NEUMANN EIGENVALUE– Consider a bounded domain Ω ⊂ R^{n} with a piecewise smooth boundary ∂Ω. μ is a Neumann eigenvalue of Ω is there exists a function u ∈ C^{2}(Ω)∩C^{0}(Ω̄) (Neumann eigenfunction) satisfying the following Neumann boundary value problem

−Δu = μu  in Ω  (1)

and

\frac{∂u}{∂n} = 0  in ∂Ω,  (2)

where Δ is the Laplace operator (i.e., Δ = ∑_{i=1}^{n} ∂^{2}/∂x_{i}^{2}). For more general definitions see [8]. Neumann eigenvalues (with n = 2) appear naturally when considering the vibrations of a free membrane In fact, for n = 2, the nonzero Neumann eigenvalues are proportional to the square of the eigenfrequencies of the membrane with free boundary. Provided Ω is bounded and the boundary ∂Ω is sufficiently regular, the Neumann Laplacian has a discrete spectrum of infinitely many nonnegative eigenvalues with no finite accumulation point

0 = μ_{1}(Ω) ≤ μ_{2}(Ω) ≤ μ_{3}(Ω) ≤ …  (3)

(μ_{k} → ∞ as k → ∞). The Neumann eigenvalues are characterized by the max–min principle [3]

μ_{k} = sup inf \frac{∫_{Ω}(∇u)^{2} dx}{∫_{Ω} u^{2} dx},  (4)

where the inf is taken over all u ∈ H^{1}(Ω) orthogonal to φ_{1}, φ_{2}, …, φ_{k−1} ∈ H^{1}(Ω), and the sup is taken over all the choices of {φ_{i}}_{i=1}^{k−1}. For simply connected domains the first eigenfunction u_{1}, corresponding to the eigenvalue μ_{1} = 0 is constant throughout the domain. All the other eigenvalues are positive. While Dirichlet Eigenvalues satisfy stringent constraints (e. g., λ_{2}/λ_{1} cannot exceed 2.539 … for any bounded domain in R^{2}, [1]), no such constraints exist for Neumann Eigenvalues, other than the fact that they are nonnegative. In fact, given any finite sequence {μ_{1} = 0 < μ_{2} < μ_{3} < … < μ_{N}} there is an open, bounded, smooth, simply connected domain of R^{2} having this sequence as the first N Neumann Eigenvalues of the Laplacian on that domain [2]. Though it is obvious from the variational characterization of both Dirichlet Eigenvalues and Neumann Eigenvalues (Eqn. (4)) that μ_{k} ≤ λ_{k}, Friedlander [4] proved the stronger result,

μ_{k+1} ≤ λ_{k},  k = 1, 2, …  (5)

How far is the first nontrivial Neumann eigenvalue from zero for a convex domain in R^{2} is given through the optimal inequality [6]

μ_{1} ≥ \frac{π^{2}}{d^{2}} ,  (6)

where d is the diameter of the domain. There are many more isoperimetric inequalities for Neumann Eigenvalues (see the article on the Rayleigh-Faber-Krahn Inequality).

For large values of k, Weyl poved [9].

μ_{k+1} ≈ \frac{4π^{2}k^{2/n}}{(C_{n}|Ω|)^{2/n}} ,  (7)

where |Ω| and C_{n} = π^{n/2}/Γ(n/2+1) are, respectively, the volumes of Ω and of the unit ball in R^{n}.

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), Pólya [7] proved that

μ_{k+1} ≤ \frac{4πk}{A}  for k = 0, 1, 2, …,  (8)

and conjectured the same bound for any bounded domain in R^{2}. This is equivalent to saying that the Weyl asymptotics of μ_{k} is an upper bound for μ_{k}. The analogous conjecture in dimension n is

μ_{k+1} ≤ \frac{4π^{2}k^{2/n}}{(C_{n}|Ω|)^{2/n}}  for k = 0, 1, 2, . . .. (9)

The most significant result towards the proof of Pólya’s conjecture for Neumann Eigenvalues is the result by Kröger [5]

∑_{i=1}^{k} μ_{i} ≤ \frac{n}{n + 2} \frac{4π^{2}k^{2/n}}{(C_{n}|Ω|)^{2/n}}  for k = 1, 2, . . .. (10)

A proof of Pólya’s conjecture for both Dirichlet and Neumann eigenvalues would imply Friedlander’s result (5).

References

[1] ASHBAUGH, M.S., AND BENGURIA, R.D.: A sharp bound for the ratio of the first two eigenvalues of Dirichlet Laplacians and extensions, Ann. of Math, 135 (1992), 601–628.

[2] COLIN DE VÉRDIERE, YVES.: Construction de laplaciens dont une partie finie du spectre est donnée. Ann. scient. Éc. Norm Sup. (4) 20 (1987), 599–615.

[3] COURANT R., AND HILBERT, D.: Methoden der mathematischen Physik, vol. I, Springer, Berlin, 1931. (English edition: Methods of Mathematical Physics, vol. I., Interscience, NY, 1953).

[4] FRIEDLANDER, L.: Some inequalities between Dirichlet and Neumann eigenvalues, Arch. Rational Mech. Anal. 116 (1991), 153–160.

[5] KRÖGER, P.: Upper bounds for the Neumann eigenvalues on a bounded domain in Euclidean Space, J. Funct. Anal. 106 (1992), 353–357.

[6] PAYNE, L.E., AND WEINBERGER, H.F.: An optimal Poincaré inequality for convex domains, Arch. Rational Mech. Anal. 5 (1960), 286–292.

[7] POLYA, G.: On the eigenvalues of vibrating membranes, Proc. London Math. Soc. (3) 11 (1961), 419–433.

[8] REED, M., AND SIMON, B.: Methods of Modern Mathematical Physics. IV: Analysis of Operators. Academic Press, New York, 1978.

[9] WEYL, H.: Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen, Math. Ann. 71 (1911), 441–479

Rafael D. Benguria Facultad de Física P. U. Católica de Chile