# Neumann eigenvalue

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

Consider a bounded domain with a piecewise smooth boundary . A number is a Neumann eigenvalue of if there exists a function (a Neumann eigenfunction) satisfying the following Neumann boundary value problem (cf. also Neumann boundary conditions): (a1) (a2)

where is the Laplace operator (i.e., ). For more general definitions, see [a8]. Neumann eigenvalues (with ) appear naturally when considering the vibrations of a free membrane (cf. also Natural frequencies). In fact, for the non-zero Neumann eigenvalues are proportional to the square of the eigenfrequencies of the membrane with free boundary. Provided is bounded and the boundary is sufficiently regular, the Neumann Laplacian has a discrete spectrum of infinitely many non-negative eigenvalues with no finite accumulation point: (a3)

( as ). The Neumann eigenvalues are characterized by the max-min principle [a3]: (a4)

where the is taken over all orthogonal to , and the is taken over all the choices of . For simply-connected domains the first eigenfunction , corresponding to the eigenvalue is constant throughout the domain. All the other eigenvalues are positive. While Dirichlet eigenvalues satisfy stringent constraints (e.g., cannot exceed for any bounded domain in , [a1]; see also Dirichlet eigenvalue), no such constraints exist for Neumann eigenvalues, other than the fact that they are non-negative. In fact, given any finite sequence , there is an open, bounded, smooth, simply-connected domain of having this sequence as the first Neumann eigenvalues of the Laplacian on that domain [a2]. Though it is obvious from the variational characterization of both Dirichlet and Neumann eigenvalues (see (a4)) that , L. Friedlander [a4] proved the stronger result (a5)

How far the first non-trivial Neumann eigenvalue is from zero for a convex domain in is given through the optimal inequality [a7] (a6)

where is the diameter of the domain. There are many more isoperimetric inequalities for Neumann eigenvalues (see Rayleigh–Faber–Krahn inequality).

For large values of , H. Weyl proved [a9] (a7)

where and are, respectively, the volumes of and of the unit ball in .

For any plane-covering domain (i.e., a domain that can be used to tile the plane without gaps, nor overlaps, allowing rotations, translations and reflections of itself), G. Pólya [a6] proved that (a8)

and conjectured the same bound for any bounded domain in . This is equivalent to saying that the Weyl asymptotics of is an upper bound for . The analogous conjecture in dimension is (a9)

The most significant result towards the proof of Pólya's conjecture for Neumann eigenvalues is the result by P. Kröger [a5]: A proof of Pólya's conjecture for both Dirichlet and Neumann eigenvalues would imply Friedlander's result (a5).

How to Cite This Entry:
Neumann eigenvalue. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Neumann_eigenvalue&oldid=12806
This article was adapted from an original article by Rafael D. Benguria (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article