Let $\Omega \subset \mathbb R^3$ be an open,bounded and connected set with boundary $\Gamma =\partial \Omega$. Consider also $u \in L^2(0,T;H^1(\Gamma))\cap H^1(0,T;H^1(\Gamma)^{\ast})$.
It holds that: $\int_{\Gamma} u^2=\int_{\Gamma} u(u-\bar u) +{\vert \vert u \vert \vert}^2_{L^1(\Gamma)}$ where $\bar u=\frac{1}{|\Gamma|}\int_{\Gamma} u$ (the average value of $u$ over $\Gamma$)
I cannot understand this equality. I only see:
$\int_{\Gamma} u(u-\bar u) +{\vert \vert u \vert \vert}^2_{L^1(\Gamma)}=\int_{\Gamma} u^2-\frac{1}{|\Gamma|}\int_{\Gamma} u \int_{\Gamma} u+(\int_{\Gamma} u)^2$
So is there any missing term or am I wrong? I think it's quite simple but my head stuck there so any help would be appreciated.
Thanks in advance