Locating the first nodal set in higher dimensions

Sunhi Choi, David Jerison, Inwon Kim

Research output: Contribution to journalArticle

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 Ω in ℝn is contained in a long and thin cylinder [0, N] × B(0) with nonempty intersections with {x1 = 0} and {x1 = N}, 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.

Original languageEnglish (US)
Pages (from-to)5111-5137
Number of pages27
JournalTransactions of the American Mathematical Society
Volume361
Issue number10
DOIs
Publication statusPublished - Oct 2009

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Keywords

  • Convex domains
  • Eigenfunctions

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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