Calculus of Variations and Geometric Measure Theory

L. Mari - Leandro Freitas Pessoa

Maximum principles at infinity and the Ahlfors-Khas'minskii duality: an overview

created by mari1 on 15 Nov 2020



Inserted: 15 nov 2020

Year: 2018

ArXiv: 1801.05263 PDF


This note is meant to introduce the reader to a duality principle for nonlinear equations that recently appeared in the literature. Motivations come from the desire to give a unifying potential-theoretic framework for various maximum principles at infinity appearing in the literature (Ekeland, Omori-Yau, Pigola-Rigoli-Setti), as well as to describe their interplay with properties coming from stochastic analysis on manifolds. The duality involves an appropriate version of these principles formulated for viscosity subsolutions of fully nonlinear inequalities, called the Ahlfors property, and the existence of suitable exhaustion functions called Khas'minskii potentials. We discuss applications, also involving the geometry of submanifolds, in the last sections, as well as the stability of these maximum principles when we remove polar sets.