Calculus of Variations and Geometric Measure Theory

F. Charro - G. De Philippis - A. Di Castro - D. Maximo

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

created by dephilipp on 29 Jul 2011
modified on 30 Oct 2017


Accepted Paper

Inserted: 29 jul 2011
Last Updated: 30 oct 2017

Journal: Calc. Var. Partial Differential Equations
Year: 2011

ArXiv: 1108.0113 PDF


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.

Keywords: Infinity Laplacian, A priori estimates, Maximum Principle