I asked the question below before.
$\Delta u$ is bounded. Can we say $u\in C^1$?
I thought I understood the discussion using the PDE theory at the time but now I am lost. I am going to a similar but different question given the answer above.
Let $\Omega\subset\mathbb{R}^d$ an open bounded set consider a function $u:\Omega\to\mathbb{R}$ we happen to have.
Following the answer above I wish to apply the discussion on the regularity of the PDE solution to see the smoothness of $u$.
Let $\partial \Omega$ be $C^2$, say.
Suppose we know that $\Delta u\in L^2(\Omega)$ with $\Delta$ being the distributional derivative.
Can we say $u\in H^2(\Omega)$?
I felt that to say the smoothness we need some behaviour on the boundary. E.g., Gilbarg--Trudinger p. 186, Theorem 8.12 says
Let us assume ... and that there exists a function $\varphi\in W^{2,2}(\Omega)$ for which $u-\varphi\in W^{1,2}_0(\Omega)$. Then, we have also $u\in W^{2,2}(\Omega)$ and
$$ \|u\|_{W^{2,2}}\le C(\|u\|_{L^2(\Omega)}+\|f\|_{L^2(\Omega)}+\|\varphi\|_{W^{2,2}(\Omega)} $$
Now, we consider $f:=\Delta u$. I thought to apply this discussion we need to have some sort of $\varphi$. But we do not have information on the behaviour on the function $u$ that we happen to have. In the answer I accepted above it is said that it suffice to consider the zero boundary case, which I think is true for the PDE, but I do not see how it is ok here.
If you do not have any information about the boundary values of $u$, then you only get local regularity $u\in H^2_{loc}(\Omega)$, but no information about the smoothness up to the boundary (check for instance Evans' book).