# American Institute of Mathematical Sciences

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
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