An identity involving exterior derivatives and applications to Gaffney inequality

Gyula Csató; Bernard Dacorogna

doi:10.3934/dcdss.2012.5.531

Article Contents

2012, Volume 5, Issue 3: 531-544. Doi: 10.3934/dcdss.2012.5.531

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An identity involving exterior derivatives and applications to Gaffney inequality

Gyula Csató^1, and
Bernard Dacorogna^2,

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

Received: July 2010

Published: June 2012

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Abstract

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.

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Mathematics Subject Classification: Primary: 58A10.

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References

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