Is $\{ u \in H^1(\Omega) \mid \Delta u \in L^2(\Omega)\}$ dense in $H^1(\Omega)$?

Question

Can it be true that the space $$\{ u \in H^1(\Omega) \mid \Delta u \in L^2(\Omega)\}$$ is dense in $H^1(\Omega)$?

If so, please give me a reference to this.

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Every $u \in H^1$ has $\Delta u \in H^{-1}$, and $L^2$ is dense in $H^{-1}$, so maybe it is true?

Answer 1

Assuming that $\Omega$ is a bounded $C^1$ domain, we can argue as follows.

I - Note that if $u\in C^\infty(\overline{\Omega})$ then, $u\in H^1(\Omega)$ and $\Delta u\in L^2(\Omega)$, therefore $$C^\infty(\overline{\Omega})\subset \{u\in H^1(\Omega):\ \Delta u\in L^2(\Omega)\}\subset H^1(\Omega).$$

II - Remember that $C^\infty(\overline{\Omega})$ is dense in $H^1(\Omega)$.

Now combine I and II to get the result.

Remark: The regularity assumption on $\partial\Omega$ is used in II, thus you can also assume that $\partial\Omega$ is only Lipschitz.