Let $\Omega$ be a bounded domain.

Question 1: Let $w\in C^{k,\alpha}(\Omega)$ satisfy \begin{equation} \begin{cases} div(A \nabla w) = 0 \quad \text{in } \Omega,\\ \partial_\nu w=0 \quad \text{on } \partial\Omega, \end{cases} \end{equation} where $A>0$ sufficiently regular (e.g. $\in C^{k,\alpha}(\bar \Omega)$) and $\partial \Omega \in C^{k+1,\alpha}$. Then $w\in C^{k,\alpha}(\bar \Omega)$.

Question 2: Let $w\in C^{\alpha}(\bar \Omega)\cap W^{1,2}(\bar\Omega)$ satisfy \begin{equation} \begin{cases} div(A \nabla w) = 0 \quad \text{in } \Omega,\\ \partial_\nu w=0 \quad \text{on } \partial\Omega, \end{cases} \end{equation} weakly. That is for $\varphi \in C_c^\infty(\bar \Omega)$, \begin{equation} \int_\Omega A \nabla w \cdot \nabla \varphi = \int_{ \partial \Omega_u} A \underbrace{\nabla w \cdot \nu}_{\partial_\nu w=0} \varphi = 0 \qquad \forall \varphi \in C_c^\infty ( \bar \Omega_u). \end{equation} Assume that $A>0$ is sufficiently regular (e.g. $\in C^{\alpha}(\bar \Omega)$) and $\partial \Omega \in C^{1,\alpha}$. Then $w\in C^{1,\alpha}(\bar \Omega)$.