This note is meant to introduce the reader to a duality principle for nonlinear equations recently discovered in cite{valtorta, marivaltorta, maripessoa}. 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. Applications, also involving the geometry of submanifolds, will be discussed in the last sections. We conclude by investigating the stability of these maximum principles when we remove polar sets.

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

Mari, Luciano;
2019-01-01

Abstract

This note is meant to introduce the reader to a duality principle for nonlinear equations recently discovered in cite{valtorta, marivaltorta, maripessoa}. 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. Applications, also involving the geometry of submanifolds, will be discussed in the last sections. We conclude by investigating the stability of these maximum principles when we remove polar sets.
2019
Contemporary research in elliptic PDEs and related topics
Bari
May 30/31, 2017
Contemporary research in elliptic PDEs and related topics
Springer INdAM Ser.
33
419
455
Mari, Luciano; Pessoa, Leandro de Freitas
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2318/1775226
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