Skip to main content
SpringerLink
Log in

Hot spots conjecture for a class of acute triangles

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

We show that the hot spots conjecture of J. Rauch holds for acute triangles if one of the angles is not larger than \(\pi /6\). More precisely, we show that the second Neumann eigenfunction on those acute triangles has no maximum or minimum inside the domain. We first simplify the problem by showing that absence of critical points on two sides implies no critical points inside a triangle. This result applies to any acute triangle and might help prove the conjecture for arbitrary acute triangles. Then we show that there are no critical points on two sides assuming one small angle. We also establish simplicity for the smallest positive Neumann eigenvalue for all non-equilateral acute triangles. This result was already known for obtuse triangles, and it fails for the equilateral case.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. Atar, R., Burdzy, K.: On Neumann eigenfunctions in lip domains. J. Am. Math. Soc. 17(2), 243–265 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bañuelos, R., Burdzy, K.: On the “hot spots” conjecture of J. Rauch. J. Funct. Anal. 164(1), 1–33 (1999)

    Article  MathSciNet  Google Scholar 

  3. Burdzy, K., Werner, W.: A counterexample to the “hot spots” conjecture. Ann. Math. 149(1), 309–317 (1999)

    Article  MathSciNet  Google Scholar 

  4. Burdzy, K.: The hot spots problem in planar domains with one hole. Duke Math. J. 129(3), 481–502 (2005)

    Article  MathSciNet  Google Scholar 

  5. Freitas, P., Siudeja, B.: Bounds for the first Dirichlet eigenvalue of triangles and quadrilaterals. ESAIM Control Optim. Calc. Var. 16(3), 648–676 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Hooker, W., Protter, M.H.: Bounds for the first eigenvalue of a rhombic membrane. J. Math. Phys. 39, 18–34 (1960/1961)

  7. Jerison, D., Nadirashvili, N.: The “hot spots” conjecture for domains with two axes of symmetry. J. Am. Math. Soc. 13(4), 741–772 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  8. Kawohl, B.: Rearrangements and convexity of level sets in PDE. Lecture Notes in Mathematics, vol. 1150 (Springer, Berlin, 1985)

  9. Laugesen, R.S., Siudeja, B.A.: Maximizing Neumann fundamental tones of triangles. J. Math. Phys. 50(11), 112903–112918 (2009)

    Article  MathSciNet  Google Scholar 

  10. Laugesen, R.S., Siudeja, B.A.: Minimizing Neumann fundamental tones of triangles: an optimal Poincaré inequality. J. Differ. Equ. 249(1), 118–135 (2010)

    Article  MathSciNet  Google Scholar 

  11. Laugesen, R.S., Siudeja, B.A.: Dirichlet eigenvalue sums on triangles are minimal for equilaterals. Comm. Anal. Geom. 19(5), 855–885 (2011)

    Article  MathSciNet  Google Scholar 

  12. Levine, H.A., Weinberger, H.F.: Inequalities between Dirichlet and Neumann eigenvalues. Arch. Ration. Mech. Anal. 94(3), 193–208 (1986)

    Article  MathSciNet  Google Scholar 

  13. McCartin, B.J.: Eigenstructure of the equilateral triangle. II. The Neumann problem. Math. Probl. Eng. 8(6), 517–539 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  14. McCartin, B.J.: On polygonal domains with trigonometric eigenfunctions of the Laplacian under Dirichlet or Neumann boundary conditions. Appl. Math. Sci. (Ruse) 2(57–60), 2891–2901 (2008)

    MathSciNet  MATH  Google Scholar 

  15. Miyamoto, Y.: The “hot spots” conjecture for a certain class of planar convex domains. J. Math. Phys. 50(10), 103530 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Miyamoto, Y.: A planar convex domain with many isolated “hot spots” on the boundary. Jpn. J. Ind. Appl. Math. 30(1), 145–164 (2013)

    Article  MathSciNet  Google Scholar 

  17. Pascu, M.N.: Scaling coupling of reflecting Brownian motions and the hot spots problem. Trans. Am. Math. Soc. 354(11), 4681–4702 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  18. Polymath7 project: The hot spots conjecture, http://michaelnielsen.org/polymath1/index.php?title=The_hot_spots_conjecture

  19. Rauch, J.: Five problems: an introduction to the qualitative theory of partial differential equations, Partial differential equations and related topics (Program, Tulane Univ., New Orleans, LA, 1974), Lecture Notes in Math., vol. 446. (Springer, Berlin, 1975), pp. 355–369

  20. Siudeja, B.: Isoperimetric inequalities for eigenvalues of triangles. Indiana Univ. Math. J. 59(3), 1097–1120 (2010)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

The work was partially supported by NCN Grant 2012/07/B/ST1/03356.

The author is grateful to Richard Laugesen for invaluable discussions on the topic of the paper, as well as suggested improvements to some arguments.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Oregon, Eugene, OR, 97403, USA

    Bartłomiej Siudeja

Authors
  1. Bartłomiej Siudeja
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bartłomiej Siudeja.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Siudeja, B. Hot spots conjecture for a class of acute triangles. Math. Z. 280, 783–806 (2015). https://doi.org/10.1007/s00209-015-1448-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-015-1448-1

Keywords

Search

Navigation