Density on Hölder spaces whose elements vanish on the boundary

Question

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 $r=1.$

Answer 1

This is already answered in this post, I think, by a simple sandwich argument.

Smooth $C^\infty_c(\Omega)$ functions are dense in $C^{r+1,\alpha}_0(\Omega)$ and $C^{r,\alpha}_0(\Omega)$ and $C^\infty_c(\Omega)\subset C^{r+1,\alpha}_0(\Omega)\subset C^{r+1,\alpha}_0(\Omega)$, so there.

Answer 2

Too long for a comment: the parametrix of the Dirichlet problem is a pseudo-differential operator with order $-2$ and thus sends $C^s$ into $C^{s+2}$ for any non-integer $s$ where $$ C^s=B^s_{\infty,\infty} \quad\text{is the Besov space}. $$ If $\Omega$ is a smooth open set, your argument should work with the estimate $$ \Vert u-u_n\Vert_{C^{r,\alpha}}\lesssim \Vert f-f_n\Vert_{C^{r-2,\alpha}}. $$ Note on Besov spaces. Using a Littlewood-Paley decomposition, $ 1=\sum_{k\ge 0}\phi_k(\xi), $ where the smooth compactly supported $\phi_k$ have support in $\{2^{k-1}\le\vert \xi\vert\le 2^{k+1}\}$ for $k\ge 1$, we get that, with $s=r+\alpha$, $$C^{r,\alpha}_0(\overline\Omega)=\{u\in L^{\infty}(\mathbb R^d),\ \sup_{k\ge 0} 2^{s k}\Vert \phi_k(D) u\Vert_{L^\infty(\mathbb R^d)}<+\infty,\ \text{supp} u\subset \overline\Omega\}=B^{r+\alpha}_{\infty,\infty}\cap \mathscr E'_{\overline\Omega}.$$