Regularity of the $p$-Poisson equation in the plane

Abstract

We study the regularity of the $p$-Poisson equation $${\Delta }_{p}u=h,h\in L^q$$ in the plane. In the case $p>2$ and $2<q<\infty$ we obtain the sharp Hölder exponent for the gradient. In the other cases we come arbitrarily close to the sharp exponent.

arXiv comment14 pages