RAYLEIGH–FABER–KRAHN INEQUAL-ITY – Inequality concerning the lowest eigenvalue of the Laplacian, with Dirichlet boundary condition, on a bounded domain in R^{n} (n ≥ 2).

Let 0 < λ_{1}(Ω) < λ_{2}(Ω) ≤ λ_{3}(Ω) ≤ … be the Dirichlet eigenvalues of the Laplacian in Ω ⊂ R^{n}, i.e.,

−Δu = λu  in Ω,  (1)
u = 0  on the boundary of Ω.  (2)

Here Δ is the Laplace operator, Ω an open bounded subset of R^{n} (n ≥ 2). If n = 2, the Dirichlet eigenvalues are proportional to the square of the eigenfrequencies of an elastic, homogeneous, vibrating membrane with fixed boundary.

The Rayleigh–Faber–Krahn inequality for the membrane (i.e., n = 2) states that

λ_{1} ≥ \frac{πj_{0,1}^{2}}{A} ,  (3)

where j_{0,1} = 2.4048 … is the first zero of the Bessel function of order zero, and A is the area of the membrane. Equality is attained in (3) if and only if the membrane is circular. In other words, among all membranes of given area, the circle has the lowest fundamental frequency. This inequality was conjectured by Lord Rayleigh [14], based on exact calculations for simple domains, and a variational argument for near circular domains. In 1918, Courant [5] proved the weaker result that among all membranes of the same perimeter L, the circular one yields the least lowest eigenvalue, i.e.,

λ_{1} ≥ \frac{4π^{2}j_{0,1}^{2}}{L^{2}} ,  (4)

with equality if and only if the membrane is circular. Rayleigh’s conjecture was proven independently by Faber [6] and Krahn [7]. The corresponding isoperimetric inequality in dimension n,

λ_{1}(Ω) ≥ (\frac{1}{|Ω|})^{2/n} C_{n}^{2/n}j_{n/2−1,1},  (5)

was proven by Krahn [8]. In (5), j_{m,1} is the first positive zero of the Bessel function J_{m}, |Ω| is the volume of the domain and C_{n} = π^{n/2}/Γ(n/2 + 1) is the volume of the n–dimensional unit ball. Equality is attained in (5) if and only if Ω is a ball.

The proof of the Rayleigh–Faber–Krahn inequality rests upon two facts: a variational characterization for the lowest Dirichlet eigenvalue and the properties of symmetric decreasing rearrangements of functions. The variational characterization of the lowest eigenvalue is given by

λ_{1}(Ω) = inf_{u∈H_{0}^{1}(Ω)} \frac{∫_{Ω}(∇u)^{2} dx}{∫_{Ω} u^{2} dx}.  (6)

Concerning decreasing rearrangements, let Ω be a measurable subset of R^{n}, then the symmetrized domain Ω^{∗} is a ball with the same measure as Ω. If u is a real valued mesaurable function defined on a bounded domain Ω ⊂ R^{n}, its spherical decreasing rearrangement u^{⋆} is a function defined on the ball Ω^{∗} centered at the origin and having the same measure as Ω, such that u^{⋆} depends only on distance from the origin, is decreasing away from the origin and is equimeasurable with u. We refer to [13, 18, 4] for properties of rearrangements of functions. Since the function u and its spherical decreasing rearrangement are equimeasurable, their L^{2} norms are the same. What Faber and Krahn actually proved is that the L^{2} norm of the gradient of a function is decreased under rearrangements (see [18] for details, and also [9] for a different approach to this fact). The fact that the L^{2} norm of the gradient of a function decreases under rearrangements, combined with the variational characterization (6) immediately gives the Rayleigh– Faber–Krahn inequality.

There are several isoperimetric inequalities for the lowest eigenvalue of boundary value problems, similar to the Rayleigh–Faber–Krahn inequality. We survey a few of them in the sequel. The lowest non trivial Neumann eigenvalue also satisfies an isoperimetric inequality. Let 0 = μ_{1}(Ω) < μ_{2}(Ω) ≤ μ_{3}(Ω) ≤ … be the Neumann eigenvalues of the Laplacian in Ω ⊂ R^{n}, i.e.,

−Δu = μu  in Ω,  (7)

\frac{∂u}{∂n} = 0  on the boundary of Ω.  (8)

If n = 2, Szegö [17] proved

μ_{2}(Ω) ≤ \frac{πp_{1}^{2}}{A} ,  (9)

where p_{1} = 1.8412 …, with equality if and only if Ω is a circle. The corresponding result for dimension n,

μ_{2}(Ω) ≤ (\frac{1}{|Ω|})^{2/n} C_{n}^{2/n}p_{n/2,1}^{2},  (10)

was proven by Weinberger [19], with equality if and only if Ω is a ball. Here C_{n} is the volume of the unit ball in dimension n. In (9) and (10), p_{m,1} denotes the first positive zero of the derivative of the Bessel function J_{m}. For n = 2 Weinberger [19] also proved

\frac{1}{μ_{2}(Ω)} + \frac{1}{μ_{3}(Ω)} ≥ \frac{2A}{πp_{1}^{2}} ,  (11)

with equality if and only if Ω is a circle.

There is also an analog of the Rayleigh–Faber– Krahn inequality for domains in spaces of constant curvature [15]. The optimal Rayleigh–Faber– Krahn inequalities for domains in S^{n} was proven by Sperner [16].

In his book, The Theory of Sound, Lord Rayleigh also conjectured an isoperimetric inequality for the lowest eigenvalue, Λ_{1}, of the clamped plate. The eigenvalue problem for the clamped plate is given by

Δ^{2}u_{1} = Λ_{1}u_{1} in Ω

with

u_{1} = |\frac{∂u}{∂n}| = 0 in the boundary of Ω.

Here Ω is a bounded open subset of R^{2}. Rayleigh’s conjecture for the clamped plate reads

Λ_{1}(Ω) ≥ Λ_{1}(Ω^{∗}),  (12)

where Ω^{∗} is a ball of the same area as Ω. Rayleigh’s conjecture was proven by N. Nadirashvili [12]. Equality is attained in (12) if and only if Ω is a circle. For dimension 3, the corresponding isoperimetric inequality was proven by Ashbaugh and Benguria [2].

To prove the analogous result for dimensions 4 and higher is still an open problem (see [3] however).

Back in the membrane problem, if we go beyond the lowest eigenvalue, there are several isoperimetric inequalities as well as a number of open problems. The simplest combination λ_{2}/λ_{1}(Ω) satisfies the following inequality [1]

\frac{λ_{2}(Ω)}{λ_{1}(Ω)} ≤ \frac{j_{n/2,1}^{2}}{j_{n/2−1,1}^{2}} ,  (13)

in n dimensions, where equality is obtained if and only if Ω is a ball. Stability results for both, the Rayleygh–Faber–Krahn inequality (3), (4), and inequality (13) have been obtained by Melas [11] (in simple words stability means that if Ω is convex and the appropriate left side on either (3), (4) or (13) is not too different from its corresponding isoperimetric value, then Ω is approximately a ball).

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] ASHBAUGH, M.S., AND BENGURIA, R.D.: On Rayleigh’s conjecture for the clamped plate and its generalization to three dimensions, Duke Math. J. 78 (1995), 1–17.

[3] ASHBAUGH, M.S., AND LAUGESEN, R.S.: Fundamental tones and buckling loads of clamped plates, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (Ser. IV), XXIII (1996), 383–402.

[4] BANDLE, C.: Isoperimetric Inequalities and Applications. Pitman Advanced Publishing Program, Boston, 1980.

[5] COURANT, R.: Beweiss des Satzes, dass von allen homogenen Membranen gegebenen Umfanges und gegebener Spannung die Kreisförmige den tiefsten Grundton besitzen, Math. Z. 1 (1918), 321–328.

[6] FABER, G.: Beweis, dass unter allen homogenen Membranen von gleicher Fläche und gleicher Spannung die kreisförmige den tiefsten Grundton gibt, Sitzungsber. Bayer. Akad. Wiss. München, Math.-Phys. Kl. (1923), 169–172.

[7] KRAHN, E.: Über eine von Rayleigh formulierte Minimaleigenschaft des Kreises, Math. Ann. 94 (1925), 97–100.

[8] KRAHN, E.: Über Minimaleigenschaft der Kugel in drei und mehr Dimensionen, Acta Comm. Univ. Tartu (Dorpat) A9 (1926), 1–44. [English translation in [10], pp.139–174].

[9] LIEB, E.H.: Existence and uniqueness of the minimizing solution of Chocquard’s nonlinear equation, Stud. Appl. Math. 57 (1977), 93–105.

[10] LUMISTE, Ü., and PEETRE, J.: Edgar Krahn, 1894–1961, A Centenary Volume. IOS Press, Amsterdam, 1994 (Chapter 6).

[11] MELAS, A.D.: The stability of some eigenvalue estimates, J. Diff. Geometry 36 (1992) 19–33.

[12] NADIRASHVILI, N.S.: Rayleigh’s conjecture on the principal frequency of the clamped plate, Arch. Rational Mech. Anal. 129 (1995), 1–10.

[13] PÓLYA, G., AND SZEGÖ, G.: Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematics Studies, Number 27, Princeton University Press, Princeton, 1951.

[14] RAYLEIGH, J.W.S.: The Theory of Sound, second edition, London, 1894/1896, pp. 339–340.

[15] SCHMIDT, E.: Beweis der isoperimetrischen Eigenschaft der Kugel im hyperbolischen und sfärischen Raum jeder Dimensionzahl, Math. Z. 49 (1943), 1–109.

[16] SPERNER, E.: Zur Symmetrisierung vonFunktionen auf Sphären, Math. Z. 134 (1973), 317– 327.

[17] SZEGÖ, G.: Inequalities for certain eigenvalues of a membrane of given area, J. Rational Mech. Anal. 3 (1954), 343–356.

[18] TALENTI, G.: Elliptic Equations and Rearrangements, Ann. Scuola Norm. Sup. Pisa (4) 3 (1976), 697–718.

[19] WEINBERGER, H.F.: An isoperimetric inequality for the N–dimensional free membrane problem, J. Rat. Mech. Anal. 5 (1956), 633–636.

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