n 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 Δ∞Nu≥g(u). Also, an ∞-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 Δ∞Nu=g(u) on relatively compact sets, that implies a uniform Lipschitz estimate for certain entire, bounded solutions without requiring the completeness of M.

Detecting the completeness of a Finsler manifold via potential theory for its infinity Laplacian / D.J. Araújo, L. Mari, L.F. Pessoa. - In: JOURNAL OF DIFFERENTIAL EQUATIONS. - ISSN 0022-0396. - 281:(2021 Apr 25), pp. 550-587. [10.1016/j.jde.2021.02.005]

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

L. Mari
Penultimo
;
2021

Abstract

n 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 Δ∞Nu≥g(u). Also, an ∞-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 Δ∞Nu=g(u) on relatively compact sets, that implies a uniform Lipschitz estimate for certain entire, bounded solutions without requiring the completeness of M.
Settore MAT/03 - Geometria
Settore MAT/05 - Analisi Matematica
25-apr-2021
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/1039372
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