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 | References | Similar Articles | Additional Information

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 in is contained in a long and thin cylinder with nonempty intersections with and , then the first nonzero eigenvalue is well approximated by the eigenvalue of an ordinary differential equation, by a bound proportional to , 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 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:
https://doi.org/10.1090/S0002-9947-09-04729-1

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