Skip to main content
SpringerLink
Log in

Multi-point Maximum Principles and Eigenvalue Estimates

  • Chapter
  • First Online:
2018 MATRIX Annals

Part of the book series: MATRIX Book Series ((MXBS,volume 3))

  • 545 Accesses

Abstract

Estimates on modulus of continuity, isoperimetric profiles of variouskinds, and quantities involving function values at several points have been central in several recent results in geometric analysis. In these lectures I will focus mostly on the applications to partial differential equations, and to estimates on eigenvalues. These lectures were presented at the MATRIX program on “Recent trends on Nonlinear PDE of Elliptic and Parabolic type” at Creswick, November 5–16, 2018.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 99.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 129.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 129.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Andrews, B.: Moduli of continuity, isoperimetric profiles, and multi-point estimates in geometric heat equations. In: Surveys in differential geometry 2014. Regularity and evolution of nonlinear equations, Surv. Differ. Geom., vol. 19, pp. 1–47. Int. Press, Somerville, MA (2015).

    Google Scholar 

  2. Andrews, B., Clutterbuck, J.: Lipschitz bounds for solutions of quasilinear parabolic equations in one space variable. J. Differential Equations 246(11), 4268–4283 (2009).

    Google Scholar 

  3. Andrews, B., Clutterbuck, J.: Time-interior gradient estimates for quasilinear parabolic equations. Indiana Univ. Math. J. 58(1), 351–380 (2009).

    Google Scholar 

  4. Andrews, B., Clutterbuck, J.: Proof of the fundamental gap conjecture. J. Amer. Math. Soc. 24(3), 899–916 (2011).

    Google Scholar 

  5. Andrews, B., Clutterbuck, J.: Sharp modulus of continuity for parabolic equations on manifolds and lower bounds for the first eigenvalue. Anal. PDE 6(5), 1013–1024 (2013).

    Google Scholar 

  6. Andrews, B., Cluuterbuck, J., Hauer, D.: Non-concavity of the robin ground state (2019), https://arxiv.org/abs/1711.02779

  7. Andrews, B., Li, H.: Embedded constant mean curvature tori in the three-sphere. J. Differential Geom. 99(2), 169–189 (2015).

    Google Scholar 

  8. Andrews, B., Ni, L.: Eigenvalue comparison on Bakry-Emery manifolds. Comm. Partial Differential Equations 37(11), 2081–2092 (2012).

    Google Scholar 

  9. Andrews, B., Xiong, C.: Gradient estimates via two-point functions for elliptic equations on manifolds. Adv. Math. 349, 1151–1197 (2019).

    Google Scholar 

  10. Ashbaugh, M.S., Benguria, R.: Optimal lower bound for the gap between the first two eigenvalues of one-dimensional Schrӧdinger operators with symmetric single-well potentials. Proc. Amer. Math. Soc. 105(2), 419–424 (1989).

    Google Scholar 

  11. Bakry, D., Qian, Z.: Some new results on eigenvectors via dimension, diameter, and Ricci curvature. Adv. Math. 155(1), 98–153 (2000).

    Google Scholar 

  12. van den Berg, M.: On condensation in the free-boson gas and the spectrum of the Laplacian. J. Statist. Phys. 31(3), 623–637 (1983).

    Google Scholar 

  13. Brascamp, H.J., Lieb, E.H.: On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation. J. Functional Analysis 22(4), 366–389 (1976).

    Google Scholar 

  14. Brendle, S.: Embedded minimal tori in S3 and the Lawson conjecture. Acta Math. 211(2), 177–190 (2013).

    Google Scholar 

  15. Caffarelli, L., Garofalo, N., Segàla, F.: A gradient bound for entire solutions of quasi-linear equations and its consequences. Comm. Pure Appl. Math. 47(11), 1457–1473 (1994).

    Google Scholar 

  16. Chen, M.,Wang, F.: Application of coupling method to the first eigenvalue on manifold. Progr. Natur. Sci. (English Ed.) 5(2), 227–229 (1995)

    Google Scholar 

  17. Chen, M.F., Wang, F.Y.: Application of coupling method to the first eigenvalue on manifold. Sci. China Ser. A 37(1), 1–14 (1994)

    Google Scholar 

  18. Esposito, L., Nitsch, C., Trombetti, C.: Best constants in Poincaré inequalities for convex domains. J. Convex Anal. 20(1), 253–264 (2013)

    Google Scholar 

  19. Farina, A., Valdinoci, E.: A pointwise gradient bound for elliptic equations on compact manifolds with nonnegative Ricci curvature. Discrete Contin. Dyn. Syst. 30(4), 1139–1144 (2011).

    Google Scholar 

  20. Farina, A., Valdinoci, E.: Gradient bounds for anisotropic partial differential equations. Calc. Var. Partial Differential Equations 49(3–4), 923–936 (2014).

    Google Scholar 

  21. He, C., We, G.: Fundamental gap of convex domains in the spheres (2017), https://arxiv.org/pdf/1705.11152.pdf

  22. Krӧger, P.: On the ranges of eigenfunctions on compact manifolds. Bull. London Math. Soc. 30(6), 651–655 (1998).

    Google Scholar 

  23. Lavine, R.: The eigenvalue gap for one-dimensional convex potentials. Proc. Amer. Math. Soc. 121(3), 815–821 (1994).

    Google Scholar 

  24. Li, P.: A lower bound for the first eigenvalue of the Laplacian on a compact manifold. Indiana Univ. Math. J. 28(6), 1013–1019 (1979).

    Google Scholar 

  25. Li, P., Yau, S.T.: Estimates of eigenvalues of a compact Riemannian manifold. In: Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979), Proc. Sympos. Pure Math., XXXVI, pp. 205–239. Amer. Math. Soc., Providence, R.I. (1980)

    Google Scholar 

  26. Li, X., Wang, K.: Moduli of continuity for viscosity solutions on manifolds. J. Geom. Anal. 27(1), 557–576 (2017).

    Google Scholar 

  27. Modica, L.: A gradient bound and a Liouville theorem for nonlinear Poisson equations. Comm. Pure Appl. Math. 38(5), 679–684 (1985).

    Google Scholar 

  28. Naber, A., Valtorta, D.: Sharp estimates on the first eigenvalue of the p-Laplacian with negative Ricci lower bound. Math. Z. 277(3–4), 867–891 (2014).

    Google Scholar 

  29. Payne, L.E., Philippin, G.A.: Some maximum principles for nonlinear elliptic equations in divergence form with applications to capillary surfaces and to surfaces of constant mean curvature. Nonlinear Anal. 3(2), 193–211 (1979).

    Google Scholar 

  30. Payne, L.E., Weinberger, H.F.: An optimal Poincaré inequality for convex domains. Arch. Rational Mech. Anal. 5, 286–292 (1960) (1960).

    Google Scholar 

  31. Seto, S., Wang, L., Wei, G.: Sharp fundamental gap estimate on convex domains of sphere. J. Differential Geom. 112(2), 347–389 (2019). https://doi.org/10.4310/jdg/1559786428. URL https://doi.org.virtual.anu.edu.au/10.4310/jdg/1559786428

  32. Singer, I.M., Wong, B., Yau, S.T., Yau, S.S.T.: An estimate of the gap of the first two eigenvalues in the Schrӧdinger operator. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 12(2), 319–333 (1985).

    Google Scholar 

  33. Sperb, R.P.: Maximum principles and their applications, Mathematics in Science and Engineering, vol. 157. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York London (1981)

    Google Scholar 

  34. Valtorta, D.: Sharp estimate on the first eigenvalue of the p-Laplacian. Nonlinear Anal. 75(13), 4974–4994 (2012).

    Google Scholar 

  35. Yu, Q.H., Zhong, J.Q.: Lower bounds of the gap between the first and second eigenvalues of the Schrӧdinger operator. Trans. Amer. Math. Soc. 294(1), 341–349 (1986).

    Google Scholar 

  36. Zhong, J.Q., Yang, H.C.: On the estimate of the first eigenvalue of a compact Riemannian manifold. Sci. Sinica Ser. A 27(12), 1265–1273 (1984)

    Google Scholar 

Download references

Acknowledgements

This survey describes work supported by Discovery Projects grants DP0985802, DP120102462, and DP120100097, and Laureate Fellowship FL150100126 of the Australian Research Council.

Author information

Authors and Affiliations

  1. Mathematical Sciences Institute, Australia National University, Canberra, Australia

    Ben Andrews

Authors
  1. Ben Andrews
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ben Andrews .

Editor information

Editors and Affiliations

  1. School of Mathematics and Statistics, University of Melbourne, Parkville, VIC, Australia

    Jan de Gier

  2. Department of Mathematics and Statistics, The University of Western Australia, Perth, WA, Australia

    Cheryl E. Praeger

  3. Department of Mathematics, University of California Los Angeles, Los Angeles, CA, USA

    Terence Tao

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Andrews, B. (2020). Multi-point Maximum Principles and Eigenvalue Estimates. In: de Gier, J., Praeger, C., Tao, T. (eds) 2018 MATRIX Annals. MATRIX Book Series, vol 3. Springer, Cham. https://doi.org/10.1007/978-3-030-38230-8_13

Download citation

Publish with us

Policies and ethics

Search

Navigation