Skip to main content
SpringerLink
Log in

Laplacian cut-offs, porous and fast diffusion on manifolds and other applications

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract

We construct exhaustion and cut-off functions with controlled gradient and Laplacian on manifolds with Ricci curvature bounded from below by a (possibly unbounded) nonpositive function of the distance from a fixed reference point, without any assumptions on the topology or the injectivity radius. Along the way we prove a generalization of the Li-Yau gradient estimate which is of independent interest. We then apply our cut-offs to the study of the fast and porous media diffusion, of \(L^q\)-properties of the gradient and of the self-adjointness of Schrödinger-type operators.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, vol. 55. Courier Corporation, New York (1964)

    MATH  Google Scholar 

  2. Abresch, U.: Lower curvature bounds, toponogov’s theorem, and bounded topology. Ann. Sci. École Norm. Sup. 18(4), 651–670 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  3. Aronson, D.G., Caffarelli, L.A.: The initial trace of a solution of the porous medium equation. Trans. Am. Math. Soc. 280(1), 351–366 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bénilan, P., Crandall, M.G., Pierre, M.: Solutions of the Porous Medium Equation in R(N) Under Optimal Conditions on Initial Values. Technical report, DTIC Document (1982)

  5. Bonforte, M., Grillo, G.: Asymptotics of the porous media equation via Sobolev inequalities. J. Funct. Anal. 225(1), 33–62 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bonforte, M., Grillo, G., Vazquez, J.L.: Fast diffusion flow on manifolds of nonpositive curvature. J. Evol. Equ. 8(1), 99–128 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Braverman, M., Milatovic, O., Shubin, M.: Essential self-adjointness of Schrödinger-type operators on manifolds. Russ. Math. Surv. 57(4), 641 (2002)

    Article  MATH  Google Scholar 

  8. Cheeger, J.: Degeneration of Riemannian metrics under Ricci curvature bounds. Accademia Nazionale dei Lincei. Scuola Normale Superiore, Lezione Fermiane (2001)

  9. Cheeger, J., Colding, T.H.: Lower bounds on Ricci curvature and the almost rigidity of warped products. Ann. Math. 144, 189–237 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chow, B., Chu, S.-C., Glickenstein, D., Guenther, C., Isenberg, J., Ivey, T., Knopf, D., Lu, P., Luo, F., Ni, L., The Ricci flow: techniques and applications. Part III. Geometric-analytic aspects. Mathematical Surveys and Monographs, vol. 163. American Mathematical Society, Providence, RI, (2010)

  11. Donnelly, H.: Exhaustion functions and the spectrum of Riemannian manifolds. Indiana Univ. Math. J. 46(2), 505–527 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gaffney, M.P.: The conservation property of the heat equation on Riemannian manifolds. Commun. Pure Appl. Math. 12(1), 1–11 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  13. Greene, R.E., Wu, H.: \(C^\infty \) approximations of convex, subharmonic, and plurisubharmonic functions. Ann. Sci. École Norm. Sup. 12, 47–84 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  14. Grillo, G., Muratori, M.: Radial fast diffusion on the hyperbolic space. Proc. Lond. Math. Soc. 109(2), 283–317 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Grillo, G., Muratori, M.: Smoothing effects for the porous medium equation on Cartan–Hadamard manifolds. Nonlinear Anal. Theory Methods Appl. 131, 346–362 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. Grillo, G., Muratori, M., Vazquez, J.L.: The porous medium equation on Riemannian manifolds with negative curvature. The large-time behaviour. Adv. Math. 314, 328–377 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  17. Grummt, R., Kolb, M.: Essential selfadjointness of singular magnetic Schrödinger operators on Riemannian manifolds. J. Math. Anal. Appl. 388(1), 480–489 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Güneysu, B.: Sequences of laplacian cut-off functions. J. Geom. Anal. 26(1), 171–184 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Herrero, M.A., Pierre, M.: The Cauchy problem for \(u_t=\delta u \) when \(0< m< 1\). Trans. Amer. Math. Soc. 291(1), 145–158 (1985)

    MathSciNet  MATH  Google Scholar 

  20. Kato, T.: Schrödinger operators with singular potentials. Isr. J. Math. 13(1–2), 135–148 (1972)

    Article  Google Scholar 

  21. Leinfelder, H., Simader, C.G.: Schrödinger operators with singular magnetic vector potentials. Math. Z. 176(1), 1–19 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  22. Li, P., Schoen, R.: \({L}_p\) and mean value properties of subharmonic functions on Riemannian manifolds. Acta Math. 153(1), 279–301 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lu, P., Ni, L., Vázquez, J.-L., Villani, C.: Local Aronson-Bénilan estimates and entropy formula for porous medium and fast diffusion equations on manifolds. J. Math. Pures Appl. 91(1), 1–19 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Pigola, S., Rigoli, M., Setti, A.G.: Vanishing and Finiteness Results in Geometric Analysis: A Generalization of the Bochner Technique, vol. 266. Springer, New York (2008)

    MATH  Google Scholar 

  25. Pigola, S., Rigoli, M., Setti, A.G.: Maximum Principles on Riemannian Manifolds and Applications, vol. 822. American Mathematical Society, Providence (2005)

    MATH  Google Scholar 

  26. Rimoldi, M., Veronelli, G.: Extremals of log Sobolev inequality on non-compact manifolds and Ricci soliton structures. ArXiv preprint arXiv:1605.09240 (2016)

  27. Schoen, R., Yau, S.-T.: Lectures on Differential Geometry. International Press, Cambridge (1994)

    MATH  Google Scholar 

  28. Shubin, M.: Essential self-adjointness for semi-bounded magnetic Schrödinger operators on non-compact manifolds. J. Funct. Anal. 186(1), 92–116 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  29. Shubin, M.A.: Spectral theory of elliptic operators on non-compact manifolds. Astérisque 207, 37–108 (1992)

    MATH  Google Scholar 

  30. Strichartz, R.S.: Analysis of the Laplacian on the complete Riemannian manifold. J. Funct. Anal. 52(1), 48–79 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  31. Vázquez, J.L.: Smoothing and Decay Estimates for Nonlinear Parabolic Equations of Porous Medium Type. Oxford Lecture Notes in Maths and Its Applications, vol. 33. Oxford University Press, Oxford (2006)

    Google Scholar 

  32. Vázquez, J.L.: The Porous Medium Equation: Mathematical Theory. Oxford University Press, Oxford (2007)

    MATH  Google Scholar 

  33. Vázquez, J.L.: Fundamental solution and long time behavior of the porous medium equation in hyperbolic space. J. Math. Pures Appl. 104(3), 454–484 (2015)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Scienza e Alta Tecnologia, Università dell’Insubria, via Valleggio 11, 22100, Como, Italy

    Davide Bianchi & Alberto G. Setti

Authors
  1. Davide Bianchi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alberto G. Setti
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alberto G. Setti.

Additional information

Communicated by A. Malchiodi.

Davide Bianchi: Member of INdAM-GNCS, partially supported by Prin 2012 “Structured Matrices in Signal and Image Processing”. Alberto G. Setti: Partially supported by INdAM-GNAMPA and Prin 2015 “Real and Complex Manifolds: Geometry, Topology and Harmonic Analysis”.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bianchi, D., Setti, A.G. Laplacian cut-offs, porous and fast diffusion on manifolds and other applications. Calc. Var. 57, 4 (2018). https://doi.org/10.1007/s00526-017-1267-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00526-017-1267-9

Mathematics Subject Classification

Search

Navigation