Let $\Omega \subset \mathbb R^3$ be an open, bounded and connected set with $C^2-$regular boundary $\Gamma$. Consider an arbitary test function $\phi \in C^2(\Gamma)$. If $\Delta$ here denotes the Laplace-Beltrami operator and $f \in W^{2,p}(\Gamma)$ for any $1 \le p \lt \infty$ then I'm wondering:
Is this $\int_{\Gamma} \Delta f \phi= \int_{\Gamma} f \Delta \phi$ valid in the sense of distributions? And if yes, could someone provide a detailed explanation?
Disclaimer: I'm mostly familiar to distributions in $\mathbb R$ and is the first time I handle with the Laplace-Beltrami operator. Thus what I'm asking may be probably silly or elementary but I would appreciate any help.
Thanks in advance!