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arXiv · 2005.04440

Detecting the completeness of a Finsler manifold via potential theory for its infinity Laplacian

Damião J. Araújo·Luciano Mari·Leandro F. Pessoa
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Abstract

In this paper, we study some potential theoretic aspects of the eikonal and infinity Laplace operator on a Finsler manifold $M$. Our main result shows that the forward completeness of $M$ can be detected in terms of Liouville properties and maximum principles at infinity for subsolutions of suitable inequalities, including $\Delta^N_\infty u \ge g(u)$. Also, an $\infty$-capacity criterion and a viscosity version of Ekeland principle are proved to be equivalent to the forward completeness of $M$. Part of the proof hinges on a new boundary-to-interior Lipschitz estimate for solutions of $\Delta^N_\infty u = g(u)$ on relatively compact sets, that implies a uniform Lipschitz estimate for certain entire, bounded solutions without requiring the completeness of $M$.

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Damião J. Araújo, Luciano Mari, Leandro F. Pessoa. 2020-05-09. Detecting the completeness of a Finsler manifold via potential theory for its infinity Laplacian. https://doi.org/10.1016/j.jde.2021.02.005

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