Revista Matemática Iberoamericana

Some Remarks on the Weak Maximum Principle

Marco Rigoli, Maura Salvatori, and Marco Vignati

Full-text: Open access


We obtain a maximum principle at infinity for solutions of a class of nonlinear singular elliptic differential inequalities on Riemannian manifolds under the sole geometrical assumptions of volume growth conditions. In the case of the Laplace-Beltrami operator we relate our results to stochastic completeness and parabolicity of the manifold.

Article information

Rev. Mat. Iberoamericana, Volume 21, Number 2 (2005), 459-481.

First available in Project Euclid: 11 August 2005

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 58J05: Elliptic equations on manifolds, general theory [See also 35-XX] 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60]

maximum principles volume growth


Rigoli, Marco; Salvatori, Maura; Vignati, Marco. Some Remarks on the Weak Maximum Principle. Rev. Mat. Iberoamericana 21 (2005), no. 2, 459--481.

Export citation


  • Coulhon, T., Holopainen, I. and Saloff-Coste, L.: Harnack inequality and hyperbolicity for subelliptic $p$-Laplacians with applications to Picard type theorems. Geom. Funct. Anal. 11 (2001), no. 6, 1139-1191.
  • Cheng, S.Y. and Yau, S.T.: Differential equations on Riemannian manifolds and their geometric applications. Comm. Pure Appl. Math. 28 (1975), no. 3, 333-354.
  • Grigor'yan, A.: 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.
  • Grigor'yan, A. and Yau, S.T.: Isoperimetric properties of higher eigenvalues of elliptic operators. Amer. J. Math. 125 (2003), no. 4, 893-940.
  • Greene, R.E. and Wu, H.: Function theory on manifolds which possess a pole. Lecture Notes in Mathematics 699. Springer, Berlin, 1979.
  • Holopainen, I.: A sharp $L^q$-Liouville theorem for $p-$harmonic functions. Israel J. Math. 115 (2000), 363-379.
  • Holopainen, I.: Volume growth, Green's function, and parabolicity of ends. Duke Math. J. 97 (1999), no. 2, 319-346.
  • Karp, L.: Differential inequalities on complete Riemannian manifolds and applications. Math. Ann. 272 (1985), no. 4, 449-459.
  • Pigola, S., Rigoli, M. and Setti, A.G.: A remark on the maximum principle and stochastic completeness. Proc. Amer. Math. Soc. 131 (2003), no. 4, 1283-1288.
  • Pigola, S., Rigoli, M. and Setti, A.G.: Volume growth, ``a priori'' estimates and geometric applications. Geom. Funct. Anal. 13 (2003), no. 6, 1302-1328.
  • Pucci, P. and Serrin, J.: A note on the strong maximum principle for elliptic differential inequalities. J. Math. Pures Appl. 79 (2000), 57-71.
  • Pucci, P., Serrin, J. and Zou, H.: A strong maximum principle and a compact support principle for singular elliptic inequalities. J. Math. Pures Appl. 78 (1999), 769-789.
  • Ratto, A., Rigoli, M. and Setti, A.G.: On the Omori-Yau maximum principle and its applications to differential equations and geometry. J. Funct. Anal. 134 (1995), no. 2, 486-510.
  • Rigoli, M. and Setti, A.G.: Liouville type theorems for $\varphi$-subharmonic functions. Rev. Mat. Iberoamericana 17 (2001), no. 3, 471-520.
  • Rigoli, M., Salvatori, M. and Vignati, M.: A Liouville type theorem for a general class of differential operators on complete manifolds. Pacific J. Math. 194 (2000), no. 2, 439-453.
  • Serrin, J.: Entire solutions to nonlinear Poisson equations. Proc. London Math. Soc. (3) 24 (1972), 348-366.
  • Sturm, K.T.: Analysis on local Dirichlet spaces I. Recurrence, conservativeness and $L^p$-Liouville properties. J. Reine Angew. Math. 456 (1994), 173-196.
  • Yau, S.T.: A general Schwarz lemma for Kähler manifolds. Amer. J. Math. 100 (1978), no. 1, 197-203.