Isoperimetric Inequalities for Eigenvalues of the Laplacianrbenguri/105.pdf · orthonormal basis of...

Proceedings of Symposia in Pure Mathematics

Isoperimetric Inequalities for Eigenvalues of the Laplacian

Mark S. Ashbaugh and Rafael D. Benguria∗

Dedicated to Professor Barry Simon on his sixtieth birthday.

Abstract. We give a survey of universal inequalities for low-lying eigenvaluesof the laplacian.

Contents

1. Introduction2. Rayleigh–Faber–Krahn Inequality3. Szego–Weinberger Inequalities4. The Payne–Polya–Weinberger Inequality in R

n

5. Monotonicity of g and B6. The PPW Bound for Domains in S

n

7. The Eigenvalue Gap λ2 − λ1

8. Steklov Eigenvalues9. Annular Domains10. Inequalities Between Dirichlet and Neumann Eigenvalues11. Polya’s Conjectures12. The Bass Tone of a Drum and Its Inradius13. The Eigenvalue Ratio λ3/λ1

14. An Isoperimetric Inequality for Ovals in the Plane15. Open ProblemsReferences

1. Introduction

The purpose of this article is to give an overview of some isoperimetric in-equalities for eigenvalues of the laplacian. Since the literature on the subject isextensive (see, e.g., the reviews [32, 138, 171]), we will restrict ourselves primarily

2000 Mathematics Subject Classification. Primary 35P15; Secondary 35J05, 49R50.Key words and phrases. eigenvalues of the laplacian, isoperimetric inequalities.∗ Supported by Fondecyt, project 102–0844.

c©0000 (copyright holder)

1

2 MARK S. ASHBAUGH AND RAFAEL D. BENGURIA

to the consideration of isoperimetric results for low-lying eigenvalues with specialattention to results connected to eigenvalue ratios. Isoperimetric inequalities havea long history in mathematics dating back to the Greeks and Dido’s problem, i.e.,the classical isoperimetric inequality in Euclidean geometry (see, e.g., [136, 170]for reviews of the subject). With the introduction of the Calculus of Variations inthe seventeenth century, isoperimetric inequalities found their way into mechanicsand physics (see, e.g., [153]). The theme of this manuscript, isoperimetric inequal-ities for eigenvalues of the laplacian, has its roots in the work of Lord Rayleighon the Theory of Sound [157]. It was determined in the nineteenth century thatthe basic equation that describes the small vibrations of an elastic medium is thewave equation. The normal modes and proper frequencies that characterize thevibrations of a fixed, homogeneous membrane correspond to particular solutions ofthe wave equation. They are determined by the solution of the eigenvalue problemfor the Dirichlet laplacian on a bounded domain in R

2.Consider a bounded domain Ω ⊂ R

n with a piecewise smooth boundary ∂Ω.Then λ is a Dirichlet eigenvalue of Ω if there exists a function u ∈ C2(Ω) ∩ C0(Ω)(Dirichlet eigenfunction) satisfying the boundary value problem

−∆u = λu in Ω, (1.1)

with u = 0 in ∂Ω, where ∆ is the Laplace operator. Dirichlet eigenvalues (forn = 2) were introduced in the study of the vibrations of the clamped membranein the nineteenth century. In fact, they are proportional to the squares of theeigenfrequencies of the membrane with fixed boundary. See [106] for a review andhistorical remarks. Provided Ω is bounded and the boundary ∂Ω is sufficientlyregular, the Dirichlet laplacian has a discrete spectrum of infinitely many positiveeigenvalues with no finite accumulation points [148],

0 < λ1(Ω) < λ2(Ω) ≤ λ3(Ω) ≤ . . . (1.2)

(with λk(Ω) → ∞ as k → ∞). The Dirichlet eigenvalues are characterized by themax-min principle [53, 158]

λk = sup inf

∫

Ω |∇u|2 dx∫

Ωu2 dx

, (1.3)

where the inf is taken over all u ∈ H10 (Ω) \ 0 orthogonal to ϕ1, ϕ2, . . . , ϕk−1 ∈

H10 (Ω), and the sup is taken over all choices of ϕik−1

i=1 . For simply connecteddomains it follows from the max-min principle (1.3) that the lowest eigenvalueλ1(Ω) is nondegenerate and the corresponding eigenfunction u1 can be taken tobe positive in the interior of Ω. For future reference, we let uk∞k=1 denote anorthonormal basis of real eigenfunctions corresponding to the Dirichlet eigenvaluesλk∞k=1 (listed with multiplicity), i.e.,

∫

Ω ui uj = δij for all i, j and −∆uk = λk uk

for each k. For higher values of k the nodal lines of the k-th eigenfunction uk divideΩ into no more than k nodal domains [53].

Dirichlet eigenvalues are completely characterized by the geometry of the do-main Ω. The inverse problem, i.e., to what extent the geometry of Ω can berecovered from knowledge of λk∞k=1, was posed by M. Kac in [96]. If n = 2, forexample, and ∂Ω is smooth (in particular ∂Ω does not have corners) the distribution

ISOPERIMETRIC INEQUALITIES FOR EIGENVALUES OF THE LAPLACIAN 3

function (trace of the heat kernel)

∞∑

k=1

e−λkt ≈ A

4π t− L

8√

π t+

1

6(1 − r) + O(t1/2), (1.4)

as t → 0+, where A is the area, L the perimeter, and r the number of holes ofΩ, so at least these features of the domain can be recovered from knowledge of allthe eigenvalues. However, complete recovery of the geometry is impossible, as waslater shown by Gordon, Webb, and Wolpert [70], who constructed two isospectraldomains in R

2 with different geometries (the same result for dimension n ≥ 4had been accomplished earlier by Urakawa in [176]). Since the area A and theperimeter L of the membrane are determined from the asymptotic expansion ofthe distribution function (1.4), it follows from the classical isoperimetric inequalitythat one can hear whether the membrane is circular.

One can make similar remarks about the Neumann eigenvalues of the laplacianfor a bounded domain Ω ⊂ R

n with smooth boundary. For further elaboration oftheir background and notation, see the first paragraph of Section 3 below.

For the Euclidean case, both the Dirichlet and Neumann eigenvalues scale asone over length squared, and therefore one can compare them with geometric prop-erties of the underlying domain, and try to find universal inequalities. The simplestsuch inequality to be considered is a universal (hopefully sharp) bound on Aλ1.This universal inequality is precisely the Rayleigh–Faber–Krahn inequality (see Sec-tion 2, below). Other universal inequalities involve bounding the first nontrivialNeumann eigenvalue by an expression involving A, the so-called Szego–Weinbergerinequality (see Section 3, below). In Section 4 we consider universal inequalities forratios of Dirichlet eigenvalues, in particular the Payne–Polya–Weinberger (hence-forth PPW) isoperimetric inequality for λ2/λ1. Section 5 is devoted to discussingmonotonicity properties of special functions involved in the proof of the PPW in-equality. In Section 6 we discuss the analog of the PPW inequality on spaces ofconstant sectional curvature. In Section 7 we discuss universal inequalities for thegap between the first two Dirichlet eigenvalues of the laplacian for convex domains.Inequalities for the Steklov eigenvalues are discussed in Section 8. In Section 9, wediscuss universal inequalities for eigenvalues of the laplacian in annular domains.In Section 10, we briefly survey inequalities between Dirichlet and Neumann eigen-values. Polya’s conjectures are discussed in Section 11. Connections between thelowest eigenvalue of the Dirichlet laplacian and the inradius of the underlying do-main are discussed in Section 12. In Section 13 we discuss universal inequalities forλ3/λ1. In Section 14 we discuss an isoperimetric inequality for ovals in the plane.Finally, in Section 15 we mention some of the most intriguing open problems con-cerning universal inequalities for low-lying eigenvalues.

2. Rayleigh–Faber–Krahn Inequality

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

λ1 ≥πj2

0,1

A, (2.1)

where j0,1 = 2.4048 . . . is the first zero of the Bessel function J0(t), and A is the areaof the membrane (Bessel functions are used throughout this manuscript, here wefollow the notation of [1]). Equality is attained in (2.1) if and only if the membrane


is circular. In simple words, among all membranes of a given area, the circular

shape gives the lowest fundamental frequency. This inequality was conjectured byLord Rayleigh (see [157], pp. 339–340), based on exact calculations for simpledomains, and a variational argument for nearly circular domains. In 1918, Courant[52] proved the weaker isoperimetric result that among all membranes of the sameperimeter L, the circular one yields the least lowest eigenvalue, i.e.,

λ1 ≥4π2j2

0,1

L2, (2.2)

with equality if and only if the membrane is circular. Rayleigh’s conjecture wasproven independently in the 1920s by Faber [64] and Krahn [101]. The correspond-ing isoperimetric inequality in dimension n,

λ1 ≥(

Cn

|Ω|

)2/n

j2n/2−1,1, (2.3)

was proven by Krahn [102] (an English translation of this article of Krahn withcommentary can be found in [124]). In (2.3), jm,1 denotes the first positive zero of

the Bessel function Jm(t), |Ω| is the volume of the domain, and Cn = πn/2/Γ(n/2+1) is the volume of the unit ball in n dimensions. Equality is attained in (2.3) ifand only if Ω is a ball.

The proof of the Rayleigh–Faber–Krahn inequality rests upon two facts: avariational characterization of the lowest Dirichlet eigenvalue and the properties ofsymmetric decreasing rearrangements of functions. The variational characterizationof the lowest eigenvalue is given by [53, 158]

λ1(Ω) = infu∈H1

0(Ω)\0

∫

Ω|∇u|2 dx

∫

Ω u2 dx. (2.4)

Concerning decreasing rearrangements, let Ω be a measurable subset of Rn, then the

symmetrized domain Ω⋆ is a ball with the same measure as Ω. If u is a real-valuedmeasurable function defined on a bounded domain Ω ⊂ R

n, its spherical decreasingrearrangement u⋆ is a function defined on the ball Ω⋆ centered at the origin, suchthat u⋆ depends only on distance from the origin, is decreasing away from theorigin, and is equimeasurable with u. The spherical increasing rearrangement ofu, denoted u⋆, is defined in the same way but with “nondecreasing” replacing“nonincreasing” in the definition above. We refer to [25, 97, 72, 118, 153, 169,171, 172] for properties of rearrangements of functions. Since the function uand its spherical decreasing rearrangement are equimeasurable, their L2-norms arethe same. What Faber and Krahn actually proved is that the L2-norm of thegradient of a function vanishing on ∂Ω is decreased under spherical decreasingrearrangement (the usual proof of this fact uses the rearrangements of the levelsets of u, Federer’s co-area formula, and the classical isoperimetric inequality; see[169, 172] for details, or [114] for a different approach). The fact that the L2-normof the gradient of a function decreases under rearrangement, combined with thevariational characterization (2.4) and the fact that u ∈ H1

0 (Ω) implies u⋆ ∈ H10 (Ω⋆),

immediately gives the Rayleigh–Faber–Krahn inequality.In S

n, the analog of the classical isoperimetric inequality was proven by Schmidt[162]. The analog of the Rayleigh–Faber–Krahn inequality for domains on S

n was


proven by Sperner [167] (see also [68]). For a discussion of Rayleigh–Faber–Krahn-type inequalities for domains in manifolds, we refer to the book by Chavel [44](pp. 86–94) and references therein.

Rayleigh–Faber–Krahn-type results have also been obtained for Schrodingeroperators (see, e.g., [125]), and more recently for Schrodinger operators with mag-netic fields [62]. A Rayleigh–Faber–Krahn result for wedge-like membranes wasobtained by Payne and Weinberger [146].

For the classical isoperimetric inequality there are well known stability results,that control the isoperimetric defect L2 − 4πA from below in terms of the radiiof the incircle and the circumcircle [36] (see also [39, 137]). In the same spirit,Melas proved a stability result for the Rayleigh–Faber–Krahn inequality for con-vex domains [131]. On the other hand, Avila [23] proved a stability result forthe Rayleigh–Faber–Krahn inequality for domains in S

n. There is also a mucholder result of Payne and Weinberger [147] which shows that Aλ1 is close to theRayleigh–Faber–Krahn lower bound if the isoperimetric defect is small.

Rayleigh–Faber–Krahn-type results have also been obtained for more generaloperators, in particular, for the p-laplacian: Let Ω ⊂ R

n be a bounded domain,and suppose that 1 < p < ∞. Let

λ1 = λ1(p, Ω) = inf

∫

Ω |∇u|p dx∫

Ω |u|p dx, (2.5)

where the inf is taken over all functions u ∈ W 1,p0 , u 6≡ 0. It is well-known that

λ1(p, Ω) > 0 and a nonzero minimizer u = u(p, Ω) exists and satisfies the Eulerequation,

div(|Du|p−2Du) + λ|u|p−2u = 0, (2.6)

in Ω, with u ∈ W 1,p0 . The operator div(|Du|p−2Du) is the p-laplacian (for p = 2

it is just the ordinary laplacian). Then, λ1 is the first eigenvalue and u the firsteigenfunction of the p-laplacian on Ω. λ1 is simple and isolated, and u has onesign (see, e.g., [34, 122]. For the p-laplacian we also have the following Rayleigh–Faber–Krahn inequality [34, 129] (see also [67]):

λ1(p, Ω) ≥ λ1(p, Ω⋆). (2.7)

Equality is obtained in (2.6) if and only if Ω is a ball.More recently, a Rayleigh–Faber–Krahn result has been proven for the lowest

eigenvalue of the laplacian for smooth (not necessarily bounded) domains on theEuclidean space R

n with a gaussian weight [33].Finally, we mention that the Rayleigh–Faber–Krahn inequality extends almost

immediately to a result for the second eigenvalue, λ2 (for this discussion we confineour attention to the case of Euclidean domains, though certainly extensions todomains in Riemannian manifolds are possible and even straightforward). Thisfollows from the fact that any eigenfunction for λ2 has exactly two nodal domains[53], and that λ2 is the first eigenvalue of either nodal domain. From this fact, thelower bound

λ2 >

(

2Cn

|Ω|

)2/n

j2n/2−1,1 (2.8)

follows immediately. In words, this just says that to minimize λ2 among domains ofequal volume one should take the limiting case of two equal balls. This inequality


was first observed by Krahn [102] in his second, longer paper proving the Rayleigh–Faber–Krahn inequality in n dimensions. There he also suggested that perhaps λk

is minimized in the (limiting) case of k identical balls in Rn. While this is generally

not true (in R2 a single disk has a lower λ3 than do three equal disks of the same

total area), it may be true for certain choices of k and n. For more information inthis direction, the reader might consult [182] or [82], among others.

Krahn’s result for λ2 was little-noticed and hence has been rediscovered byothers several times since its original publication in 1926. For example, it was laterfound by Hong [92] and, independently, by Peter Szego (see p. 336 of Polya’s paper[151]). For recent discussions on minimizing λ2 see, e.g., [84, 85].

3. Szego–Weinberger Inequalities

The lowest nontrivial eigenvalue for the free membrane also satisfies an isoperi-metric inequality. Let Ω be a a bounded domain in R

n with smooth boundary.Then we can list the Neumann eigenvalues of the laplacian on Ω as 0 = µ0(Ω) <µ1(Ω) ≤ µ2(Ω) ≤ . . . , i.e., this is a listing, with multiplicities, of the eigenvalues ofthe problem

−∆u = µu in Ω, (3.1)

with∂u

∂n= 0 on ∂Ω, (3.2)

where n denotes the outward normal to ∂Ω.For the free membrane problem (i.e., for n = 2), Szego [168] proved, for a

simply-connected domain Ω,

µ1(Ω) ≤πp2

1,1

A= µ1(Ω

⋆), (3.3)

where p1,1 = 1.8412 . . . is the first positive zero of the derivative of the Besselfunction J1(t). This isoperimetric inequality had been conjectured by Kornhauserand Stakgold [100]. The corresponding result for dimension n,

µ1(Ω) ≤(

Cn

|Ω|

)2/n

p2n/2,1 = µ1(Ω

⋆), (3.4)

was proven by Weinberger [178] for arbitrary bounded domains. Here, Cn is thevolume of the unit ball in dimension n, as above, and pm,1 denotes the first positivezero of the derivative of the “Bessel” function t1−mJm(t). In dimension n = 2,Weinberger [178] also noted (based on a conversation with Szego) that Szego’sproof yields

1

µ1(Ω)+

1

µ2(Ω)≥ 2A

πp21,1

=1

µ1(Ω⋆)+

1

µ2(Ω⋆)(3.5)

for a simply-connected domain Ω. In (3.3) and (3.5) equality is attained if and onlyif Ω is a disk, and in (3.4) if Ω is a ball.

Although the Szego–Weinberger inequality appears to be the analog for Neu-mann eigenvalues of the Rayleigh–Faber–Krahn inequality, its proof is significantlydifferent. In fact, the proof of the Szego–Weinberger isoperimetric inequality ismuch closer in spirit to that of the PPW inequality that we discuss in the next sec-tion. We recall that the Rayleigh–Faber–Krahn inequality was done in two steps: (i)the Rayleigh–Ritz characterization of λ1, and (ii) the use of spherically decreasingrearrangement, in particular, the fact that the L2-norm of an H1

0 function remains


the same while the L2-norm of its gradient decreases under spherically decreas-ing rearrangement. In the Neumann problem, the eigenfunction corresponding tothe first nontrivial eigenvalue of the ball is not spherically symmetric, and thusthe method we just described cannot be used in exactly the same way. On theother hand one can exploit the fact that the first nontrivial Neumann eigenvaluefor the ball in n dimensions is n-fold degenerate. The main steps in the proof ofthe Szego–Weinberger result are thus the following:

(i) The Rayleigh–Ritz inequality for the first nontrivial Neumann eigenvalue

µ1 ≤∫

Ω |∇P |2 dx∫

ΩP 2 dx

, (3.6)

provided∫

Ω P dx = 0, and P 6≡ 0. Here P is a trial function for µ1, orthogonal tothe constants.

(ii) Center of Mass result. Use (3.6) written as

µ1

∫

Ω

P 2 dx ≤∫

Ω

|∇P |2 dx (3.7)

n times with the n different trial functions

Pi = g(r)xi

ri = 1, 2, . . . , n, (3.8)

where the xi’s denote the usual Cartesian coordinates for Rn and r denotes radial

distance from the origin. Weinberger [178] (see also [25], pp. 153–154) used atopological argument based on the Brouwer fixed point theorem to show that onecan always choose the origin so that the trial functions Pi defined by (3.8) areorthogonal to the constants. We call this a center of mass result since it generalizesaway from the case with g(r) = r, where one would choose the origin to be thecenter of mass of Ω when Ω is viewed as a uniform mass density in R

n.

(iii) Choice of the variational function g. Substituting each of the Pi’s defined by(3.8) into (3.7) and summing, one arrives at the basic estimate

µ1 ≤∫

Ω

[

(g′)2 + (n − 1)(g/r)2]

dx∫

Ω g2 dx. (3.9)

The choice of g is dictated by the fact that we want equality in (3.6) when the Pi’sare of the form (3.8) and Ω is a ball. Based on the knowledge of the eigenfunctionsfor the first nontrivial Neumann eigenvalue for a ball, one thus takes

g(r) = r1−n/2Jn/2(pn/2,1r/R), (3.10)

for 0 ≤ r ≤ R, and g(r) = g(R) for r ≥ R, where R is the radius of the ball Ω⋆ andpm,1 denotes the first positive zero of the derivative of t1−mJm(t).

(iv) Monotonicity properties. From the definition of g and properties of Besselfunctions, one can prove that g(r) is a nondecreasing function of r and

B(r) ≡ (g′)2 + (n − 1)(g/r)2 (3.11)

is a nonincreasing function of r (see [178] or [25] for details).

(v) Symmetric rearrangement. Using the basic estimate (3.9), the monotonicityproperties of g and B, and properties of rearrangements, one has

µ1(Ω) ≤∫

Ω B(r) dx∫

Ωg2 dx

≤∫

Ω⋆ B(r) dx∫

Ω⋆ g2 dx=

p2n/2,1

R2= µ1(Ω

⋆), (3.12)


which is precisely the Szego–Weinberger inequality.The analog of the Szego–Weinberger inequality in spaces of constant sectional

curvature was considered by Ashbaugh and Benguria in [17]. In particular, weproved the analog of (3.4) when Ω is a domain contained in a hemisphere of Sn.Earlier variants of this result are due to Bandle (see her book [25]) and Chavel [43](see also [44]). In [17] we also gave a proof of the analog of (3.4) for any bounded,smooth domain Ω in H

n, a result which had been noted earlier by Chavel [43].

4. The Payne–Polya–Weinberger Inequality in Rn

The next simplest isoperimetric inequality concerns the quotient of the two low-est Dirichlet eigenvalues λ2/λ1. In 1955, Payne, Polya, and Weinberger (henceforthPPW) proved the universal bound,

λ2(Ω)

λ1(Ω)≤ 3, (4.1)

for the quotient between the first two Dirichlet eigenvalues of a bounded smoothdomain Ω ⊂ R

2 (using twice the Rayleigh–Ritz characterization of λ2 with trialfunctions xu1 and yu1, and combining the two inequalities appropriately) [143,144]. This was extended to domains in R

n in [174], where the corresponding boundreads λ2(Ω)/λ1(Ω) ≤ 1+4/n. Based on exact calculations for simple domains, PPWconjectured the isoperimetric inequality

λ2(Ω)

λ1(Ω)≤ λ2(Ω

⋆)

λ1(Ω⋆)=

j21,1

j20,1

≈ 2.539 . . . for Ω ⊂ R2. (4.2)

The bound (4.1) was successively improved in [37, 57, 49]. Finally, the conjec-tured inequality (4.2) and the corresponding inequalities in n dimensions for all nwere proven by Ashbaugh and Benguria [7, 9] (see also [10]). Notice that whileDirichlet eigenvalues satisfy stringent constraints (like the one embodied by (4.2)for any bounded domain in R

2), no such constraints exist for Neumann eigenvalues,other than the fact that they are nonnegative. In fact, given any finite sequence0 = µ0 < µ1 < µ2 < · · · < µk−1, there is an open bounded, smooth, simplyconnected domain of R

2 having this sequence as the first k Neumann eigenvaluesof the laplacian on that domain [51].

The proof of the PPW inequality (4.2) has several features in common withthe proof of the Szego–Weinberger inequality. However, the necessary facts aboutspecial functions are much harder to establish for the PPW inequality. A keyfact which goes some ways towards explaining this difference is that in the Szego–Weinberger case, the first (Neumann) eigenfunction is explicitly known (it is justa constant), while in the proof of PPW, neither the first (Dirichlet) eigenfunctionu1 nor its symmetric decreasing rearrangement is explicitly known. Since in thecourse of the proof integrals occur which involve u1 and its symmetric decreasingrearrangement u⋆

1 explicitly, a comparison technique, due to Chiti, is needed toestimate these integrals in terms of integrals of known functions.

The proof of (4.2) is performed in the following six steps:

(i) The Rayleigh–Ritz characterization of the second Dirichlet eigenvalue, through

the use of the gap formula [158, 88]:

λ2 − λ1 ≤∫

Ω |∇P |2u21 dx

∫

ΩP 2u2

1 dx, (4.3)


where∫

ΩP u2

1 dx = 0, and P 6≡ 0. To get the isoperimetric result out of (4.3), wemust make very special choices of the function P , in particular, choices for which(4.3) is an equality if Ω is a ball.

(ii) A center of mass result: In n dimensions one uses (4.3) for n trial functions

Pi = g(r)xi

ri = 1, 2, . . . , n, (4.4)

where the xi’s represent the usual Cartesian coordinates for Rn and r denotes

radial distance from the origin. To insure orthogonality of Piu1 to u1 for 1 ≤ i ≤ n,one must make a special choice of origin, using a topological argument due toWeinberger [178] based on the Brouwer fixed point theorem. We call this a center

of mass result since it generalizes away from the case with g(r) = r, where onemust choose the origin to be precisely the center of mass of Ω thought of as a massdistribution with density function u2

1.

(iii) Averaging over the Pi’s and choosing g: Using (4.3) with the special choicesgiven by (4.4) (following similar ideas as in the proof of the Szego–Weinbergerinequality), we arrive at the inequality

λ2 − λ1 ≤∫

Ω

[

(g′)2 + (n − 1)(g/r)2]

u21 dx

∫

Ωg2u2

1 dx. (4.5)

Once we have arrived at (4.5) it is convenient to use results on rearrangements.For that, it is crucial to understand the behavior of the two functions g(r) and(g′)2 + (n − 1)(g/r)2. The choice of g is dictated by the fact that we need tohave equality in (4.3) when Ω is a ball (but precisely which ball has yet to bedetermined). This determines g and (g′)2 + (n − 1)(g/r)2 up to a scaling of theindependent variable and their behavior for large r (which does not matter if Ωis a ball but is necessary for the general situation). Given the knowledge of theeigenfunctions corresponding to λ1 and λ2 for the ball, one takes

g(r) = w(γr) (4.6)

where w is defined by

w(t) =Jn/2(jn/2,1t)

Jn/2−1(jn/2−1,1t)for 0 ≤ t < 1, (4.7)

and w(t) = limt→1− w(t), for t ≥ 1. That is, g needs to be the ratio of the “radialparts” of the eigenfunctions u1, u2 for some n-ball (so a scaling of that same ratiow for the unit ball). Since g(r) = w(γr), we have (g′)2 + (n− 1)(g/r)2 = γ2B(γr),with

B(t) ≡ w′(t)2 + (n − 1)(w(t)/t)2. (4.8)

(iv) Monotonicity properties of special functions: From the definition of w and theproperties of Bessel functions one can prove that w(t) is a nondecreasing functionof t and B(t) is a nonincreasing function of t (see Section 5, below).

(v) Use of symmetric rearrangement: Using rearrangements and the monotonicityproperties of B and w, one has

∫

Ω

B(γr)u21 dx ≤

∫

Ω⋆

B(γr)∗u∗21 dx ≤

∫

Ω⋆

B(γr)u∗21 dx (4.9)

and∫

Ω

w(γr)2u21 dx ≥

∫

Ω⋆

w(γr)2∗u∗21 dx ≥

∫

Ω⋆

w(γr)2u∗21 dx. (4.10)


(vi) Chiti comparison argument: Let B1 be the ball with radius chosen so that thefirst Dirichlet eigenvalue of B1 is also λ1. It is clear that this radius is jn/2−1,1/

√λ1.

Let z be the first Dirichlet eigenfunction of B1 normalized such that∫

B1

z2 dx =∫

Ωu2

1. The final step, essential to proving (4.2), is a result of Chiti [47, 48, 49] com-paring the spherical decreasing rearrangement u⋆

1 of u1 with z. Chiti’s comparisontheorem implies that

∫

Ω⋆

f(r)u⋆12 dx ≥

∫

B1

f(r)z2 dx, (4.11)

if f is increasing, and the reverse inequality if f is decreasing.The result of Chiti builds upon the work of Talenti [169], which makes use

of rearrangements results, Federer’s co-area formula [65], and the classical isoperi-metric inequality. The Rayleigh–Faber–Krahn inequality implies that B1 ⊂ Ω⋆ (infact B1 is a proper subset of Ω⋆ unless Ω itself starts out as a ball). It follows from(4.11) and the monotonicity properties of B and w that

∫

Ω⋆

B(γr)u∗21 dx ≤

∫

B1

B(γr)z2 dx (4.12)

and∫

Ω⋆

w(γr)2u∗21 dx ≥

∫

B1

w(γr)2z2 dx. (4.13)

Combining these inequalities with (4.9), (4.10), (4.5), (4.7), (4.8), and the definitionof z, and taking γ =

√λ1/jn/2−1,1, we finally get

λ2 − λ1 ≤ λ1

α2

∫

B1

B(γr)z2 dx∫

B1

w(γr)2z2 dx=

λ1

α2

(

β2 − α2)

, (4.14)

where α = jn/2−1,1 and β = jn/2,1. From here the inequality

λ2

λ1≤

j2n/2,1

j2n/2−1,1

=λ2(Ω

⋆)

λ1(Ω⋆)(4.15)

follows immediately.

Remarks. (i) Note that inequality (4.2) and the fact that equality holds if andonly if Ω is a ball means that one can “hear” the shape of a circular drum from justits first two frequencies. Moreover, if all one seeks is to know if the drum is circularor not, one needs only the ratio of the first two frequencies and not their individualvalues. And, as Melas [131] tells us, we can “hear” that a drum is close to circularif the ratio of its first two tones is close to its value for a circular drum. That is,using an equivalent characterization in terms of eigenvalues, if λ2/λ1 is very closeto its value for a disk, then in a certain rigorous sense the boundary of the domainis very close to being circular, at least for convex domains.

(ii) There are by now several classes of (not necessarily sharp) universal in-equalities between Dirichlet eigenvalues. For example, one has the general boundof Payne, Polya, and Weinberger,

λm+1 − λm ≤ 4

m n(λ1 + λ2 + · · · + λm) , (4.16)


which holds for m = 1, 2, 3, . . . and for any bounded domain in Rn [143, 144]. One

also has the stronger Hile–Protter bound [91],m

∑

i=1

λi

λm+1 − λi≥ m n

4, (4.17)

and, in addition, an even stronger result due to Yang [183]. We will not review themany generalizations of these bounds that have been obtained (instead, we referthe reader to the review articles [15, 2], and also to the recent articles [4, 21]).See also [183, 45], and also [12, 13].

(iii) A very useful approach to proving universal inequalities for eigenvalues ofthe laplacian (such as those given in (ii) above) makes use of commutators (see,e.g., [73, 93, 74, 78, 79, 80, 111, 21]). These take as their starting points theworks of PPW, Hile and Protter, Yang, and others, as mentioned in the precedingremark. Interestingly, a connection can also be made to the sum rules studied inquantum mechanics from its early days [111].

(iv) There are a number of other interesting isoperimetric results which arecorollaries of (4.2). Some of the most interesting involve the geometric objectknown as the maximal inner conformal radius r, or conformal radius for short.Polya and Szego (see [153]) had established that for a simply-connected domainr2λ1 is maximized at the disk (or, equivalently, that among all domains of a givenconformal radius r the one giving the largest λ1 is the disk). Combining this with(4.2) now shows that λ2 enjoys the same property: λ2 is maximized by the diskamong all simply-connected domains of a given conformal radius. Beyond that,Banuelos has remarked that these results also combine to show that under thesame conditions, the fundamental gap λ2 − λ1 is maximized at a disk. Indeed, thesame is true of λ2 − cλ1 for any constant c less than or equal to λ2/λ1 of a ball.Thus, for any c ≤ j2

1,1/j20,1, r

2(λ2 − cλ1) ≤ j21,1 − c j2

0,1, which is isoperimetric, withequality if and only if Ω is a disk.

5. Monotonicity of g and B

In our original proof of the PPW inequality (4.15) [7, 9], we used the productrepresentation for Bessel functions and a certain inequality between ratios of zerosof Bessel functions to prove the monotonicity properties of the functions w (org) and B introduced above. The proof of these properties can be simplified byanalyzing a Riccati differential equation satisfied by the function

q(t) =t w′(t)

w(t). (5.1)

The main advantage of this proof is that it can be generalized to other situations(e.g., to a proof of the analog of the Szego–Weinberger bound for domains in spacesof constant sectional curvature, or to a proof of the PPW inequality for domainsin S

n). In this section we will sketch the proof of the monotonicity properties of wand B in this approach. In terms of q, we can write

B = [q2 + (n − 1)]w2

t2(5.2)

and

B′ = 2[q q′ + (q − 1)(q2 + (n − 1))/t]w2

t2. (5.3)


From (5.1) and (5.3) it is clear that the monotonicity properties of w and B willfollow if we can show that

0 ≤ q ≤ 1 (5.4)

and

q′ ≤ 0 (5.5)

hold for 0 ≤ t ≤ 1. Our strategy for proving these inequalities is as follows. ByTaylor expansion at 0 one finds that q(0) = 1 and that q < 1 on an interval just tothe right of 0. Similarly, by Taylor expanding at t = 1, one finds q(1) = 0 and q > 0just to the left of t = 1. The expansions here are straightforward though the keycomparisons depend on additional facts concerning eigenvalues for balls. In fact,

q(0) = 1, q′(0) = 0, q′′(0) =2j2

n/2−1,1

n−

2j2n/2,1

n + 2(5.6)

and one must show that

j2n/2−1,1

n<

j2n/2,1

n + 2,

for n ≥ 2, a result that follows using reasonably simple trial functions in a Rayleighquotient for j2

n/2−1,1 (see [10], Lemma 2.2). Similarly,

q(1) = 0, q′(1) = ((n − 1) − (j2n/2,1 − j2

n/2−1,1))/3 (5.7)

and one can show that q′(1) < 0 via another Rayleigh quotient argument. Again,see [10] for details. The next step is to show that q cannot cross below 0, for0 < t < 1. If it did, one could find values 0 < a < b < 1 such that q′(a) ≤ 0, andq′(b) ≥ 0. On the other hand, the differential (i.e., the Riccati) equation for q isgiven by

q′ = −(β2 − α2)t +1

t(1 − q)(q + n − 1) + 2α

Jn/2(αt)

Jn/2−1(αt)q, (5.8)

with α = jn/2−1,1 and β = jn/2,1. Thus, whenever q = 0 for t ∈ (0, 1) we have

q′ = −(β2 − α2)t +n − 1

t. (5.9)

The right side of (5.9) is strictly decreasing with increasing t, so it is not possible tohave points a, b, with a < b, q(a) = q(b) = 0 and q′(a) ≤ 0, and q′(b) ≥ 0. Thereforeq ≥ 0 in (0, 1). In an analogous way one can prove that q ≤ 1 (see [10]). Finally,to prove that q is decreasing in (0, 1), we argue by contradiction. Assume that q′

is somewhere positive on (0, 1). Then, there are three points, t1, t2, t3, say, with0 < t1 < t2 < t3 < 1, such that q(t1) = q(t2) = q(t3), and q′(t1) < 0, q′(t2) > 0,q′(t3) < 0. Think of the right side of (5.8) as a function of q and t. For fixed q(with 0 < q < 1) the right side of (5.8) is a convex function of t, which followsfrom the convexity of Jn/2(αt)/Jn/2−1(αt), a fact which is not entirely trivial toestablish (see [10], Lemma 2.3). Since t2 above is a convex combination of t1 andt3, this immediately yields a contradiction. Thus q′ ≤ 0 in (0, 1).


6. The PPW Bound for Domains in Sn

The analog of the PPW bound for domains contained in a hemisphere of Sn also

holds [19]. This follows from Sperner’s isoperimetric inequality (i.e., the analog ofthe Rayleigh–Faber–Krahn inequality for domains in S

n) and the following theoremof [19] (a fixed λ1 result): Let Ω be contained in a hemisphere and let B1 denotethe geodesic ball in S

n having the same value of λ1 as Ω (i.e., λ1(Ω) = λ1(B1)).Then,

λ2(Ω) ≤ λ2(B1), (6.1)

with equality if and only if Ω is itself a geodesic ball in Sn. The proof of (6.1)

follows a similar sequence of steps as that sketched in the proof of the PPW boundin Euclidean space. The form of the trial functions Pi must be modified slightly,again with optimality for the case that Ω is itself a geodesic ball as the motivatingfactor. Thinking of S

n as the unit sphere in Rn+1 and with the center of mass point

for Ω fixed at the north pole, one considers

Pi = g(θ)xi

sin θ, i = 1, 2, . . . , n, (6.2)

where θ is the azimuthal angle of a point from the axis through the north pole (inS

n, θ plays the role of a geodesic radial variable). We saw that in the Euclideancase, the special functions that enter into the solution of the Dirichlet problem fora ball were Bessel functions. In the case at hand, they are Legendre or associatedLegendre functions. Then one defines as before the function g(θ) as the quotientbetween the radial parts of the first two Dirichlet eigenfunctions of the Laplace–Beltrami operator for the geodesic ball. Following the same scheme as we didearlier, one must prove monotonicity properties of g(θ) and the function

B(θ) ≡ g′(θ)2 + (n − 1)g(θ)2

sin2(θ), (6.3)

which appears naturally in the problem when using the averaged version of the gapformula (the analog of (4.5)) with trial functions Pi given by (6.2). In order toprove the monotonicity properties one proceeds as in Section 5, above, but nowusing the function

q(θ) ≡ sin θg′(θ)

g(θ). (6.4)

This time the monotonicity properties of g and B follow from these two propertiesof q,

0 ≤ q ≤ cos θ; and q′ ≤ 0, (6.5)

for θ out to an appropriate radius θ1. The proof of (6.5) is obtained by a carefulanalysis of the properties of a Riccati equation satisfied by q, analysis that involvescertain key convexity and monotonicity properties of some combinations of asso-ciated Legendre functions (see [19] for details). The rest of the proof proceedsas before, using symmetric decreasing rearrangements (with respect to the northpole), and the analog of the Chiti comparison argument. Our proof only holds fordomains contained in a hemisphere of S

n because the proof of the properties (6.5)only holds if θ1 ≤ π/2.

Remarks. (i) In fact, in addition to (6.1), we prove that for a domain Ω containedin a hemisphere, λ2/λ1 is maximized by the spherical cap of the same area (volume).


This follows from (6.1) and the fact that for spherical caps λ2/λ1 is an increasingfunction of radius (see [19]), at least up to the hemisphere.

(ii) To prove the analog of the sharp PPW bound for λ2 when λ1 is fixed fordomains in H

n is still open. That is, no one has yet proved the Hn analog of (6.1).

We also note that an isoperimetric ratio result for Ω ⊂ Hn saying that λ2/λ1 is

maximized at the ball of the same volume Ω⋆ is impossible, for reasons detailed inthe introductory section of [19].

(iii) The question of finding sharp estimates for the ratio between the first twoDirichlet eigenvalues in other more general settings can also be addressed. Forexample, while a sharp PPW bound has been conjectured for the p-laplacian onbounded domains in R

n (for 2 ≤ p < ∞), it has only been proven for p = 2 [9],and recently for the limiting case p = ∞ [95]. For some particular Schrodingeroperators, a sharp bound of this type has also been proven [71].

7. The Eigenvalue Gap λ2 − λ1

For an arbitrary domain Ω in Rn, the gap between the first two eigenvalues,

known as the fundamental gap, can be very small. (Think, for example, of a domainmade out of two equal balls joined by a thin tube.) However, if we restrict thedomain Ω to be convex, this gap can be bounded from below in terms of thediameter of the domain. In 1983, van den Berg [177] (see also [184], Problem 44,and [5]), made the following conjecture: Consider a bounded, convex subset Ω ofR

n, and let λ1(Ω) and λ2(Ω) be the first two Dirichlet eigenvalues of the laplacianon Ω. Then

λ2(Ω) − λ1(Ω) ≥ 3π2

d2, (7.1)

where d is the diameter of the set Ω (one can get as close as desired to the con-jectured lower bound by choosing cigar-like domains; the lower bound is attainedfor an interval of length d in one dimension). Since 1983, many authors have ob-tained lower bounds on this eigenvalue gap [165, 187, 123, 166]. The best generalresults to date give (7.1) but without the factor 3 on the right. Recently, a num-ber of authors have obtained results for the multidimensional case which are sharpbut require certain symmetry conditions on the domain. These authors include[30, 56, 29, 185] (see also [59]). We note, however, that the general conjectureremains open for an arbitrary, bounded, convex domain Ω.

Related to this inequality is the Payne–Weinberger inequality [145]: The firstnonzero eigenvalue µ1 for the free membrane problem for a convex region Ω satisfiesthe inequality

µ1 ≥ π2

d2, (7.2)

where d is the diameter of Ω. This inequality is sharp in the sense that µ1d2 tends

to π2 for a parallelepiped all but one of whose dimensions shrink to zero (or in thelimit of an infinite strip).

Remarks. (i) The original conjectures for the sharp lower bound to the funda-mental gap in terms of the diameter were actually for Schrodinger operators on aconvex domain with a convex potential, i.e., for the operator H ≡ −∆ + V (x) on aconvex domain Ω ⊂ R

n with potential V a convex function on Ω. This more generalconjecture also remains open, though the results of many of the authors cited abovealso apply to this case, giving partial results.


(ii) Lower bounds for the gap between the first two Dirichlet eigenvalues havealso been considered for convex domains in S

n (see, e.g., [108] and [123]). Forspectral gap estimates on compact manifolds, see, e.g., [134].

8. Steklov Eigenvalues

The Steklov eigenvalue problem on a bounded smooth domain Ω ⊂ Rn consists

of finding nontrivial solutions u to

∆u = 0 in Ω,∂u

∂n= ν u on ∂Ω. (8.1)

Here ∂/∂n denotes the outward normal derivative. One seeks values of the constantν for which a nontrivial solution u exists. Such ν’s are called the Steklov eigenvalues

of Ω. This problem is also the eigenvalue problem for the Dirichlet-to-Neumann mapfor the laplacian on Ω. Further general information on this subject can be foundin [25], pp. 95–96, or [161], pp. 37 (Ex. 3.11) and 94 (Ex. 11.2), for example. Thereader should be aware that “Steklov” is also often transliterated as “Stekloff.”

The setting for this problem may be usefully generalized in various ways. Forone, we could consider other operators replacing the laplacian ∆ in the first partof (8.1). For example, the eigenvalue problem for the Dirichlet-to-Neumann mapfor the operator ∆ + λ has been employed to very good effect by Friedlander [69]in his comparison of Dirichlet and Neumann eigenvalues (his results are recountedin Section 10, below). However, in the sequel we consider only the case of thelaplacian.

We can also consider modifying the second part of the definition by introducinga nonnegative (and nontrivial) weight function g defined on ∂Ω and asking that

∂u

∂n= ν g u on ∂Ω. (8.2)

A reasonable class of functions from which to take g is L∞(∂Ω).The Steklov problem for the laplacian, with or without a weight function, has

a countable sequence of eigenvalues νi∞i=1 (counted according to their multiplic-ity) with corresponding eigenfunctions ui∞i=1. These eigenvalues can be listed inincreasing order as 0 = ν1 < ν2 ≤ ν3 ≤ · · · → ∞ (see, for example, [25]). We can(and will) assume that the corresponding eigenfunctions are real and orthonormalin the L2 sense on ∂Ω. Thus, ui∞i=1 forms a real orthonormal basis for L2(∂Ω).

For g a positive weight function, the Steklov eigenvalues ν can be convenientlystudied variationally via the Rayleigh quotient

∫

Ω|∇ϕ|2

∫

∂Ω gϕ2, (8.3)

where ϕ represents a real trial function (one can allow complex-valued trial functionsϕ if one interprets |∇ϕ|2 in the numerator as ∇ϕ · ∇ϕ and if one replaces ϕ2 in thedenominator by |ϕ|2 = ϕϕ, where − denotes complex conjugation).

For the Steklov problem, ν1 = 0 (with u1 = const.; note that the variationalformulation precludes ν1 < 0), and the first eigenvalue of real interest is ν2. Anumber of authors have given estimates of ν2 for more or less arbitrary boundeddomains, or for domains in suitably restricted classes (convex, star-shaped, simply-connected, . . . ). We recount here some of the highlights in these development. Wefocus, in particular, on upper bounds, since these can be obtained using Rayleigh–Ritz estimates much as the inequalities for the other eigenvalue problems already


discussed. For lower bounds, one might consult various works of Kuttler and/orSigillito, for example. For convex planar domains with analytic boundary, there isalso the nice isoperimetric result of Payne [141] (or see [25], pp. 161–163), thatν2 is greater than or equal to the minimum curvature κmin of the boundary, i.e.,ν2 ≥ κmin, with equality if and only if the domain is a disk.

Weinstock’s inequality ([179]): Consider the Steklov eigenvalue problemfor a bounded simply-connected smooth domain Ω ⊂ R

2 with weight g, and totalmass M ≡

∫

∂Ωg. The eigenvalue ν2 satisfies the inequality

ν2 ≤ 2π

M, (8.4)

with equality if and only if Ω is a disk and g is constant on ∂Ω.This inequality was subsequently strengthened in several stages. What follows

will all be in the setting of bounded, simply-connected planar domains with suf-ficiently smooth boundary. First, it was observed by Hersch and Payne [89] thatWeinstock’s proof actually yields the stronger inequality, including both ν2 and ν3,

1

ν2+

1

ν3≥ M

π. (8.5)

Later, by an improved argument, Hersch, Payne, and Schiffer [90] were able tostrengthen this inequality even further to

ν2ν3 ≤ 4π2

M2. (8.6)

Again, there is equality if and only if Ω is a disk and g is constant. That thisinequality is stronger than the one before is a simple consequence of the elementaryinequality between the geometric and harmonic means of two positive numbers.Various other inequalities for Steklov eigenvalues in two dimensions are proved inthe papers already cited. For a good summary including the discussion of somefurther developments, see [25].

Finally, for a generalization to n dimensions, we have the recent result of Brock[38]

n+1∑

i=2

1

νi(Ω)≥ nRg, (8.7)

where R is the radius of the n-dimensional ball BR (= Ω⋆) having the same measureas Ω and g is defined by

1

g≡

∫

∂Ω(1/g)

|∂BR|. (8.8)

In this result, too, equality obtains if and only if Ω is a ball and g is constanton ∂Ω. Using the notation Cn ≡ πn/2/Γ(n/2 + 1), we have, by the well-knownformulas for volume and surface area of an n-dimensional ball, |BR| = CnRn =|Ω| and |∂BR| = n CnRn−1(≤ |∂Ω|, this last inequality by virtue of the classicalisoperimetric inequality).

Brock’s result holds for arbitrary bounded smooth domains in Rn. Note that

while on the one hand its two-dimensional specialization is somewhat weaker thanthe two-dimensional result observed by Hersch and Payne in that its right-handside is easily shown to be less than or equal to that of the other inequality (bya straightforward application of the Cauchy–Schwarz inequality and use of theclassical isoperimetric inequality), it is more general since it applies to all bounded


smooth domains and not just simply-connected ones. And, of course, the bounditself applies in all dimensions. The reason the two-dimensional results are restrictedto simply-connected domains is that their proofs rely on the existence of conformalmaps taking arbitrary simply-connected domains into a disk. That this is essentiallya two-dimensional phenomenon explains both the merits and the drawbacks of themethod.

Another recent result, which is an isoperimetric lower bound to the sum of thesquared reciprocals of all the Steklov eigenvalues save the first, can be found inDittmar [58]. This result holds for bounded simply-connected plane domains withsufficiently smooth boundary and equality obtains if and only if the domain is adisk. There is also an interesting isoperimetric result due to Edward [61] whichinvolves all the Steklov eigenvalues.

9. Annular Domains

In this section we discuss certain problems for the eigenvalues of the laplacianfor annular domains. We start with domains in R

n. A problem that made therounds in the 1990s (and perhaps even earlier) is,

Suppose you have a Dirichlet eigenvalue problem inside a disk(or ball) and you have a smaller disk (ball) that you may placeas an “obstacle” anywhere inside the larger disk (ball). Whereshould you place the obstacle so as to maximize (or minimize)the first eigenvalue?

To fix notation, let D1 denote the larger disk and D2 the smaller. Then thequestion is, among all placements D2 ⊂ D1 to find that which maximizes λ1(D1 \D2) and that which minimizes it (here our disks are taken as open sets, and thenotation A denotes the closure of the set A).

This question is answered rather easily by the Hadamard variational formula,which tells how a simple eigenvalue can be expected to change at first order due toa specified change in the underlying domain, and a simple reflection and domainmonotonicity argument (followed by use of the boundary maximum principle).

The answer is that to maximize λ1 one should place the smaller disk in thecenter of the larger one, while to minimize one should do the opposite, i.e., oneshould push the small disk to the boundary, so that D2 remains in D1 but D2 justtouches the boundary of D1 (so, in fact, the two boundary circles will be tangent).Also, the analogous statement is true in higher dimensions.

The proof proceeds as follows. One considers a given placement of the smalldisk inside the larger one, then one reflects about the center of the smaller disk(or ball) in the hyperplane perpendicular to the line connecting the centers, andthen one uses domain monotonicity of Dirichlet eigenvalues, applies the boundarymaximum principle, and finishes off by applying the Hadamard variational formula.Assuming that the small disk is in “general position,” with neither its center co-inciding with that of the larger disk or with its boundary touching that of thelarger disk, then the reflection hyperplane cuts our domain D1 \ D2 into a smallerpiece and a larger piece. We choose to view the smaller piece (and the part of theeigenfunction that lives on it) as reflected into the larger piece and the two eigen-functions combined by subtraction (the one from the smaller piece is subtracted).By a domain monotonicity argument (and the maximum principle), this difference


must be positive on the reflection of the smaller piece of the domain (we assumehere, and throughout, that the first eigenfunction u1 has been taken to be positiveon Ω, as is always possible by a choice of sign). Since the difference is 0 along thereflection hyperplane, and, most crucially, along the part of the boundary of D2

which bounds the reflected piece, one sees that the inward normal derivative of u1

on the “larger side” of the reflection hyperplane is pointwise larger than the inwardnormal derivative of u1 at the corresponding point on the “smaller side.” One thenhas only to invoke the Hadamard variational formula (for λ a simple eigenvalue of−∆ with u a normalized real eigenfunction corresponding to λ),

∂λ

∂t

∣

∣

∣

t=0= −

∫

∂Ω

|∇u|2 n · ~v (9.1)

where n denotes the outward normal to the boundary and ~v is a vector field (os-tensibly defined on Ω, but ultimately needed only on ∂Ω) detailing how, at firstorder in t at t = 0, our domain variation occurs. Thus, behind the scenes we havea mapping, Mt, defined for t small, that sends Ω to a new domain, Ωt. In detail,each x ∈ Ω gets sent to y = Mt(x) ∈ Ωt and, under the assumption that Mt is C1

in t, say, the vector field ~v is gotten via the identification Mt(x) = x+ t ~v(x)+ o(t).Thus to leading order Mt leaves points alone (i.e., M0 is the identity mapping and,in particular, Ω0 = Ω) and then ~v tells us how points are instantaneously pushedat first order by the mapping. In the present case, of course, the vector field ~v isjust a constant on ∂D2 (and vanishes on ∂D1) since all we are considering are rigidmotions of the obstacle D2.

Sorting out the signs in the Hadamard variational formula and using the point-wise comparison of normal derivatives at the points on ∂D2 identified by reflec-tion (these pointwise comparisons being the result of the reflection argument givenabove), and noting that rigid motions of D2 parallel to the line connecting thecenters lead to “equal and opposite” dot products n · ~v, we see that movement ofD2 towards the center (so moving D2 so that its center moves directly towards thatof D1) causes λ1 to increase, while movement of D2 away from the center causesλ1 to decrease. Since movement along a radius in this fashion covers all possiblepositions of D2 relative to D1 (up to symmetries, i.e., rigid movements of the en-tire configuration D1 \ D2), we see that λ1 is maximized when D2 is centered andminimized when D2 is pushed to its most extreme eccentric position, i.e., when itjust touches the boundary of D1.

These arguments have been written up, and applied in greater generality, in thepaper of Harrell, Kroger, and Kurata [76] (for some further discussion, see [63]),in the paper [156] of Ramm and Shivakumar, which briefly recounts the argumentabove, citing a personal communication between one of the present authors (MSA)and Professor Ramm (see also the unpublished manuscript [20]), and in the paper ofKesavan [99]. Some further discussion occurs in Henrot’s paper [82], pp. 448–449.

Remarks. 1. These investigations were prompted by queries of Davies ([76]) and,independently, Ramm ([20]). We do not know where Kesavan learned of the prob-lem, though it may have been through the paper of Ramm and Shivakumar (quitepossibly in an early version), as this paper is referenced in [99]. It is also notedthere, but as an afterthought suggested by the referee, that a web-based version ofRamm and Shivakumar’s paper cited as “www.math.ksu.edu/ramm/r.html, publi-cation 383,” does sketch a proof of their conjecture, i.e., the eigenvalue monotonicityresult proved above.


2. Note that the result presented above can be viewed as a (rather special)equal areas (volumes) rearrangement result.

There are many other results in the literature that address eigenvalue problemsfor the laplacian for annular domains (and their higher dimensional analogs). Webriefly discuss several of these. An early result of Payne and Weinberger [147] saysthat among all annular domains in R

2 of given area A and with outer boundary∂Ω0 of fixed length L0, if we impose Dirichlet boundary conditions along ∂Ω0 andleave the rest of the boundary (∂Ω \ ∂Ω0) “free” (or we could impose Neumannboundary conditions on the rest of the boundary), then λ1 is maximized when theouter boundary is a circle of length L0 and the inner boundary is the concentriccircle such that the area between the circles is A (in fact, for this result to holdthere could be many “inner boundaries” to the original domain). Also, see [25],pp. 144–146, for a generalization of this result.

A counterpart to the Payne–Weinberger result is the result of Hersch [87] thatfor a bounded multiply-connected domain in R

2 of fixed area A and with Dirichletboundary condition imposed along one “inner boundary” Γ0 of fixed length L0

and all other boundaries free, the first eigenvalue λ1 is maximized when the innerboundary is a circle (of length L0), there are no other inner boundaries, and theouter boundary is a circle concentric with the inner boundary of such radius thatthe area between the two boundaries is A. The outer boundary is free, i.e., oneshould impose Neumann conditions along it. Again, Bandle [25] discusses thisresult (pp. 147–149) and gives some extensions.

Note that both of the theorems above allow one to obtain computable upperbounds to the first eigenvalue in appropriate situations, as both of the maximizingproblems are amenable to exact solution (via separation of variables and the use ofBessel functions).

Some more recent results with a similar flavor are those of Exner, Harrell,and Loss [63]. These authors consider what we shall call “thickened” curves andsurfaces in R

n. For example, in R2 one could consider a closed, nonintersecting

curve and thicken it by including all points within a distance d of the curve to geta two-dimensional domain Ω. If d is small and the curve is nice (C2, say), thenΩ will also be nice and we can hope to understand its eigenvalues in terms of theunderlying curve. (That all the curves and surfaces we consider are nice enoughin this sense, and d small enough, will be an implicit assumption in our entirediscussion of the results of Exner, Harrell, and Loss.) In particular, within certainrestricted families of curves we may be able to determine the curve or curves thatmaximize (or minimize) a given eigenvalue of the laplacian on Ω, especially thefirst. In variants of this overall setting, one may decide to include only those pointswithin distance d of the curve and lying either inside or outside. If one moves up indimension (to R

n, n > 2, say), but stays with curves, then the thickening processleads to tubes, for which we again might hope to be able to say something abouteigenvalues based on the underlying geometric structure (and, indeed, there areresults in this direction). But, in keeping with our focus on domains of “annulartype,” the generalization to which we shall adhere in the present discussion is thatwhere a bounded closed hypersurface is considered in R

n with the hypersurfacehomeomorphic to S

n−1. We can then thicken this hypersurface into a shell byincluding all points of R

n within distance d of it. Again, in this case, as for a


closed curve in R2, we can choose to include only the points within distance d of

the surface and inside (resp., outside) it, should we so choose.We are now in a position to state some of the results of Exner, Harrell, and Loss.

First, in the setting of closed curves in R2, among all curves of a given length and

for a fixed choice of d (suitably small), the first eigenvalue of the thickened curve isuniquely maximized when the curve is a circle. (This holds whether the thickeningis done to both sides, or in either one-sided sense. But one should make a choiceand adhere to it.) Second, if we move up to closed surfaces in R

3 homeomorphicto S

2 and we fix the area of the surface and the volume of Ω, and insist that thesurface be convex and that the thickening be done only towards the inside of thesurface, then again the first eigenvalue λ1 is uniquely maximized when the surfaceis a (Euclidean) sphere.

Remarks. 1. Note that in the case of closed curves in R2 the constraint on the

length and specification of d (and how the thickening is done) completely determinethe area of Ω. Thus the result for closed curves is actually an “equal areas” result andas such could be compared or contrasted with the Rayleigh–Faber–Krahn inequalityand other such results. Obviously the second result is an “equal volumes” result asstated, and as such bears similar comparisons. Note that equal volume in this caseentails changing the value of d when one changes from one surface to another.

2. There is a sense in which the results of Exner, Harrell, and Loss discussedabove relate directly to eigenvalue problems on the underlying curves or surfacesthemselves. This can be seen if one considers the limiting situation when d → 0and renormalizes the eigenvalues (this works because with d going to infinity theeigenvalues also go to infinity, but in a controlled and uniform way). The basicoperator then becomes the laplacian on the curve or surface as induced from theEuclidean metric of R

n, possibly modified by potential terms coming from curvatureeffects. Thus, one arrives at the study of a sort of geometric Schrodinger operator,some cases of which are discussed in Section 14 of this paper. These also havephysical relevance in the setting of electrical properties of nanoscale structures (theconsideration of quantum wires, quantum optics, and the like).

Finally, we mention that some interesting results for the first (resp., second)Dirichlet eigenvalue of concentric annular domains on the sphere S

2 have beenobtained by Shen and Shieh [163] (resp., Shieh [164]). By “concentric annulardomains” in this setting we mean domains defined as the region between two con-centric circles on the sphere. For the purposes of our descriptions, we shall speak interms of the north and south poles, the equator, and lines of latitude and longitude.Furthermore, we shall always consider the circles that bound our annular domainsto have their centers at the poles so that these circles become lines of latitude.Obviously there is no loss of generality in this restriction, and once the results arestated the reader will have no difficulty applying them to annular domains in arbi-trary position. For λ1, the result is that among spherical bands (belts) of fixed areaon S

2, the one which maximizes λ1 is the symmetric band centered on the equator(an equatorial belt). Indeed, Shen and Shieh show that λ1 increases monotonicallyas one pushes the belt from its extreme position as a polar cap to the centeredequatorial position. This fits nicely with the Rayleigh–Faber–Krahn result on S

2,as the polar cap is the minimizer of λ1 among all domains of a given area, andthus initially as one slides the belt away from the pole λ1 must go up. Shen andShieh add to that by showing that it just keeps increasing until its other extreme


position is reached. (Note that if one removes the restriction that the boundarycircles be centered at the poles, but insists that they remain concentric, then themain result is that λ1 is maximized by any belt of the prescribed area centered ona great circle.)

The second paper shows that the same main statement can be made for λ2 ifone restricts the total area of the band to be less than or equal to 2π (the areaof a hemisphere). That is, Shen shows that among concentric annuli (i.e., bands)on S

2 of a given area A ≤ 2π, the band centered on the equator maximizes λ2.Though monotonicity is not shown, it is conjectured, and monotonicity is shownfor the analogous problem in R

2. Moreover, it is shown that λ2 for any band isdoubly degenerate (i.e., λ2 has multiplicity 2) and that the nodal line of any secondeigenfunction is given by two pieces of longitudes (parts of the same great circlethrough the poles, where again we are restricting our description to the case oflatitudinal bands).

10. Inequalities Between Dirichlet and Neumann Eigenvalues

A different type of universal inequality that one can consider involves a com-parison between Dirichlet and Neumann eigenvalues. Several inequalities betweenDirichlet and Neumann eigenvalues have been established. Let 0 < λ1 < λ2 ≤ . . .denote the Dirichlet eigenvalues and 0 = µ0 < µ1 ≤ µ2 ≤ . . . denote the Neumanneigenvalues for a bounded domain Ω ⊂ R

n, with a smooth boundary. The min-maxcharacterization of Dirichlet and Neumann eigenvalues gives the easy comparison,

µk−1 ≤ λk, for all k ≥ 1. (10.1)

It was suggested by Payne (see [139] or, for example, [142], p. 155) that

µk ≤ λk, for all k ≥ 1, (10.2)

and perhaps more, especially for convex domains. The first such inequality,

µ1 < λ1, (10.3)

was proven by Polya [149] (motivated by the work of Kornhauser and Stakgold[100]). This is an immediate consequence of the Szego–Weinberger inequality, theRayleigh–Faber–Krahn inequality, and an explicit calculation for balls, through thechain of inequalities,

µ1(Ω) ≤ µ1(Ω⋆) < λ1(Ω

⋆) ≤ λ1(Ω), (10.4)

as first suggested in [100]. The first proof of (10.3), however, as given by Polya[149], used a different intermediate result, which circumvented any direct needfor the Szego–Weinberger inequality. (In connection with (10.4), we note thatµ1(Ω

⋆)/λ1(Ω⋆) = p2

n/2/j2n/2−1 < 1, for all n.) A little later Payne [139] proved that

if in two dimensions Ω is convex,

µk+1 < λk, k ≥ 1. (10.5)

Then, Aviles [24] showed the strict inequality,

µk+1 < λk, k=1,2, . . . (10.6)

for domains in Rn with a C2+α boundary having nonnegative mean curvature.

Concurrently and independently, Levine and Weinberger [110] proved this sameresult as part of a family of results for domains in R

n based on hypotheses on theprincipal curvatures of ∂Ω. Their first result was (10.6) under the same nonnegative


mean curvature condition that Aviles used, but then they took progressively morerestrictive hypotheses on the principal curvatures, obtaining stronger and strongerinequalities, until reaching the convex case (and the result, (10.8), given explicitlybelow). Finally, in 1991, Friedlander [69] proved Payne’s suggestion,

µk ≤ λk, for k ≥ 1, (10.7)

for domains Ω ⊂ Rn with a C1 boundary. (For an alternative proof see [130], and

more recently [66].) Levine and Weinberger [110] had shown earlier, among theirother results, that for Ω a bounded, smooth, convex domain in R

n (n ≥ 2),

µk+n−1 < λk, for all k ≥ 1. (10.8)

Inequalities between Dirichlet and Neumann eigenvalues have also been con-sidered for domains in S

n [22, 94]. In particular, one has

µ1(Ω) < λ1(Ω), (10.9)

for smooth domains Ω contained in a hemisphere of Sn, which follows from Sperner’s

inequality [167], our analog of the Szego–Weinberger inequality (valid for domainscontained in a hemisphere) [17], and exact calculations for geodesic balls. Noticethat (10.9) cannot hold for general domains in S

n, since µ1 = λ1 = n at thehemisphere in S

n, and µ1 > λ1 for geodesic balls larger than the hemisphere. In[22], the analog of the Aviles bound is proven for domains in S

n, more precisely, thatµk(Ω) ≤ λk(Ω) for a domain Ω ⊂ S

n whose boundary is everywhere of nonnegativemean curvature (with strict inequality if the mean curvature is ever positive). Thisand further results, based on a more differential-geometric approach, are to befound in [94].

11. Polya’s Conjectures

For large values of k, if Ω ⊂ Rn, Weyl [180] (see also [181, 155]) proved

λk ≈ 4π2k2/n

(Cn|Ω|)2/n, (11.1)

where Ω and Cn = πn/2/Γ(n/2 + 1) are, respectively, the volumes of Ω and of theunit ball in R

n. Relation (11.1) is usually referred to as Weyl asymptotics or asWeyl’s law. For any plane-covering domain (i.e., a domain that can be used to tilethe plane without gaps or overlaps, allowing rotations, translations, and reflectionsof itself), Polya [152] proved that

λk ≥ 4πk

A, for all k = 1, 2, . . . (11.2)

and conjectured the same bound for any bounded domain in R2 (here A denotes

the area of the domain). Polya’s conjecture in n dimensions is equivalent to sayingthat the Weyl asymptotics of λk (11.1) is in fact a lower bound for λk, i.e.,

λk ≥ 4π2k2/n

(Cn|Ω|)2/n, (11.3)

for k = 1, 2, . . . . This can be shown for tiling domains in Rn, following Polya’s

proof of the two-dimensional result. The best result to date towards a proof of the


Polya conjecture (11.3) is the bound of Li and Yau [113]

k∑

i=1

λi ≥n

n + 2

4π2k1+2/n

(Cn|Ω|)2/n, (11.4)

for all k = 1, 2, . . . , proven using the asymptotic behavior of the heat kernel ofΩ and the connection between the heat kernel and the Dirichlet eigenvalues of adomain (see, e.g., [55] for a review and related results; see also [132] for a recentimprovement on (11.4) that includes a correction term). From (11.4) it followseasily that the individual eigenvalues λk satisfy

λk >n

n + 2

4π2k2/n

(Cn|Ω|)2/n(11.5)

for all k ≥ 1. In a sense (11.4) is an “integrated” form of the conjectured bound(11.3).

There are related results also for Neumann eigenvalues. For large values of k,Weyl [180] proved

µk ≈ 4π2k2/n

(Cn|Ω|)2/n. (11.6)

For any plane-covering domain, Polya [152] and, in full generality, Kellner [98]proved

µk ≤ 4πk

A, for all k = 1, 2, . . . (11.7)

and conjectured the same bound for any bounded domain in R2. The analogous

conjecture in n dimensions is

µk ≤ 4π2k2/n

(Cn|Ω|)2/n, (11.8)

for k = 1, 2, . . . . Thus far, Kroger [103, 104, 105] has obtained the most significantresults towards a proof of Polya’s conjecture for Neumann eigenvalues, (11.8). Inparticular, he has proven the following integrated version of (11.8)

k∑

i=1

µi ≤n

n + 2

4π2k1+2/n

(Cn|Ω|)2/n, (11.9)

for all k = 1, 2, . . . . He has also proven that

µk ≤(

n + 2

2

)2/n4π2k2/n

(Cn|Ω|)2/n, k = 0, 1, 2, . . . , (11.10)

for the individual µk’s. We note that a proof of Polya’s conjectures for both Dirich-let and Neumann eigenvalues would imply Friedlander’s result (10.7). Polya’s con-jectures for Ω ⊂ R

2 were first published in [150] (in a weaker and preliminary formfor the Neumann case), and later, in definitive form, in [152].

Remark. Inequalities of the form

λk ≥ αn4π2k2/n

(Cn|Ω|)2/n, (11.11)

for some constants αn < n/(n + 2) (cf. (11.5)) were proven for bounded domainsin [35, 50] and later for arbitrary domains in [159, 160, 133, 116]. The result of


Li and Yau gives (11.11) with αn = n/(n + 2). Polya’s conjecture says that (11.11)should hold with αn = 1 for an arbitrary bounded domain Ω.

12. The Bass Tone of a Drum and Its Inradius

The inradius rΩ of a domain Ω ⊂ R2 is the radius of the largest disk that

can be inscribed in Ω. It follows from domain monotonicity that among all planar

regions of fixed inradius, the disk has the highest bass tone (throughout this section

we will call the square root of the lowest Dirichlet eigenvalue of Ω,√

λ1(Ω), its bass

tone; this represents the actual lowest vibrational frequency of a uniform drum ofshape Ω up to an overall constant factor determined by physical parameters). In1951, Polya and Szego [153] found the first lower bounds for r2

Ω λ1(Ω), for convexdomains, and raised the question of finding the best lower bound

Λ ≡ infΩ

r2Ω λ1(Ω) (12.1)

for general simply connected domains. In 1965, Makai [127], proved the lowerbound

Λ ≥ 14 (12.2)

(see also [126, 81, 135, 173, 154, 54]). If Ω is an infinite strip of width 2, oneobtains r2

Ω λ1(Ω) = π2/4, hence,

Λ ≤ π2

4= 2.467 . . . (12.3)

(π2/4 is, in fact, a sharp lower bound for r2Ω λ1(Ω) among all convex domains [86]).

Using probabilistic methods, Banuelos and Carroll [26, 27, 28, 42], improved thelower bound on Λ and obtained

Λ ≥ 0.6197 . . . (12.4)

On the other hand, by considering an explicit, nonconvex domain, they also showedthat

Λ ≤ 2.1292 . . . (12.5)

While (12.4) and (12.5) are the best estimates on Λ to date, determining its exactvalue, and characterizing the optimizing domain(s), are still open problems.

It was long ago observed by Polya that the inradius rΩ is a much better geo-metric indicator of the bass tone (or λ1) of a membrane (drum) than its area A, theRayleigh–Faber–Krahn inequality notwithstanding, in the following sense. Specifi-cally, even though one has the Rayleigh–Faber–Krahn lower bound, there can be noupper bound in terms of A since a membrane of given area can have arbitrarily large

first eigenvalue (consider a × b rectangles with fixed area ab). On the other hand,the results above show that a simply-connected domain has its bass tone boundedboth above and below by constant multiples of 1/rΩ, and, in fact, the constantsare not that far apart. This was quite true already in Polya’s time, and he is saidto have remarked that rΩ determines the bass tone of a drum to within an octave.Indeed, with the improved constant due to Banuelos and Carroll there is only a fac-tor of approximately 3.05 between the upper and lower bounds (

√0.6197 ≈ 0.7872

vs. j0,1 ≈ 2.4048). For convex domains, of course, the agreement is even better:The ratio between j0,1 and π/2 is approximately 1.53. Thus if one estimates thebass tone by taking the geometric mean of the two bounds for a convex drum, onecannot be off by more than about 25% either way.


It might also be observed that the fundamental gap λ2 − λ1 of a simply-connected domain has an isoperimetric upper bound in terms of inradius, viz.,λ2 − λ1 ≤ (j2

1,1 − j20,1)/r2

Ω. This follows at once from the result (due to Banuelos)quoted at the end of Section 4, using c = 1 and rΩ ≤ r (see [153]). Unfortunately,a rectangle with one dimension arbitrarily large shows that no corresponding lowerbound is possible (i.e., there can be no “inradius lower bound” for the fundamen-tal gap). Moreover, this remains true even if we restrict consideration to convexdomains.

13. The Eigenvalue Ratio λ3/λ1

The problem of maximizing the ratio λ3/λ1 in two dimensions, has been consid-ered by several authors. The maximum is certainly not achieved by the disk (for the

disk, λ3/λ1 ≈ 2.539, while for the√

3 by√

8 rectangle, λ3/λ1 = 35/11 ≈ 3.182).Marcellini [128], found the upper bound 3.917, which was improved in 1994 byAshbaugh and Benguria [14] to 3.905 and then in 1996 to 3.831 [18], so the op-timal bound is between 3.182 and 3.831. Levitin and Yagudin [112], via a nu-merical search, found a certain dumbbell shape for which λ3/λ1 is approximately3.202. Recently, Trefethen and Betcke [175] have computed numerically this ratiofor a domain consisting of the union of the disks of radius 2.002323 centered at±1.191322. For this particular domain, λ3/λ1 falls short of the value of Levitin

and Yagudin by approximately 0.35%. For each of the√

3 by√

8 rectangle, thedomain considered by Trefethen and Betcke, and several domains found by Levitinand Yagudin (maximizing λ3/λ1 within one class or another), one has the propertythat λ3 = λ4. It has been conjectured by Ashbaugh and Benguria [15] that thedomain that maximizes λ3/λ1 should satisfy λ3 = λ4 (notice also that the optimalupper bound for λ2/λ1 is attained for the ball, at which λ2 = λ3). This conjec-ture is still open. Some indications in its support may be found in [112] (see, inparticular, Theorem 4.2).

14. An Isoperimetric Inequality for Ovals in the Plane

The next problem we will consider is a conjectured isoperimetric inequalityfor closed, smooth curves in the plane. It has attracted considerable attention inthe literature during the last decade (see, e.g., [60, 77, 63, 75, 40]), and it hasmany interesting connections in geometry and physics. In particular, Benguria andLoss [31] have shown a connection between this problem and a special case of theLieb–Thirring inequalities [120, 117], inequalities which play a fundamental rolein Lieb and Thirring’s proof of the stability of matter (see, in particular, [119] andthe review article [115]).

Denote by C a closed curve in the plane, of length 2π, with positive curvatureκ, and let

H(C) ≡ − d2

ds2+ κ2 (14.1)

acting on L2(C) with periodic boundary conditions. Let λ1(C) denote the lowesteigenvalue of H(C). Certainly, λ1(C) depends on the geometry of the curve C. Ithas been conjectured that

λ1(C) ≥ 1, (14.2)

with equality if and only if C belongs to a one-parameter family of ovals whichinclude the circle (in fact, the one-parameter family of curves is characterized by a


curvature given by κ(s) = 1/(a2 cos2(s)+a−2 sin2(s))[31]). It is a simple matter tosee that if C is a circle of length 2π, the lowest eigenvalue of H(C) is precisely 1.The fact that there is degeneracy of the conjectured minimizers makes the problemmuch harder.

The conjecture (14.2) is still open. Concerning (nonoptimal) lower bounds,Benguria and Loss proved [31]

λ1(C) ≥ 12 , (14.3)

and more recently Linde [121] found the best lower bound to date,

λ1(C) >

(

1 +π

π + 8

)−2

≈ 0.60847. (14.4)

Even though this is the best bound to date, it is still some distance from theconjectured optimal value 1.

Very recently Burchard and Thomas [41] have shown that the ovals that giveλ1(C) = 1, alluded to before, minimize λ1(C) at least locally, i.e., there is no smallvariation around these curves that reduces λ1(C). This is an important indicationthat the conjecture (14.2) is true, though, of course, it is not enough to prove itand thus it remains open.

In recent years several authors have obtained isoperimetric inequalities for thelowest eigenvalues of a variant of H(C), and we give a short summary of the mainresults in the sequel. Consider the Schrodinger operator

Hg(C) ≡ − d2

ds2+ gκ2 (14.5)

defined on L2(C) with periodic boundary conditions. As before, C denotes a closedcurve in R

2 with positive curvature κ, and length 2π. Here, s denotes arclength. Ifg < 0, the lowest eigenvalue of Hg(C), say λ1(g, C), is uniquely maximized whenC is a circle [60]. When g = −1, the second eigenvalue, λ2(−1, C), is uniquelymaximized when C is a circle [77]. If 0 < g ≤ 1

4 , λ1(g, C) is uniquely minimizedwhen C is a circle [63]. It is an open problem to determine the curve C thatminimizes λ1(g, C) in the cases, 1

4 < g ≤ 1, and g < 0, g 6= −1. If g > 1 the circleis not a minimizer for λ1(g, C) (see, e.g., [63, 75] for more details on the subject).

15. Open Problems

As we have seen throughout the paper, there are many open problems concern-ing universal and isoperimetric bounds for low-lying eigenvalues of the laplacian.To finish, we will list a few more of the most interesting conjectures in the field(see [140, 142, 184, 15, 3] and references therein for additional open problems).Other interesting open problems (and much else of interest) can be found in theforthcoming book by Henrot [83].

Perhaps the most significant outstanding problems concerning eigenvalue ratiosare the following conjectures of Payne, Polya, and Weinberger:

(i) For Ω ⊂ R2

λ2 + λ3

λ1(Ω) ≤ λ2 + λ3

λ1(Ω⋆) ≈ 5.077, (15.1)

and its n-dimensional analog (i.e., for Ω ⊂ Rn)

λ2 + λ3 + · · · + λn+1

λ1(Ω) ≤ λ2 + λ3 + · · · + λn+1

λ1(Ω⋆). (15.2)


For nonoptimal bounds on these quantities, see [144, 37, 128, 13, 14, 18]. Inparticular, the best bound proved to date for (λ2 + λ3)/λ1 in two dimensions isapproximately 5.507 [18], while the conjectured best bound is given by (15.1).

(ii) For Ω ⊂ Rn, and m = 1, 2, 3, . . . ,

λm+1

λm(Ω) ≤ λ2

λ1(Ω⋆). (15.3)

This inequality has been proved for the cases m = 1, 2, and 3 [8, 11].

(iii) Indeed, it might well be true that for integers m > 1,

λ2m

λm(Ω) ≤ λ2

λ1(Ω⋆). (15.4)

This was not conjectured by Payne, Polya, and Weinberger, although it wouldrepresent a substantial strengthening of (ii) above. The inequality has been provedfor m = 2 (see [11]). For further discussion and indications in this direction, see[11, 16]. A similar conjecture could be made for λkm/λm(Ω), which would be

that λkm/λm(Ω) ≤ supΩλk/λ1(Ω), but for this to be truly useful one would need

to know or have good estimates for supΩλk/λ1(Ω), and the present state of ourknowledge for k > 2 is not nearly as good as it is for k = 2 (the case addressed by(15.4)).

(iv) For Ω ⊂ R2, show that

λ4

λ1(Ω) ≤ λ4

λ1(Ω⋆) ≈ 4.5606. (15.5)

The best bound to date is (λ4/λ1)(Ω) ≤ (λ2/λ1(Ω⋆))2 ≈ 6.4452 [11].

(v) For Ω ⊂ R2, show that

λ3(Ω) ≥ λ3(Ω⋆), (15.6)

or even that

(λ2 + λ3)(Ω) ≥ (λ2 + λ3)(Ω⋆). (15.7)

Corresponding inequalities can be formulated for Ω ⊂ Rn; see [3] for details.

(vi) Friedlander’s conjecture: It has been observed by Friedlander that possibly

µk+n−1 < λk, for all k ≥ 1. (15.8)

This is (10.8) from Section 10, so it is known for all convex domains in Rn. The

novelty here is that this inequality, with index shift n − 1 in dimension n, mightindeed be true for all domains (one should disregard a claimed example to thecontrary mentioned near the end of [110] and in [109], p. 126). This conjecture, iftrue, would represent a substantial improvement upon (10.7).

(vii) For a convex domain Ω ⊂ Rn, show that

µ2

µ1≤ 4. (15.9)

This bound is the analog of van den Berg’s conjecture for the case of Neumanneigenvalues. It would be saturated by any rectangular parallelepiped having all butone of its dimensions tiny.


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(Mark S. Ashbaugh) Department of Mathematics, University of Missouri, Columbia,

Missouri 65211-4100

E-mail address: [email protected]

(Rafael D. Benguria) Departamento de Fısica, P. Universidad Catolica de Chile,

Casilla 306, Santiago 22, Chile

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