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 language | English (US) |
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Pages (from-to) | 5111-5137 |
Number of pages | 27 |
Journal | Transactions of the American Mathematical Society |
Volume | 361 |
Issue number | 10 |
DOIs | |
State | Published - Oct 2009 |
Keywords
- Convex domains
- Eigenfunctions
ASJC Scopus subject areas
- General Mathematics
- Applied Mathematics