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An Endpoint Alexandrov Bakelman Pucci Estimate in the Plane

Part of: Partial differential equations Classical measure theory Manifolds

Published online by Cambridge University Press:  11 January 2019

Stefan Steinerberger
Stefan Steinerberger*
Affiliation:
Department of Mathematics, Yale University, New Haven, CT 06511, USA Email: stefan.steinerberger@yale.edu
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Abstract

The classical Alexandrov–Bakelman–Pucci estimate for the Laplacian states

$$\begin{eqnarray}\max _{x\in \unicode[STIX]{x03A9}}|u(x)|\leqslant \max _{x\in \unicode[STIX]{x2202}\unicode[STIX]{x03A9}}|u(x)|+c_{s,n}\text{diam}(\unicode[STIX]{x03A9})^{2-\frac{n}{s}}\Vert \unicode[STIX]{x0394}u\Vert _{L^{s}(\unicode[STIX]{x03A9})},\end{eqnarray}$$
where $\unicode[STIX]{x03A9}\subset \mathbb{R}^{n}$, $u\in C^{2}(\unicode[STIX]{x03A9})\cap C(\overline{\unicode[STIX]{x03A9}})$ and $s>n/2$. The inequality fails for $s=n/2$. A Sobolev embedding result of Milman and Pustylnik, originally phrased in a slightly different context, implies an endpoint inequality: if $n\geqslant 3$ and $\unicode[STIX]{x03A9}\subset \mathbb{R}^{n}$ is bounded, then
$$\begin{eqnarray}\max _{x\in \unicode[STIX]{x03A9}}|u(x)|\leqslant \max _{x\in \unicode[STIX]{x2202}\unicode[STIX]{x03A9}}|u(x)|+c_{n}\Vert \unicode[STIX]{x0394}u\Vert _{L^{\frac{n}{2},1}(\unicode[STIX]{x03A9})},\end{eqnarray}$$
 where $L^{p,q}$ is the Lorentz space refinement of $L^{p}$. This inequality fails for $n=2$, and we prove a sharp substitute result: there exists $c>0$ such that for all $\unicode[STIX]{x03A9}\subset \mathbb{R}^{2}$ with finite measure,
$$\begin{eqnarray}\max _{x\in \unicode[STIX]{x03A9}}|u(x)|\leqslant \max _{x\in \unicode[STIX]{x2202}\unicode[STIX]{x03A9}}|u(x)|+c\max _{x\in \unicode[STIX]{x03A9}}\int _{y\in \unicode[STIX]{x03A9}}\max \left\{1,\log \left(\frac{|\unicode[STIX]{x03A9}|}{\Vert x-y\Vert ^{2}}\right)\right\}|\unicode[STIX]{x0394}u(y)|dy.\end{eqnarray}$$
 This is somewhat dual to the classical Trudinger–Moser inequality; we also note that it is sharper than the usual estimates given in Orlicz spaces; the proof is rearrangement-free. The Laplacian can be replaced by any uniformly elliptic operator in divergence form.

Keywords

Alexandrov–Bakelman–Pucci estimatesecond order Sobolev inequalityTrudinger-Moser inequality

MSC classification

Primary: 35A23: Inequalities involving derivatives and differential and integral operators, inequalities for integrals
Secondary: 35B50: Maximum principles 28A75: Length, area, volume, other geometric measure theory 49Q20: Variational problems in a geometric measure-theoretic setting
Type
Article
Information
Canadian Mathematical Bulletin , Volume 62 , Issue 3 , September 2019 , pp. 643 - 651
Copyright
© Canadian Mathematical Society 2018 

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