Detecting the completeness of a Finsler manifold via potential theory for its infinity Laplacian. (English) Zbl 1461.53057
Summary: 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 \(\Delta_\infty^N u \geq g(u)\). 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 \(\Delta_\infty^N u = g(u)\) on relatively compact sets, that implies a uniform Lipschitz estimate for certain entire, bounded solutions without requiring the completeness of \(M\).
MSC:
| 53C60 | Global differential geometry of Finsler spaces and generalizations (areal metrics) |
| 31C12 | Potential theory on Riemannian manifolds and other spaces |
Keywords:
eikonal; forward completeness; maximum principles at infinity; subsolutions; \(\infty\)-capacity; viscosity; Ekeland principleReferences:
| [1] | Ahlfors, L. V.; Sario, L., Riemann Surfaces, Princeton Mathematical Series, vol. 26 (1960), Princeton Univ. Press · Zbl 0196.33801 |
| [2] | Alías, L. J.; Mastrolia, P.; Rigoli, M., Maximum Principles and Geometric Applications, Springer Monographs in Mathematics (2016), Springer: Springer Cham, xvii+570 pp · Zbl 1346.58001 |
| [3] | Alías, L. J.; Miranda, J.; Rigoli, M., A new open form of the weak maximum principle and geometric applications, Commun. Anal. Geom., 24, 1, 1-43 (2016) · Zbl 1348.58010 |
| [4] | Araújo, D. J.; Leitão, R.; Teixeira, E. V., Infinity Laplacian equation with strong absorptions, J. Funct. Anal., 270, 6, 2249-2267 (2016) · Zbl 1344.35034 |
| [5] | Armstrong, S. N.; Smart, C., An easy proof of Jensen’s theorem on the uniqueness of infinity harmonic functions, Calc. Var. Partial Differ. Equ., 37, 3-4, 381-384 (2010) · Zbl 1187.35104 |
| [6] | Armstrong, S. N.; Smart, C., A finite difference approach to the infinity Laplace equation and Tug-of-war games, Trans. Am. Math. Soc., 364, 2, 595-636 (2012) · Zbl 1239.91011 |
| [7] | Aronsson, G., Extension of functions satisfying Lipschitz conditions, Ark. Mat., 6, 551-561 (1967) · Zbl 0158.05001 |
| [8] | Aronsson, G., On the partial differential equation \(u_{x^2} u_{x x} + 2 u_x u_y u_{x y} + u_{y^2} u_{y y} = 0\), Ark. Mat., 7, 395-425 (1968) · Zbl 0162.42201 |
| [9] | Aronsson, G.; Crandall, M. G.; Juutinen, P., A tour of the theory of absolutely minimizing functions, Bull. Am. Math. Soc., 41, 4, 439-505 (2004) · Zbl 1150.35047 |
| [10] | Bao, D.; Chern, S.-S.; Shen, Z., An Introduction to Riemann-Finsler Geometry, Graduate Texts in Mathematics, vol. 200, 431 (2000), Springer-Verlag: Springer-Verlag New York · Zbl 0954.53001 |
| [11] | Barles, G.; Busca, J., Existence and comparison results for fully nonlinear degenerate elliptic equations without zeroth-order term, Commun. Partial Differ. Equ., 26, 11-12, 2323-2337 (2001) · Zbl 0997.35023 |
| [12] | Barron, E. N.; Evans, L. C.; Jensen, R., The infinity Laplacian, Aronsson’s equation and their generalizations, Trans. Am. Math. Soc., 360, 77-101 (2008) · Zbl 1125.35019 |
| [13] | Barron, E. N.; Jensen, R. R.; Wang, C. Y., The Euler equation and absolute minimizers of \(L^\infty\) functionals, Arch. Ration. Mech. Anal., 157, 255-283 (2001) · Zbl 0979.49003 |
| [14] | Bianchini, B.; Mari, L.; Pucci, P.; Rigoli, M., On the interplay among maximum principles, compact support principles and Keller-Osserman conditions on manifolds, available at |
| [15] | Calabi, E., An extension of E. Hopf’s maximum principle with an application to Riemannian geometry, Duke Math. J., 25, 1, 45-56 (1958) · Zbl 0079.11801 |
| [16] | Camilli, F.; Siconolfi, A., Maximal subsolutions for a class of degenerate Hamilton-Jacobi problems, Indiana Univ. Math. J., 48, 1111-1131 (1999) · Zbl 0939.49019 |
| [17] | Caponio, E.; Javaloyes, M. A.; Sánchez, M., On the interplay between Lorentzian causality and Finsler metrics of Randers type, Rev. Mat. Iberoam., 27, 3, 919-952 (2011) · Zbl 1229.53070 |
| [18] | Champion, T.; de Pascale, L., Principles of comparison with distance functions for absolutely minimizers, J. Convex Anal., 14, 3, 515-541 (2007) · Zbl 1128.49024 |
| [19] | Cheng, S. Y.; Yau, S. T., Differential equations on Riemannian manifolds and their geometric applications, Commun. Pure Appl. Math., 28, 3, 333-354 (1975) · Zbl 0312.53031 |
| [20] | Cohn-Vossen, S., Existenz kürzester Wege, Compos. Math., Groningen, 3, 441-452 (1936) · JFM 62.0861.01 |
| [21] | Crandall, M. G., A visit with the ∞-Laplace equation, (Calculus of Variations and Nonlinear Partial Differential Equations. Calculus of Variations and Nonlinear Partial Differential Equations, Lecture Notes in Mathematics, vol. 1927 (2008), Springer: Springer Berlin), 75-122 · Zbl 1357.49112 |
| [22] | Crandall, M. G.; Evans, L. C.; Gariepy, R. F., Optimal Lipschitz extensions and the infinity Laplacian, Calc. Var. Partial Differ. Equ., 13, 2, 123-139 (2001) · Zbl 0996.49019 |
| [23] | Crandall, M. G.; Ishii, H.; Lions, P. L., User’s guide to viscosity solutions of second-order partial differential equations, Bull. Am. Math. Soc., 27, 1-67 (1992) · Zbl 0755.35015 |
| [24] | Ekeland, I., On the variational principle, J. Math. Anal. Appl., 47, 324-353 (1974) · Zbl 0286.49015 |
| [25] | Ekeland, I., Nonconvex minimization problems, Bull. Am. Math. Soc. (N.S.), 1, 3, 443-474 (1979) · Zbl 0441.49011 |
| [26] | Evans, L. C., Estimates for smooth absolutely minimizing Lipschitz extensions, Electron. J. Differ. Equ., 03 (1993), approx. 9 pp. (electronic only) · Zbl 0809.35032 |
| [27] | García-Azorero, J.; Manfredi, J. J.; Peral, I.; Rossi, J. D., The Neumann problem for the ∞-Laplacian and the Monge-Kantorovich mass transfer problem, Nonlinear Anal., 66, 349-366 (2007) · Zbl 1387.35286 |
| [28] | Grigor’yan, A., Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Am. Math. Soc., 36, 135-249 (1999) · Zbl 0927.58019 |
| [29] | Guo, C.-Y.; Chang, C.-L.; Yang, D., \( L^\infty \)-variational problems associated to measurable Finsler structures, Nonlinear Anal., 132, 126-140 (2016) · Zbl 1329.49004 |
| [30] | Harvey, F. R.; Lawson, H. B., Dirichlet duality and the nonlinear Dirichlet problem on Riemannian manifolds, J. Differ. Geom., 88, 395-482 (2011) · Zbl 1235.53042 |
| [31] | Jensen, R., Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient, Arch. Ration. Mech. Anal., 123, 1, 51-74 (1993) · Zbl 0789.35008 |
| [32] | Juutinen, P., Minimization problems for Lipschitz functions via viscosity solutions, Ann. Acad. Sci. Fenn., Math. Diss., 115 (1998), 53 pp · Zbl 0902.35037 |
| [33] | Lindqvist, P.; Manfredi, J. J., The Harnack inequality for ∞-harmonic functions, Electron. J. Differ. Equ., 04 (1995), approx. 5 pp. (electronic) · Zbl 0818.35033 |
| [34] | Lindqvist, P.; Manfredi, J. J., Note on ∞-superharmonic functions, Rev. Mat. Univ. Complut. Madr., 10, 2, 471-480 (1997) · Zbl 0891.35043 |
| [35] | Lu, G.; Wang, P., Inhomogeneous infinity Laplace equation, Adv. Math., 217, 1838-1868 (2008) · Zbl 1152.35042 |
| [36] | Lu, G.; Wang, P., A PDE perspective of the normalized infinity Laplacian, Commun. Partial Differ. Equ., 33, 10-12, 1788-1817 (2008) · Zbl 1157.35388 |
| [37] | Mari, L.; Pessoa, L. F., Duality between Ahlfors-Liouville and khas’minskii properties for non-linear equations, Commun. Anal. Geom., 28, 2, 395-497 (2020) · Zbl 1453.53047 |
| [38] | Mari, L.; Pessoa, L. F., Maximum principles at infinity and the Ahlfors-Khas’minskii duality: an overview, Contemporary Research in Elliptic PDEs and Related Topics. Contemporary Research in Elliptic PDEs and Related Topics, Springer INdAM Ser., vol. 33, 419-455 (2019), Springer: Springer Cham · Zbl 1477.35049 |
| [39] | Mari, L.; Valtorta, D., On the equivalence of stochastic completeness, Liouville and Khas’minskii condition in linear and nonlinear setting, Trans. Am. Math. Soc., 365, 9, 4699-4727 (2013) · Zbl 1272.31012 |
| [40] | Mebrate, B.; Mohammed, A., Infinity-Laplacian type equations and their associated Dirichlet problems, Complex Var. Elliptic Equ., 31 pp (2018) |
| [41] | Mebrate, B.; Mohammed, A., Comparison principles for infinity-Laplace equations in Finsler metrics, Nonlinear Anal., 190, Article 111605 pp. (2020) · Zbl 1433.35100 |
| [42] | Mennucci, A. G.C., Regularity and variationality of solutions to Hamilton-Jacobi equations. Part II: variationality, existence, uniqueness, Appl. Math. Optim., 63, 191-216 (2011) · Zbl 1216.49025 |
| [43] | Mennucci, A. G.C., On asymmetric distances, Anal. Geom. Metric Spaces, 1, 200-231 (2013) · Zbl 1293.53081 |
| [44] | Omori, H., Isometric immersions of Riemannian manifolds, J. Math. Soc. Jpn., 19, 205-214 (1967) · Zbl 0154.21501 |
| [45] | Peres, Y.; Schramm, O.; Sheffield, S.; Wilson, D., Tug-of-war and the infinity Laplacian, J. Am. Math. Soc., 22, 167-210 (2009) · Zbl 1206.91002 |
| [46] | Pigola, S.; Rigoli, M.; Setti, A. G., A remark on the maximum principle and stochastic completeness, Proc. Am. Math. Soc., 131, 4, 1283-1288 (2003) · Zbl 1015.58007 |
| [47] | Pigola, S.; Rigoli, M.; Setti, A. G., Maximum principles on Riemannian manifolds and applications, Mem. Am. Math. Soc., 174, 822 (2005) · Zbl 1075.58017 |
| [48] | Pigola, S.; Rigoli, M.; Setti, A. G., Maximum principles at infinity on Riemannian manifolds: an overview, Workshop on Differential Geometry (Portuguese). Workshop on Differential Geometry (Portuguese), Mat. Contemp., 31, 81-128 (2006) · Zbl 1145.58009 |
| [49] | Pigola, S.; Setti, A. G., Global Divergence Theorems in Nonlinear PDEs and Geometry, Ensaios Matemáticos, vol. 26 (2014), Sociedade Brasileira de Matemática: Sociedade Brasileira de Matemática Rio de Janeiro, ii+77 pp · Zbl 1295.31002 |
| [50] | Rossi, J. D., The infinity Laplacian and Tug-of-War games (games that PDE people like to play) |
| [51] | Shen, Z., Lectures on Finsler Geometry, 307 (2001), World Scientific Publishing: World Scientific Publishing Singapore · Zbl 0974.53002 |
| [52] | Sullivan, F., A characterization of complete metric spaces, Proc. Am. Math. Soc., 83, 2, 345-346 (1981) · Zbl 0468.54021 |
| [53] | Weston, J. D., A characterization of metric completeness, Proc. Am. Math. Soc., 64, 1, 186-188 (1977) · Zbl 0368.54007 |
| [54] | Wu, Y., Absolute minimizers in Finsler metrics (1995), UC Berkeley, Ph.D. dissertation |
| [55] | Wu, B. Y.; Xin, Y. L., Comparison theorems in Finsler geometry and their applications, Math. Ann., 337, 177-196 (2007) · Zbl 1111.53060 |
| [56] | Yau, S. T., Harmonic functions on complete Riemannian manifolds, Commun. Pure Appl. Math., 28, 201-228 (1975) · Zbl 0291.31002 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
