# An Endpoint Alexandrov Bakelman Pucci Estimate in the Plane

@article{Steinerberger2018AnEA,
title={An Endpoint Alexandrov Bakelman Pucci Estimate in the Plane},
author={S. Steinerberger},
year={2018},
volume={62},
pages={643-651}
}
• S. Steinerberger
• Published 2018
• Mathematics
• The classical Alexandrov-Bakelman-Pucci estimate for the Laplacian states $$\max_{x \in \Omega}{ |u(x)|} \leq \max_{x \in \partial \Omega}{|u(x)|} + c_{s,n} \mbox{diam}(\Omega)^{2-\frac{n}{s}} \left\| \Delta u\right\|_{L^s(\Omega)}$$ where $\Omega \subset \mathbb{R}^n$, $u \in C^2(\Omega) \cap C(\overline{\Omega})$ and $s > n/2$. The inequality fails for $s = n/2$. A Sobolev embedding result of Milman & Pustylink, originally phrased in a slightly different context, implies an endpoint… CONTINUE READING