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 / L. Mari, L.D.F. Pessoa (SPRINGER INDAM SERIES). - In: Contemporary research in elliptic PDEs and related topics / [a cura di] S. Dipierro. - Cham : Springer INdAM Ser., 2019. - ISBN 978-3-030-18920-4. - pp. 419-455 (( convegno Contemporary research in elliptic PDEs and related topics tenutosi a Bari nel 2017 [10.1007/978-3-030-18921-1_10].

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

L. Mari;
2019

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.
Brownian motion; Completeness; Ekeland; Liouville theorem; Martingale; Maximum principles; Omori-Yau; Potential theory; Stochastic completeness
Settore MAT/05 - Analisi Matematica
Settore MAT/06 - Probabilita' e Statistica Matematica
2019
Book Part (author)
File in questo prodotto:
File Dimensione Formato  
1801.05263.pdf

accesso aperto

Tipologia: Pre-print (manoscritto inviato all'editore)
Dimensione 415.51 kB
Formato Adobe PDF
415.51 kB Adobe PDF Visualizza/Apri
Pubblicazioni consigliate

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/1039253
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 6
  • ???jsp.display-item.citation.isi??? ND
social impact