I have given functions $f \in L^2(0,T;L^2(U))$ and $g \in L^2(0,T;H^1(U))$ and a function $H \in C^1([0,T]\times U)$ such that $$1 \leq H(t,x) \leq 2$$ for a.e. $(t,x)$. Here, $U=\partial\Omega$ is the boundary of a bounded, smooth domain.
Suppose I know that for a.e. $t \in [0,T]$, $$\int_U f(t,x)\varphi(x)H(t,x) + \nabla_x g(t,x)\cdot \nabla_x (\varphi(x)H(t,x))\;\mathrm{d}x =0$$ for every $\varphi \in H^1(U)$.
Can I conclude that $$\int_U f(t,x)\phi(x) + \nabla_x g(t,x)\cdot \nabla_x \phi(x)\;\mathrm{d}x =0$$ for every $\phi \in H^1(U)$ and a.e. $t \in [0,T]$?
Do I need $f \in L^2(0,T;H^1(U))$ for this?
