×

On the equivalence of stochastic completeness and Liouville and Khas’minskii conditions in linear and nonlinear settings. (English) Zbl 1272.31012

Summary: Set in the Riemannian enviroment, the aim of this paper is to present and discuss some equivalent characterizations of the Liouville property relative to special operators, which in some sense are modeled after the \(p\)-Laplacian with potential. In particular, we discuss the equivalence between the Liouville property and the Khas’minskii condition, i.e. the existence of an exhaustion function which is also a supersolution for the operator outside a compact set. This generalizes a previous result obtained by one of the authors [D. Valtorta, Math. Z. 270, No. 1–2, 165–177 (2012; Zbl 1242.53040)].

MSC:

31C12 Potential theory on Riemannian manifolds and other spaces
35B53 Liouville theorems and Phragmén-Lindelöf theorems in context of PDEs
58J65 Diffusion processes and stochastic analysis on manifolds
58J05 Elliptic equations on manifolds, general theory

Citations:

Zbl 1242.53040
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Paolo Antonini, Dimitri Mugnai, and Patrizia Pucci, Quasilinear elliptic inequalities on complete Riemannian manifolds, J. Math. Pures Appl. (9) 87 (2007), no. 6, 582 – 600 (English, with English and French summaries). · Zbl 1178.58008
[2] Anders Björn and Jana Björn, Boundary regularity for \?-harmonic functions and solutions of the obstacle problem on metric spaces, J. Math. Soc. Japan 58 (2006), no. 4, 1211 – 1232. · Zbl 1211.35109
[3] Felix E. Browder, Existence theorems for nonlinear partial differential equations, Global Analysis (Proc. Sympos. Pure Math., Vol. XVI, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 1 – 60.
[4] Roberta Filippucci, Patrizia Pucci, and Marco Rigoli, Non-existence of entire solutions of degenerate elliptic inequalities with weights, Arch. Ration. Mech. Anal. 188 (2008), no. 1, 155 – 179. · Zbl 1151.35110
[5] Roberta Filippucci, Patrizia Pucci, and Marco Rigoli, On weak solutions of nonlinear weighted \?-Laplacian elliptic inequalities, Nonlinear Anal. 70 (2009), no. 8, 3008 – 3019. · Zbl 1165.35488
[6] Ronald Gariepy and William P. Ziemer, A regularity condition at the boundary for solutions of quasilinear elliptic equations, Arch. Rational Mech. Anal. 67 (1977), no. 1, 25 – 39. · Zbl 0389.35023
[7] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. · Zbl 1042.35002
[8] Alexander Grigor\(^{\prime}\)yan, Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Amer. Math. Soc. (N.S.) 36 (1999), no. 2, 135 – 249. · Zbl 0927.58019
[9] Juha Heinonen, Tero Kilpeläinen, and Olli Martio, Nonlinear potential theory of degenerate elliptic equations, Dover Publications, Inc., Mineola, NY, 2006. Unabridged republication of the 1993 original. · Zbl 1115.31001
[10] J. B. Keller, On solutions of \Delta \?=\?(\?), Comm. Pure Appl. Math. 10 (1957), 503 – 510. · Zbl 0090.31801
[11] R. Z. Has\(^{\prime}\)minskiĭ, Ergodic properties of recurrent diffusion processes and stabilization of the solution of the Cauchy problem for parabolic equations, Teor. Verojatnost. i Primenen. 5 (1960), 196 – 214 (Russian, with English summary). · Zbl 0093.14902
[12] Tero Kilpeläinen, Singular solutions to \?-Laplacian type equations, Ark. Mat. 37 (1999), no. 2, 275 – 289. · Zbl 1018.35028
[13] David Kinderlehrer and Guido Stampacchia, An introduction to variational inequalities and their applications, Pure and Applied Mathematics, vol. 88, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. · Zbl 0457.35001
[14] Takeshi Kura, The weak supersolution-subsolution method for second order quasilinear elliptic equations, Hiroshima Math. J. 19 (1989), no. 1, 1 – 36. · Zbl 0735.35056
[15] Zenjiro Kuramochi, Mass distributions on the ideal boundaries of abstract Riemann surfaces. I, Osaka Math. J. 8 (1956), 119 – 137. · Zbl 0071.07303
[16] Olga A. Ladyzhenskaya and Nina N. Ural’tseva, Linear and quasilinear elliptic equations, Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis, Academic Press, New York-London, 1968. · Zbl 0164.13002
[17] Marco Magliaro, Luciano Mari, Paolo Mastrolia, and Marco Rigoli, Keller-Osserman type conditions for differential inequalities with gradient terms on the Heisenberg group, J. Differential Equations 250 (2011), no. 6, 2643 – 2670. · Zbl 1223.35308
[18] Jan Malý and William P. Ziemer, Fine regularity of solutions of elliptic partial differential equations, Mathematical Surveys and Monographs, vol. 51, American Mathematical Society, Providence, RI, 1997. · Zbl 0882.35001
[19] Luciano Mari, Marco Rigoli, and Alberto G. Setti, Keller-Osserman conditions for diffusion-type operators on Riemannian manifolds, J. Funct. Anal. 258 (2010), no. 2, 665 – 712. · Zbl 1186.53051
[20] Mitsuru Nakai, On Evans potential, Proc. Japan Acad. 38 (1962), 624 – 629. · Zbl 0197.08604
[21] Robert Osserman, On the inequality \Delta \?\ge \?(\?), Pacific J. Math. 7 (1957), 1641 – 1647. · Zbl 0083.09402
[22] S. Pigola, M. Rigoli, and A. G. Setti, Maximum principles at infinity on Riemannian manifolds: an overview, Mat. Contemp. 31 (2006), 81 – 128. Workshop on Differential Geometry (Portuguese). · Zbl 1145.58009
[23] Stefano Pigola, Marco Rigoli, and Alberto G. Setti, A remark on the maximum principle and stochastic completeness, Proc. Amer. Math. Soc. 131 (2003), no. 4, 1283 – 1288. · Zbl 1015.58007
[24] Stefano Pigola, Marco Rigoli, and Alberto G. Setti, Maximum principles on Riemannian manifolds and applications, Mem. Amer. Math. Soc. 174 (2005), no. 822, x+99. · Zbl 1075.58017
[25] Stefano Pigola, Marco Rigoli, and Alberto G. Setti, Some non-linear function theoretic properties of Riemannian manifolds, Rev. Mat. Iberoam. 22 (2006), no. 3, 801 – 831. · Zbl 1112.31004
[26] Stefano Pigola, Marco Rigoli, and Alberto G. Setti, Aspects of potential theory on manifolds, linear and non-linear, Milan J. Math. 76 (2008), 229 – 256. · Zbl 1205.31005
[27] Patrizia Pucci, Marco Rigoli, and James Serrin, Qualitative properties for solutions of singular elliptic inequalities on complete manifolds, J. Differential Equations 234 (2007), no. 2, 507 – 543. · Zbl 1143.35099
[28] Patrizia Pucci and James Serrin, The maximum principle, Progress in Nonlinear Differential Equations and their Applications, vol. 73, Birkhäuser Verlag, Basel, 2007. · Zbl 1134.35001
[29] Patrizia Pucci, James Serrin, and Henghui Zou, A strong maximum principle and a compact support principle for singular elliptic inequalities, J. Math. Pures Appl. (9) 78 (1999), no. 8, 769 – 789. · Zbl 0952.35045
[30] L. Sario and M. Nakai, Classification theory of Riemann surfaces, Die Grundlehren der mathematischen Wissenschaften, Band 164, Springer-Verlag, New York-Berlin, 1970. · Zbl 0199.40603
[31] Chiung-Jue Sung, Luen-Fai Tam, and Jiaping Wang, Spaces of harmonic functions, J. London Math. Soc. (2) 61 (2000), no. 3, 789 – 806. · Zbl 0963.31004
[32] Peter Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 (1984), no. 1, 126 – 150. · Zbl 0488.35017
[33] Daniele Valtorta, Reverse Khas’minskii condition, Math. Z. 270 (2012), no. 1-2, 165 – 177. · Zbl 1242.53040
[34] Daniele Valtorta and Giona Veronelli, Stokes’ theorem, volume growth and parabolicity, Tohoku Math. J. (2) 63 (2011), no. 3, 397 – 412. · Zbl 1232.26011
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.