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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.

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  • $\begingroup$ I would suggest that you specify the sense in which you are applying the Laplacian to your function. You're asking if $$\Delta u =f$$ and $$f$$ is bounded implies that $$u \in C^1$$. What you're not specifying is in what sense $$\Delta u =f$$. Is it a weak solution, a strong solution, a viscosity solution, a distributional solution, a classical solution..? $\endgroup$ – Glitch Nov 5 '15 at 20:55
  • $\begingroup$ @Glitch Thank you and sorry for my late reply. I have been busy and I wanted to understand you comment as accurate as possible. I updated the question and I hope it makes sense. $\endgroup$ – shall.i.am Nov 17 '15 at 2:51
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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).

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  • $\begingroup$ That is what I thought. But @ tankonetoone commented that "The requirement on trace zero is fine....", then I replied to that, still confused, and leaded me to the current question. $\endgroup$ – shall.i.am Nov 17 '15 at 2:31

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