I would like to ask the following problem.

Let $\Omega$ be a $C^{r+1,\alpha}$ domain, $r\in \mathbb{N}, 0<\alpha<1.$ We denote $$C^{r,\alpha}_{0}(\overline{\Omega})=\{f\in C^{r,\alpha}(\overline{\Omega}): f=0 \mbox{ on }\partial \Omega\},$$ here $C^{r,\alpha}(\overline{\Omega})$ is Holder spaces. Is $C^{r+1,\alpha}_0(\overline{\Omega})$ dense in $C^{r,\alpha}_0(\overline{\Omega})?$

We see that when $r\geq 2$, the answer is positive. For any $u\in C^{r,\alpha}_{0}(\overline{\Omega}),$ we have $$\Delta u := f\in C^{r-2,\alpha}_{0}(\overline{\Omega}).$$ Then there exists a sequence $f_n\in C^{r-1,\alpha}_{0}(\overline{\Omega})$ such that $f_n\rightarrow f$ in $C^{r-2,\alpha}_{0}(\overline{\Omega}).$ With each $f_n,$ there exists unique $u_n\in C^{r+1,\alpha}_{0}(\overline{\Omega})$ such that

$$\left\{\begin{array}{ll}\Delta u_n=f_n &\mbox{ in }\Omega\\ u_n =0 &\mbox{ on }\partial \Omega. \end{array}\right.$$ Therefore, by eliptic regularity, we obtain that $||u_n-u||_{C^{2,\alpha}}\leq C||f_n-f||_{C^{r-2,\alpha}}.$ It implies the conclution. It seems that we can not apply the above method for the case $r=1.$