An Endpoint Alexandrov Bakelman Pucci Estimate in the Plane

@article{Steinerberger2018AnEA,
  title={An Endpoint Alexandrov Bakelman Pucci Estimate in the Plane},
  author={S. Steinerberger},
  journal={Canadian Mathematical Bulletin},
  year={2018},
  volume={62},
  pages={643-651}
}
  • S. Steinerberger
  • Published 2018
  • Mathematics
  • Canadian Mathematical Bulletin
  • 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
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