A subspace of the $H_{0}^1(\Omega) \cap H^2(\Omega)$ is dense in a subspace of $L^2(\Omega)$?

Question

Let $n \in \mathbb{N}$ and $\Omega \subset \mathbb{R}^n$ bounded open and of class $C^1$. Consider the spaces $L^2(\Omega)$, $H_0^1(\Omega)$ and $H^2(\Omega)$.

I know that $H_0^1(\Omega) \cap H^2(\Omega)$ is densely embedded in $L^2(\Omega)$ and $H_0^1(\Omega)$ is also densely embedded in $L^2(\Omega)$. Now, define $$H_{0,m}^1(\Omega):=\left\{ f \in H_0^1(\Omega) \; ; \; \int_{\Omega} f(x) \;dx=0\right\} \quad \text{and} \quad L_{m}^2(\Omega):=\left\{ f \in L^2(\Omega) \; ; \; \int_{\Omega} f(x) \;dx=0\right\} .$$

I know that $H_{0,m}^1(\Omega)$ is closed subspace of the $H_0^1(\Omega)$, $L_{m}^2(\Omega)$ is a closed subspace of the $L^2(\Omega)$ and $H_{0,m}^1(\Omega) \cap H^2(\Omega)$ is a subspace of $L_{m}^2(\Omega)$ . Moreover $H_{0,m}^1(\Omega)$ continuously embedded in $L_{m}^2(\Omega)$ (right?).

Question. We have that $H_{0,m}^1(\Omega) \cap H^2(\Omega)$ is densely embedded in $L_{m}^2(\Omega)$? And $H_{0,m}^1(\Omega)$ is also densely embedded in $L^2_m(\Omega)$?

Answer 1

Yes. Let $\psi \in C_c^\infty(\Omega)$ with $\int_\Omega \psi \, \mathrm{d}x = 1$ be given.

For any $f \in L^2_m(\Omega)$ take a sequence $(f_n) \subset C_c^\infty(\Omega)$ with $f_n \to f$ in $L^2(\Omega)$. Then, $\int f_n \, \mathrm dx \to \int f \, \mathrm dx$. Thus, we can define $$ g_n := f_n - \psi \, \int_\Omega f_n \, \mathrm dx$$ and obtain $g_n \to f$ in $L^2(\Omega)$.