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Nodal inequalities on surfaces

Published online by Cambridge University Press:  01 September 2007

LEONID POLTEROVICH  and
MIKHAIL SODIN
LEONID POLTEROVICH
Affiliation:
School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel. email: polterov@post.tau.ac.il, sodin@post.tau.ac.il
MIKHAIL SODIN
Affiliation:
School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel. email: polterov@post.tau.ac.il, sodin@post.tau.ac.il
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Abstract

Given a Laplace eigenfunction on a surface, we study the distribution of its extrema on the nodal domains. It is classically known that the absolute value of the eigenfunction is asymptotically bounded by the 4th root of the eigenvalue. It turns out that the number of nodal domains where the eigenfunction has an extremum of such order, remains bounded as the eigenvalue tends to infinity. We also observe that certain restrictions on the distribution of nodal extrema and a version of the Courant nodal domain theorem are valid for a rather wide class of functions on surfaces. These restrictions follow from a bound in the spirit of Kronrod and Yomdin on the average number of connected components of level sets.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 143 , Issue 2 , September 2007 , pp. 459 - 467
Copyright
Copyright © Cambridge Philosophical Society 2007

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References

REFERENCES

[1]Agranovich, M. S.. Elliptic boundary problems. Encyclopaedia Math. Sci. 79, Partial differential equations, IX, 1–144 (1997).CrossRefGoogle Scholar
[2]Arnold, V. I.. On the topology of the eigenfields. Topol. Methods Nonlinear Anal. 26 (2005), 9–16.Google Scholar
[3]Cabré, X.. On the Alexandroff–Bakelman–Pucci estimate and the reversed Hölder inequality for solutions of elliptic and parabolic equations. Comm. Pure Appl. Math. 48 (1995) no. 5, 539–570.CrossRefGoogle Scholar
[4]Chavel, I.. Eigenvalues in Riemannian geometry. Pure Appl. Math. 115 (1984).Google Scholar
[5]Gladwell, G. M. L. and Zhu, Hongmei. The Courant–Herrmann conjecture. Z. Angew. Math. Mech. 83 (2003), 275–281.CrossRefGoogle Scholar
[6]Kronrod, A. S.. On functions of two variables. (Russian) Uspehi Matem. Nauk (N.S.) 35 (1950), 24–134.Google Scholar
[7]Ladyzhenskaya, O. A.. The boundary value problems of mathematical physics. Appl. Math. Sci. 49 (1985).Google Scholar
[8]Mangoubi, D.. On the inner radius of nodal domains. arXiv math.SP/0511329. to appear in Canad.Math.Bull.Google Scholar
[9]Smith, H. F. and Sogge, Chr. D.. On the Lp norm of spectral clusters for compact manifolds with boundary. arXiv math.AP/0605682. Acta Math. 198 (2007), 107–153.CrossRefGoogle Scholar
[10]Sogge, Chr. D.. Fourier integrals in classical analysis. Cambridge Tracts in Math. 105 (1993).Google Scholar
[11]Yomdin, Y.. Global bounds for the Betti numbers of regular fibers of differentiable mappings. Topology 24 (1985), 145–152.CrossRefGoogle Scholar