Let $F : \Omega \times [0,T] \to \mathbb{R}$ be a locally Lipschitz continuous function with respect to $C_{0}(\Omega)$-norm and $F(0)=0$. Assume that $F \in C([0,T],C_{0}(\Omega))\cap C((0,T],H_{0}^{1}(\Omega))\cap C([0,T],L^{2}(\Omega))$. Here, $\Omega$ is a bounded open subset of $\mathbb{R}^{N}$. Then, I want to show that
$\forall i \in \{1,2,...N\},\forall t\in [0,T], \frac{\partial}{\partial x_{i}}\int_{0}^{t}F(x,s)ds = \int_{0}^{t}\frac{\partial}{\partial x_{i}}F(x,s)ds$
My problem is that I don't have continuity of partial derivative of $F$ so I can't use the Leibniz Integral Rule in the classical sense. So, I try to apply Leibniz Integral Rule in Measure Theory.
In order to do so, I have to assure $F$ satisfies:
1. $\forall x \in \Omega, F$ is Lebesgue Integrable w.r.t $t$.
2. $\forall t \in [0,T] \text{ a.e. }, \forall x \in \Omega, \exists \frac{\partial}{\partial x_{i}}F$
3. $\exists \theta : [0,T] \to \mathbb{R} \forall x \in\Omega \forall t \in [0,T]\text{ a.e. }, |\frac{\partial}{\partial x_{i}}F(x,t)|\leq \theta(t)$
So, I have succeeded in showing that at least $F$ satisfies assumption 1. However, I want to ask the meaning of partial derivative of $F$ in assumption 2 and assumption 3. Does that mean the partial derivative in the weak sense? Moreover, how do I show assumption 2 and assumption 3 here?
Any help will be much appreciated! Thank you!
