On the Aleksandrov-Bakelman-Pucci estimate for the infinity Laplacian (Q376589)

In this article, \(L^\infty\) bounds and estimates of modulus of continuity of solutions for the normalized infinity and \(p\)-Laplacian, i.e., \(-\Delta^N_p u=f\) for \(n<p\leq \infty\), are proven. The authors are able to provide a stable family of results that depend continuously on the parameter \(p\). Additionally, they show the failure of the classical Alexandrov-Bakelman-Pucci estimate for the normalized infinity Laplacian, but they propose alternate estimates.