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Take a weak solution $u$ of the Poisson equation on $\mathbb{R}^d$ $$ \Delta u = f $$ By standard elliptic regularity theory we have (for some $\alpha\in (0,1]$) $f\in C^{0, \alpha}_{\text{loc}}(\mathbb{R}^d)$, then $u \in C^{2, \alpha}_{\text{loc}}(\mathbb{R}^d)$. Now my (naive) question is:

Are there some weaker assumptions that on $f$ that ensure that $u \in C^2(\mathbb{R}^d)$?

By weaker assumptions I mean something like $f\in C^0(\mathbb{R}^d)$ plus some condition which is not of Hölder type (I am aware that $f\in C^0(\mathbb{R}^d)$ is not sufficient).

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    $\begingroup$ There is a standard counterexample that shows the existence of $f \in C^0$ such that $u\not\in C^2$ (which I guess you are aware of); so I guess you are looking for an assumption of the form $C^0$ + "something not measured on the Holder scale"? $\endgroup$ Jul 12 '19 at 16:30
  • $\begingroup$ @WillieWong Exactly. I will edit such that it becomes more transparent. $\endgroup$ Jul 12 '19 at 16:52
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Dini continuity may be what you are looking for: if $f = \Delta u$ is Dini continuous, that is, $$\int_0^1 \frac{\omega_f(t)}{t} \, dt < \infty,$$ then $u$ is $C^2$. This is a rather old result, but I do not know the reference. A quick Google search leads to Poisson's equation by T. Gantumur, see Theorem 42 and Corollary 43 there.

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  • $\begingroup$ Thanks. I'll check it out. $\endgroup$ Jul 12 '19 at 18:04
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    $\begingroup$ Additional references: the Laplacian version seems to have been due to Shapiro, and there is a version with more general elliptic operators due to Burch. $\endgroup$ Jul 15 '19 at 13:49
  • $\begingroup$ @WillieWong: Interesting, thanks! (I learned about the result from a colleague of mine. When I meet him, I will ask if he knows any relevant references.) $\endgroup$ Jul 15 '19 at 19:41
  • $\begingroup$ @MateuszKwaśnicki: and I learned about the result from you :-) I just looked up the keywords on MathSciNet for those references. $\endgroup$ Jul 15 '19 at 21:19

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