Let $D \subset \mathbb R^d$ be a $C^2$ bounded domain. I consider the following boundary value problem for the Helmholtz equation $$ (\Delta+k^2)u = 0 \quad \text{in $D$}, \\ u|_{\partial D} = u_0, $$ for some given continuous $u_0$, $k^2 > 0$. I suppose that $-k^2$ is not a Dirichlet eigenvalue for $\Delta$ in $D$. It is known that if $u_0 \in C^1(\partial D)$ then there exists $\widetilde u_0 \in H^1(D)$ with $\widetilde u_0|_{\partial D} = u_0$. This implies that there exists the unique solution to the above problem in $H^1(D)$. But what are the simplest conditions under which this solution will belong to $C^2(D) \cap C^1(\overline D)$?
For example, the condition $u_0 \in C^3(\partial D)$ is sufficient since then there exists $\widetilde u_0 \in C^3(\overline D)$ with $(\Delta + k^2)\widetilde u = f \in C^1(D)$, and the problem $$ (\Delta+k^2)u = -f, \quad \text{in $D$}, \\ u|_{\partial D} = 0, $$ has a solution from $C^2(D) \cap C^1(\overline D)$. But does this result hold if we replace the assumption $u_0 \in C^3$ by, e.g., $u_0 \in C^{1,\alpha}$?