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An identity involving exterior derivatives and applications to Gaffney inequality

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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.
    Mathematics Subject Classification: Primary: 58A10.


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