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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Locating the first nodal set in higher dimensions


Authors: Sunhi Choi, David Jerison and Inwon Kim
Journal: Trans. Amer. Math. Soc. 361 (2009), 5111-5137
MSC (2000): Primary 35J25; Secondary 35J05
Published electronically: May 5, 2009
MathSciNet review: 2515805
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Abstract: We extend the two-dimensional results of Jerison (2000) on the location of the nodal set of the first Neumann eigenfunction of a convex domain to higher dimensions. If a convex domain $ \Omega$ in $ \mathbb{R}^n$ is contained in a long and thin cylinder $ [0,N] \times B_{\epsilon}(0)$ with nonempty intersections with $ \{x_1= 0\}$ and $ \{x_1=N\}$, then the first nonzero eigenvalue is well approximated by the eigenvalue of an ordinary differential equation, by a bound proportional to $ \epsilon$, whose coefficients are expressed in terms of the volume of the cross sections of the domain. Also, the first nodal set is located within a distance comparable to $ \epsilon$ near the zero of the corresponding ordinary differential equation.


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Additional Information

Sunhi Choi
Affiliation: Department of Mathematics, University of Arizona, Tucson, Arizona 85721
Email: schoi@math.arizona.edu

David Jerison
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Email: jerison@math.mit.edu

Inwon Kim
Affiliation: Department of Mathematics, University of California, Los Angeles, Los Angeles, California 90095
Email: ikim@math.ucla.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04729-1
PII: S 0002-9947(09)04729-1
Keywords: Convex domains, eigenfunctions
Received by editor(s): March 21, 2007
Published electronically: May 5, 2009
Additional Notes: The first author was partially supported by NSF grant DMS 0713598.
The second author was partially supported by NSF grant DMS 0244991.
The third author was partially supported by NSF grant DMS 0627896
Article copyright: © Copyright 2009 American Mathematical Society