In the article https://arxiv.org/pdf/1211.0800.pdf the author uses two variational equalities (in my case $L=L(\nabla\varphi)$ only): $$ \int_{\delta\Omega}L(\nabla\varphi){\rm d}x=\int_{\partial\Omega}L(\nabla\varphi)\boldsymbol n(x)\cdot\delta\boldsymbol u(x) {\rm d}x, $$ $$ \int_{\delta(\partial\Omega)}f(x){\rm d}x=2\int_{\partial\Omega}H(x)f(x)\boldsymbol n(x)\cdot\delta\boldsymbol u(x) {\rm d}x, $$ refering to Ou-Yang and Helfrich`s proof, which, as I understand, is something different.
Here $L$ is the Lagrangian, $\Omega\subset\mathbb{R}^3$, $H$ is the mean curvature of the boundary $\partial\Omega$, $\boldsymbol n$ is the outward normal to $\partial\Omega$, $\boldsymbol u$ is the displacement of the area element of $\partial\Omega$.
How to proove those equalities?