An Endpoint Alexandrov Bakelman Pucci Estimate in the Plane

Abstract

The classical Alexandrov-Bakelman-Pucci estimate for the Laplacian states $$ \max_{x \in Ω}{ |u(x)|} \leq \max_{x \in \partial Ω}{|u(x)|} + c_{s,n} \mbox{diam}(Ω)^{2-\frac{n}{s}} \left\| Δu\right\|_{L^s(Ω)}$$ 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 inequality: if $n\ge 3$ and $\Omega \subset {\mathbb{R}}^n$ is bounded, then $$\underset{x\in \Omega }{\max }|u(x)|\le \underset{x\in \partial \Omega }{\max }|u(x)|+c_n{\Vert \Delta u\Vert }_{L^{\frac{n}{2},1}(\Omega )},$$ 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 $\Omega \subset {\mathbb{R}}^2$ with finite measure $$\underset{x\in \Omega }{\max }|u(x)|\le \underset{x\in \partial \Omega }{\max }|u(x)|+c\underset{x\in \Omega }{\max }{\int }_{y\in \Omega }\max \left\{1,\log \left(\frac{|\Omega |}{\Vert x-y{\Vert }^2}\right)\right\}\left|\Delta u(y)\right|dy.$$ 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.