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Maximum Principles at Infinity and the Ahlfors-Khas’minskii Duality: An Overview

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Contemporary Research in Elliptic PDEs and Related Topics

Part of the book series: Springer INdAM Series ((SINDAMS,volume 33))

Abstract

This note is meant to introduce the reader to a duality principle for nonlinear equations recently discovered in Valtorta (Reverse Khas’minskii condition. Math Z 270(1):65–177, 2011), Mari and Valtorta (Trans Am Math Soc 365(9):4699–4727, 2013), and Mari and Pessoa (Commun Anal Geom, to appear). 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.

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Notes

  1. 1.

    Here, as usual, if we write “w(x) → +∞ as x diverges” we mean that the sublevels of w have compact closure in X, that is, that w is an exhaustion.

  2. 2.

    The radial sectional curvature is the sectional curvature restricted to 2-planes containing ∇ϱ. Inequality \(\mathrm {Sect}_{\mathrm {rad}} \geqslant - \mathrm {G}^2(\varrho )\) means that \(\mathrm {Sect}(\uppi _{\mathrm {x}}) \geqslant - \mathrm {G}^2\big (\varrho (\mathrm {x})\big )\) for each x∉{o}∪cut(o) and \(\uppi _{\mathrm {x}} \leqslant \mathrm {T}_{\mathrm {x}} \mathrm {X}\) 2-plane containing ∇ρ.

  3. 3.

    That is, the opposite of the roots of .

  4. 4.

    In [48], the uniform continuity of the Pucci operators in \((\mathcal {E} 6)\) is not explicitly stated but can be easily checked. For instance, in the case of \(\mathcal {P}^+_{\uplambda ,\Lambda }\), referring to Definition 2.23 in [48] and using the min-max definition,

    If ∥(A −B)+∥ < δ, then \(\mathcal {P}^+_{\uplambda ,\Lambda }(\mathrm {B}) \geqslant \mathcal {P}^+_{\uplambda ,\Lambda }(\mathrm {A}) - \mathrm {m}\Lambda \updelta \), that proves the uniform continuity of \(\mathrm {F}_{\uplambda ,\Lambda }^+\). The case of \(\mathrm {F}_{\uplambda , \Lambda }^-\) is analogous.

  5. 5.

    Theorem 3.11 can be stated for F locally jet-equivalent to a universal example, provided that the strong maximum principle in \((\mathcal {H} 1')\) holds for each manifold Y and each \(\widetilde {\mathrm {F}} \subset \mathrm {J}^2(Y)\) constructed by gluing as in the theorem.

  6. 6.

    By definition, a function u is semiconcave if and only if, locally, there exists v ∈C2, such that u + v is concave when restricted to geodesics.

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Acknowledgements

This work was completed when the second author was visiting the Abdus Salam International Center for Theoretical Physics (ICTP), Italy. He is grateful for the warm hospitality and for financial support. The authors would also like to thank the organizing and local committees of the INdAM workshop “Contemporary Research in elliptic PDEs and related topics” (Bari, May 30/31, 2017) for the friendly and pleasant environment.

The first author “Luciano Mari” is supported by the grants SNS17_B_MARI and SNS_RB_MARI of the Scuola Normale Superiore.

The second author “Leandro F. Pessoa” was partially supported by CNPq-Brazil.

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Authors and Affiliations

  1. Dipartimento di Matematica, Scuola Normale Superiore, Pisa, Italy

    Luciano Mari

  2. Departamento de Matemática, Universidade Federal do Piauí-UFPI, Teresina, Brazil

    Leandro F. Pessoa

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  1. Luciano Mari
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Correspondence to Luciano Mari .

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  1. Dept. of Mathematics and Statistics, University of Western Australia, Crawley, Perth, WA, Australia

    Serena Dipierro

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Mari, L., Pessoa, L.F. (2019). Maximum Principles at Infinity and the Ahlfors-Khas’minskii Duality: An Overview. In: Dipierro, S. (eds) Contemporary Research in Elliptic PDEs and Related Topics. Springer INdAM Series, vol 33. Springer, Cham. https://doi.org/10.1007/978-3-030-18921-1_10

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