzerland

2. 

Section de Mathématiques, Station 8, EPFL, 1015 Lausanne

Received  July 2010 Published  October 2011

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.
Citation: Gyula Csató, Bernard Dacorogna. An identity involving exterior derivatives and applications to Gaffney inequality. Discrete & 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 (1976).

[3]

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

[4]

M. P. Gaffney, Hilbert space methods in the theory of harmonic integrals,, Trans. Amer. Math. Soc., 78 (1955), 426. 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. doi: 10.1007/BF02505905.

[6]

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

[7]

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

[8]

C. B. Morrey, Jr., "Multiple Integrals in the Calculus of Variations,", Die Grundlehren der mathematischen Wissenschaften, (1966).

[9]

G. Schwarz, "Hodge Decomposition-A Method for Solving Boundary Value Problems,", Lecture Notes in Math., 1607 (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 (1976).

[3]

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

[4]

M. P. Gaffney, Hilbert space methods in the theory of harmonic integrals,, Trans. Amer. Math. Soc., 78 (1955), 426. 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. doi: 10.1007/BF02505905.

[6]

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

[7]

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

[8]

C. B. Morrey, Jr., "Multiple Integrals in the Calculus of Variations,", Die Grundlehren der mathematischen Wissenschaften, (1966).

[9]

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

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