I am looking for a reference (better a book) that contain integral Bochner formulas for domains with boundary (I need it for 1-forms and functions only).

For example I will need the following formula: $$\int\limits_\Omega |\Delta f|^2 -|\mathrm{Hess}f|^2 +\langle\mathrm{Ric}(\nabla f),\nabla f\rangle =\int\limits_{\partial\Omega} H\cdot|\nabla f|^2,$$ where $H$ denotes mean curvature of $\partial \Omega$ and $f$ vanish on the boundary, but I will also need its analog for 1-forms and yet general boundary condition.

So far I found "Spin geometry" by Lawson and Michelsohn --- based on II/§5, it is easy to derive any formula I need (but I am sure someone did it somewhere).

relativeBochner formula? IIRC, Wu's "The Bochner Technique in Differential Geometry" (bookstore.ams.org/ctm-6) states the Bochner-Weitzenböck formulas for 1-forms and functions as corollaries. $\endgroup$1more comment