On the Aleksandrov-Bakelman-Pucci estimate for the infinity Laplacian

TL;DR

This paper establishes $L^$ bounds and continuity estimates for solutions to the Poisson problem involving the normalized infinity and $p$-Laplacian, highlighting the failure of classical estimates for the infinity case and proposing alternatives.

Contribution

It provides a continuous family of estimates depending on $p$ and introduces new bounds for the normalized infinity Laplacian where classical estimates fail.

Findings

Established $L^$ bounds for solutions.

Demonstrated failure of classical ABP estimate for infinity Laplacian.

Proposed alternative estimates for the infinity Laplacian.

Abstract

We prove $L^\infty$ bounds and estimates of the modulus of continuity of solutions to the Poisson problem for the normalized infinity and $p$-Laplacian, namely \[ -\Delta_p^N u=f\qquad\text{for $n<p\leq\infty$.} \] We are able to provide a stable family of results depending continuously on the parameter $p$. We also prove the failure of the classical Alexandrov-Bakelman-Pucci estimate for the normalized infinity Laplacian and propose alternate estimates.