Maximum principles at infinity and the Ahlfors-Khas'minskii duality: an overview
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arXiv
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| Format: | Preprint |
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2018
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| _version_ | 1866909346911223808 |
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| author | Mari, Luciano Pessoa, Leandro Freitas |
| author_facet | Mari, Luciano Pessoa, Leandro Freitas |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1801_05263 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Maximum principles at infinity and the Ahlfors-Khas'minskii duality: an overview Mari, Luciano Pessoa, Leandro Freitas Differential Geometry Analysis of PDEs 31C12, 35B50 (Primary), 35B53, 58J65, 58J05, 53C42 (Secondary) 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. |
| title | Maximum principles at infinity and the Ahlfors-Khas'minskii duality: an overview |
| topic | Differential Geometry Analysis of PDEs 31C12, 35B50 (Primary), 35B53, 58J65, 58J05, 53C42 (Secondary) |
| url | https://arxiv.org/abs/1801.05263 |