Locating the first nodal set in higher dimensions
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- by Sunhi Choi, David Jerison and Inwon Kim PDF
- Trans. Amer. Math. Soc. 361 (2009), 5111-5137 Request permission
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.References
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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
- MR Author ID: 684869
- Email: ikim@math.ucla.edu
- 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 - © Copyright 2009 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 361 (2009), 5111-5137
- MSC (2000): Primary 35J25; Secondary 35J05
- DOI: https://doi.org/10.1090/S0002-9947-09-04729-1
- MathSciNet review: 2515805