Remarks on functions with bounded Laplacian

Abstract

$\Delta \psi:=\frac{\partial^2 \psi}{\partial x_1^2}+\frac{\partial^2 \psi}{\partial x_2^2}$ being locally bounded does not imply that $D^2\psi$ is locally bounded. However, we prove that if $\psi$ is invariant under rotation by $\frac{2\pi}{m}$, for some $m\geq 3$, and $\Delta \psi$ is locally bounded, then $$\sup_{x\in B_1(0)}\frac{|\nabla \psi(x)|}{|x|}<\infty.$$ This is sharp in that there are examples of functions $\psi$ for which $\Delta \psi$ is locally bounded, which are invariant under rotation by $\pi$ with $|\psi(x)-\psi(0)|\approx |x|^2 |\log|x||$ as $|x|\rightarrow 0$. This bound and its generalizations could be of use in different contexts, particularly for questions about singularity formation in evolution equations. We came upon it while studying certain singular solutions of the incompressible Euler equations in two dimensions (see \cite{E}). One other application is to prove boundedness of $D^2 \psi$ when $\Delta \psi$ is the characteristic function of a set with self-intersection points (see Section 5). In fact, if $\Delta \psi=\chi_{A}$ and $A$ is the union of sectors emanating from a single point, one can give necessary and sufficient conditions on $A$ for $D^2 \psi$ to be locally bounded (see Section 6).