Let $U \subset \mathbb R^3$ be an open,bounded and connected set with a $C^2-$ regular boundary $\Gamma=\partial U $ and consider $f \in L^2(0,T;H^1(\Gamma))\cap H^1(0,T;H^1(\Gamma)^{\ast})$ .I have trouble understanding what the following norm means:
${\vert \vert f \vert \vert}_{L^{\infty}(0,T;L^2(\Gamma)) \cap L^2(0,T;H^1(\Gamma))}$
I'm confused on how I should "read" this. Does this mean that $f$ has uniform bounds but in $L^2-$ norm? If so, then what's the interpretation of $L^2(0,T;H^1(\Gamma))$?
Maybe it is a silly question to make but I haven't seen this notation before. Any help is appreciated!
Thanks in advance
The norm in the intersection is the sum (or the max) of the two norms. For $f\in L^2(0,T;H^1(\Gamma))$ we have $$\|f\|_{L^2(0,T;H^1(\Gamma))}=\left(\int_0^T \|f(t)\|_{H^1(\Gamma)}^2 dt \right)^{1/2}.$$