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Suppose that $M$ is a compact manifold with corners, where each boundary hypersurface is an embedded submanifold. Then, do we have an integration by parts identity? i.e. \begin{align*} \int_M g(\nabla f, X) \,d\textrm{vol} = \int_{\partial M} f\left\langle X,N \right\rangle \,d\textrm{vol}_{\tilde{g}} - \int_M f\cdot (\operatorname{div} X ) \,d\textrm{vol} \end{align*} with $f\in C^\infty(M), X\in \Gamma(M,TM)$. I know that this identity hold for domains with lipschitz boundary, but it is not very clear to me if a domain with corners is a special case of a lipschitz domain.

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    $\begingroup$ I think the second term on the RHS should read $\int_Mf\text{div}X d\text{vol}$. $\endgroup$ – S.Surace May 16 at 13:58
  • $\begingroup$ Thanks, I fixed it. $\endgroup$ – Δημήτρης Ο May 16 at 14:02

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