SpringerLink
Log in

The Compact Support Principle

  • Chapter
  • First Online:
Geometric Analysis of Quasilinear Inequalities on Complete Manifolds

Part of the book series: Frontiers in Mathematics ((FM))

  • 453 Accesses

Abstract

Consider the problem

$$\displaystyle \left \{ \begin {array}{l} \Delta _\varphi u \ge b(x)f(u)l(|\nabla u|) \qquad \text{on } \, \Omega \, \text{ end of }M, \\[0.2cm] \displaystyle u \ge 0, \qquad \lim _{x \in \Omega , \, x \rightarrow \infty } u(x) = 0. \end {array}\right . $$

We recall that an end Ω ⊂ M is a connected component with non-compact closure of M∖K, for some compact set K.

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 44.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 59.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free ship** worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. P. Antonini, D. Mugnai, P. Pucci, Quasilinear elliptic inequalities on complete Riemannian manifolds. J. Math. Pures Appl. 87, 582–600 (2007)

    Article  MathSciNet  Google Scholar 

  2. R. Azencott, Behavior of diffusion semi-groups at infinity. Bull. Soc. Math. France 102, 193–240 (1974)

    MathSciNet  MATH  Google Scholar 

  3. D. Bianchi, A.G. Setti, Laplacian cut-offs, porous and fast diffusion on manifolds and other applications. Calc. Var. Partial Differ. Equ. 57, 1 (2018), 33 pp. Art. 4

    Google Scholar 

  4. E.B. Davies, Heat kernel bounds, conservation of probability and the Feller property. Festschrift on the occasion of the 70th birthday of Shmuel Agmon. J. Anal. Math. 58, 99–119 (1992)

    Google Scholar 

  5. J. Dodziuk, Maximum principle for parabolic inequalities and the heat flow on open manifolds. Indiana Univ. Math. J. 32(5), 703–716 (1983)

    Article  MathSciNet  Google Scholar 

  6. P. Felmer, M. Montenegro, A. Quaas, A note on the strong maximum principle and the compact support principle. J. Differ. Equ. 246, 39–49 (2009)

    MathSciNet  MATH  Google Scholar 

  7. E.-P. Hsu, Heat semigroup on a complete Riemannian manifold. Ann. Probab. 17(3), 1248–1254 (1989)

    Article  MathSciNet  Google Scholar 

  8. E.-P. Hsu, Stochastic Analysis on Manifolds. Graduate Studies in Mathematics, vol. 38 (American Mathematical Society, New York, 2002)

    Google Scholar 

  9. T. Kura, The weak supersolution-subsolution method for second order quasilinear elliptic equations. Hiroshima Math. J. 19(1), 1–36 (1989)

    Article  MathSciNet  Google Scholar 

  10. P. Li, L. Karp, The heat equation on complete Riemannian manifolds. Unpublished

    Google Scholar 

  11. S. Pigola, A.G. Setti, The Feller property on Riemannian manifolds. J. Funct. Anal. 262(5), 2481–2515 (2012)

    MathSciNet  MATH  Google Scholar 

  12. P. Pucci, J. Serrin, Dead cores and bursts for quasilinear singular elliptic equations. SIAM J. Math. Anal. 38, 259–278 (2006)

    Article  MathSciNet  Google Scholar 

  13. P. Pucci, J. Serrin, The Maximum Principle. Progress in Nonlinear Differential Equations and their Applications, vol. 73 (Birkhäuser Verlag, Basel, 2007), x+235 pp.

    Google Scholar 

  14. P. Pucci, J. Serrin, H. Zou, A strong maximum principle and a compact support principle for singular elliptic inequalities. J. Math. Pures Appl. 78, 769–789 (1999)

    MathSciNet  MATH  Google Scholar 

  15. P. Pucci, M. Rigoli, J. Serrin, Qualitative properties for solutions of singular elliptic inequalities on complete manifolds. J. Differ. Equ. 234(2), 507–543 (2007)

    MathSciNet  MATH  Google Scholar 

  16. M. Rigoli, M. Salvatori, M. Vignati, On the compact support principle on complete manifolds. J. Differ. Equ. 246, 870–894 (2009)

    MathSciNet  MATH  Google Scholar 

  17. M. Rigoli, M. Salvatori, M. Vignati, The compact support principle for differential inequalities with gradient terms. Nonlinear Anal. 72, 4360–4376 (2010)

    MathSciNet  MATH  Google Scholar 

  18. R. Schoen, S.-T. Yau, Lectures on Differential Geometry, vol. 1 (International Press Cambridge, Cambridge, 1994)

    MATH  Google Scholar 

  19. P. Tolksdorf, Regularity of a more general class of quasilinear elliptic equations. J. Differ. Equ. 51, 126–150 (1984)

    Article  MathSciNet  Google Scholar 

  20. X. Wang, L. Zhang, Local gradient estimate for p-harmonic functions on Riemannian manifolds. Commun. Anal. Geom. 19(4), 759–771 (2011)

    Article  MathSciNet  Google Scholar 

  21. S.T. Yau, Harmonic functions on complete Riemannian manifolds. Commun. Pure Appl. Math. 28, 201–228 (1975)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica Pura e Applicata, Università degli Studi di Padova, Padova, Italy

    Bruno Bianchini

  2. Dipartimento di Matematica, Università degli Studi di Torino, Torino, Italy

    Luciano Mari

  3. Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, Perugia, Italy

    Patrizia Pucci

  4. Dipartimento di Matematica, Università degli Studi di Milano, Milano, Italy

    Marco Rigoli

Authors
  1. Bruno Bianchini
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Luciano Mari
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Patrizia Pucci
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Marco Rigoli
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Bianchini, B., Mari, L., Pucci, P., Rigoli, M. (2021). The Compact Support Principle. In: Geometric Analysis of Quasilinear Inequalities on Complete Manifolds. Frontiers in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-62704-1_9

Download citation

Publish with us

Policies and ethics

Search

Navigation