Let $U$ be a bounded open domain in $\mathbb{R}^n$. Does the space $C^2(\overline{U})$ (the bar over $U$ means closure) mean the set of twice-differentiable functions $u$ such that $u, u_t, u_{x_i}$ are continuous on $\overline{U}$?
Or is it something else? If it's not what I think it is, what space of functions do I want to use?