# Minimal assumptions such that the solution of Poisson equation is $C^2$

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).

• 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"? – Willie Wong Jul 12 at 16:30
• @WillieWong Exactly. I will edit such that it becomes more transparent. – Severin Schraven Jul 12 at 16:52

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.

• Thanks. I'll check it out. – Severin Schraven Jul 12 at 18:04
• 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. – Willie Wong Jul 15 at 13:49
• @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.) – Mateusz Kwaśnicki Jul 15 at 19:41
• @MateuszKwaśnicki: and I learned about the result from you :-) I just looked up the keywords on MathSciNet for those references. – Willie Wong Jul 15 at 21:19