Regularity of ultra-weak solutions

Question

Suppose to have in the sense of distributions $$ -\Delta u = f \in L^q (B_1(0)) $$

where $B_1(0)\subset R^2$ and $q>1.$

Can I infere that $u\in L^{\infty}(B_1(0))$? Which results could I use? Any references?

Answer 1

You do not get global boundedness $u\in L^\infty(B_1(0))$ in general. For example, $u(z)=\operatorname{Re} (1-z)^{-1}$ satisfies $-\Delta u=0$ in $B_1(0)$ and is unbounded. (Using complex variable $z$ here).

If you just want $f$ to be locally bounded, the answer is yes: it even has a locally Hölder continuous representative. Convolution of $f$ with the fundamental solution $\frac{1}{2\pi}\log \frac{1}{|x|}$ gives a function $v\in W^{2,q}_{\rm loc}$ such that $-\Delta v=f$. Then $\Delta(u-v)=0$ in the sense of distributions, which by Weyl's lemma implies that $u-v$ is harmonic. Hence, $u=v+h$ where $v$ is locally in $W^{2,q}$ and $h$ is harmonic. In two dimensions, $W^{2,q}\subset C^{\alpha}$ for some $\alpha>0$ depending on $q>1$, by the Morrey-Sobolev embedding.

Some references:

• When can one expect a classical solution of a PDE?
• Elliptic Regularity for solutions in distributional sense
• Bootstrap argument to prove regularity of a special solution
• Is there an elliptic regularity for the whole space?