Bounds for functions of eigenvalues of vibrating systems

Bounds

for Functions

of Eigenvalues

of Vibrating

Systems

K. D. GENTRY Department

of Mathematics and Statistics, l?kwsity Guelph, Ontario,

Canada, NlG

of Guelph,

.?Wl

AND

D. 0. BANKS Department

of Mathematics,

Uninmersity

of Califorka,

Davis, Daz,is, California

95616

Submitted by Peter D. Las

1.

INTRODUCTION

The natural frequencies of a vibrating string of length I, under unit tension, and with density given by an integrable function p defined on [0, I] are determined by the eigenvalues of the differential system J” + hp(s) 3’ = 0,

s E [O, Z],

y(0) -= y(Z) = 0.

(1.1)

The eigenvalues of the system (1.1) are positive and may be enumerated in a strictly increasing sequence h,(p), X,(p), AS(p),... having no finite limit. The ith eigenvalue is denoted by hi(p) to emphasize that it is a functional of the density p. In this paper, we will be concerned with finding bounds for functions of the eigenvalues using only general information about the density function p. M. G. Rrein [15] solved the following problem. Let E(M, 0, H), where ill and H are given constants, denote the class of all integrable functions p on [0, I] such that sip(x) dx = M and 0 < p(x) < H for all x E [0, I]. Then for all p E E(M, 0, H) and each integer n

where x(t) is the least positive root of the equation

Copyright All rights

C 1975 by Academic Press, Inc. of reproduction in any form reserved.

BOUNDS

FOR EIGENVALUES

101

The above inequalities are sharp, equality being attained for known functions in E(M, H) for each n. For other bounds of this type see the results of Banks [l-4], Barnes [5], and the references given there. The main results of this paper can be considered an extension of the type of results obtained by Krein. Here we obtain a sharp lower bound for the set of values of the ratio of the second and first eigenvalues X,(p)/&(p) determined by letting the density function p range over the set E(M, 0, H). In particular, we show that

where xl(t) is the least positive root of the equation tan z = ~(1 - t). The unique density function p,, at which this minimum is attained is defined by

PO(X)-

H, i 0, ! H,

x 6 [0, M/2H], .TG (M/2H, 1 - M/2H), s G [1 - M/2H, 11.

We then examine the ratio of the n + 1st and nth eigenvalues, h,+l(p)/X,(p), and show that this ratio is maximized over all densities p E E(M, 0, H), by a density p, which assumes the constant value H except on a set, consisting of at least one and at most n + 1 subintervals, where it vanishes identically. We conjecture that such a maximizing density is in fact unique, symmetric about x = Z/2 and vanishes on exactly n + 1 subintervals. For the case n = 1 we show that the maximizing density does vanish on two subintervals and if we consider only symmetric functions that it is unique. We note here that by the same methods we can show that the function p. which minimizes h,(p)/X,(p) over E(M, 0, H) also minimizes l/X,[p] - l/X,(p). It follows that

where Z, is the same number defined above and depends on HE/M. These problems are special cases of a more general problem which we now define. Let F be a continuous function of n variables si , ,ta ,..., s,,, defined for Nf > 0, i = l,..., n. Then replacing xi by the ith eigenvalue of (1.1) for a given density function p, i = I,..., n, we can define the functional F(p) by letting F(P)

=W~P),

UPL

UP)).

Furthermore, we define a class of functions E(M, h, H) as follows. Let h and H denote two nonnegative integrable functions on the interval [0, I]

102

GENTRY

which satisfy the condition constant Ilf such that

II(X) I<
AND

BANIts:

We then define the set of functions

We may thus consider the problem of finding extreme values of the functional F(p) over the class E(M, h, H) when these extreme values exist. The primary motivation for the study of this type of problem comes from a result of Payne, Polya and Weinberger 1171. They showed that the ratio &+J&, of consecutive eigenvalues associated with homogeneous vibrating membrane satisfy the inequality

They conjectured that the lower bound for the ratio ha/h, is 2.539 and is attained when the boundary of the membrane is circular. In the case II = 2, Brands [6] and DeVries [8] have improved on the upper bound of 3 and Thompson [21] h as extended the bound of [17] to higher dimensions for the ratio ha/A, . However, none of these bounds are sharp so that more sensitive techniques are needed, even for the one-dimensional case considered here. We also note that our results on the lower bound for ha/X, can be used in computing the error in the numerical computation of the second eigenvalue A, by iteration. (See Temple and Bicklep [20].) Other functions for F have been considered. For example, Banks [2] and Schwarz [19] have also considered the sum of the reciprocals of finitely many eigenvalues of the vibrating string. Hersch [lO-121, has given bounds for numerous functions of eigenvalues using conformal mapping techniques. Our method of determining the density functions at which the extremals occur is based upon a variational technique which is developed in Section 2. In the third section we prove a theorem to guarantee that extremizing densities exist over the class E(M, h, H). In Section 4 we give a comparison theorem and some immediate consequences of it, which we then use to find the minimum of ha/h,. The maximum of Xn+i,/An is examined in Section 5. In the last section, we add a few remarks on the limitations of our method and provide a table indicating approximate bounds for the functionals considered for particular values of the ratio of the mass to the upper bound, M/M.

BOUNDS FOR EIGENVALUES Finally, we mention the following notational conveniences. (1.1) we let x1 = x/Z to get the equivalent system d + p(x) u = 0,

103 In the system

x 6 [O, 11,

u(0) = U(1) = 0, where we have dropped the subscript 1. The eigenvalues of (1.2) are 1 times those of (1.1) and p(x) = Zp(h). It follows that if p E E(M, h, H) then p E E(M, hi , Hi) where hi = I/r and Hi = 1H. Hereafter, we consider only the system (1.2) with p E E(M, hi , HI) and drop the use of the subscript 1. In numerous instances in our development we refer to the reflected problem or to the “reflected functions.” By a reflection of a given function f defined on [0, I] we mean the function f defined by j(x) = f (1 - x). Hence in reference to the reflected case, we mean that the density p in (1.2) should be replaced by the density function F. It is clear that with this substitution the eigenvalues of the system (1.2) remain unchanged whereas the eigenfunctions u, are transformed into the new eigenfunctions given by ZC,.

2. VARIATIONAL CHARACTERIZATION In this section, we develop the variational techniques by which we determine the nature of the extremal density functions for certain types of functionals F. Our approach is to use the first variation of the functional with respect to an admissible variation of a density function in order to derive necessary conditions for a density function to be extremal. We utilize the fact that a necessary condition for a function to be a maximal or minimal element of a functional over a given class is that all admissible variations from the element result in nonpositive or respectively nonnegative variations of the functional. We consider any class E of density functions p, defined on the unit interval [0, l] and satisfying the equality ji p(x) dLx = ill. An admissible variation of a function p E E is defined to be a function Sp such that (p + Sp) E E. Associated with each density function p E E are the eigenvalues of (1.2), Banks [2] has characterized the variation of these eigenvalues UP), UP)Y. in the THEOREM 1. The$rst variation of the nth eigenvalue of a vibrating string of unit length, constant tension, jked endpoints and whose density is given by p is SUP)

= -UP)

JO1G’(X) &P(N) dbv,

where 8p is an admissible ,z>ariation of the densit~l p in the class E and II,, is the 12th eigenfunction corresponding to h,,(p) normalized to satisjlq

We consider a differentiable tional F defined by

function F of r/-variables

F(P) = F@,(P), UP),...,

and form the func-

UP)),

where p is a nonnegative density function defined on the unit interval X,(p) denotes the ith eigenvalue of the differential system (1.2). We then have the following generalization of Theorem 1.

(2.1) and

THEOREM 2. Let the functional F be defined on a class E of nonnegatizle density functions p, by F(p) = F(A(p)) where F is a dzflerentiable function of n-oariables, h(p) = (h,(p), X,(p),..., X,Jp)) and h,(p),..., h,(p) are the first n eigenaalues of the vibrating string of unit length with fixed endpoints and density p(x). If Sp is an admissible e?ariation of a p in the class E, then the variation of F is giz>en by

where ui = u,(p), the eigenfunction associated with h,(p), is normalized so that

s

’ zQ(x) p(x) dx = 1. 0

This theorem is a simple consequence of applying functional calculus to the functional F (see [18]). It is apparent that the nature of 6F(p), f or any given variation 8p of the density p, will be completely determined by the explicit behavior of the function g(p) defined on [0, l] by

.g(p; 4=[i - h,(p) w 2$2(x)] , i=l

(2.2)

I

which is independent of Sp. In particular, for a density p E E to yield a maximum or minimum value of the functional F over the class E, the function g = g(p) must be such that si g(x) 6p(x) d.r is nonpositive or nonnegative, respectively, for all admissible variations Sp of p in the class E.

BOUNDS FOR EIGENVAJ..UES

105

THEOREM 3. Let F be a functional of the form F(p) = F(h,(p),..., h,(p)) where p E E(M, h, H) and h,(p) denotes the ith ezgenvalue of a vibrating string of density p. If the maximum value of F over E(M, h, H) is attained at a density p, , then p, must be of the form

(2.3) where So is a measurable subset of [0, I] such that

jolpo(x)

dx = Is0 H(x) dx + jsoC h(x) d.z = ~1

(2.4)

and on So , g(po , x) satisfies the inequalities

g(Po; -42 to g(p0; 4 -i to

for

x E so ,

for

x E s,c,

for some to in the range of g. If the minimum value of F over E(M, h, A!) is attained at a density p, , then p, must be of the form (44, p1(x) = IH(x),

.t E s, ) .VE SIC,

(2.5)

where S, is a measurable subset of [0, l] such that jol p,(Gv) dx = $,, h(.r) dx + jslC H(x) ds = M

(2.6)

and on S, , g(p,; x) satisjies the inequalities

g(p,; x) .‘- t,

fOY x E s,

g(p,; x) < t1

fOY x E SIC,

,

for some t, in the range of g. Proof.

We first define two set-valued functions for a given density p by

S(t)= (xEco,11Ig(p;x) 2 t>, and

T(t)=

(x E [O, l] I g(p; x) > t}.

GENTRY Ah-DILOKS

106

Note that for t, < t, , S(t,) & S(t,) and that the set on which g(p) has value t is S(t)\,T(t). For a given function p E E(M, h, H), we define

M,(t) = ( H(s) tix L j- h(x) dx, ‘SC(t) -S(t) where SC(t) = [0, l]‘$(t). From its definition (2.2), it is clear that g(p) is a continuous function of x on [0, I]. If t' is a number less than the minimum of g(p) and t" is a number greater than the maximum of g(p) on [0, 11, then we note that S(t') = [0, I] and S(t") = ‘7. It follows that

M,(t’)

= j1 H(r) ds 3 M,(t) 2 j1 h(x) ci.r = N,(t”)

for all

t

E

[t',

t"].

0

0

Also, since p E E(M, A, H), we have Mp(t’) ,) M > MJt”). Since s(t) is a decreasing set function, it is clear that M, is a decreasing function of t. Hence, it can only have jump discontinuities and there is a number to such that either ilID = 31 or MD(to-) > 111> MD(to+). In the latter case, we note that

M,,(t,-)

= jsct ) H(x) dx + jscct ) h(x) ds 0 0

and

We define S, = T(t,) + [0, 01 n (S(t,)\,,T(t,)) and note that

is a continuous function of 0 with the properties that MO = MJt,+) and Ml = MD(to-). Hence, there is a number 0, such that J$, = dl, i.e., there is a set SO, such that

j

II(.r) dx + j s.90

h(x) d.z-= M

and

ses0

Vo)

c so0 c S(to).

Now let p =p, , the maximizing density of F over E(M, h, H) and let So = SO, , the corresponding set determined by the above analysis. Define

p. on P, 11by d”)

(H(x), N E so ) = /Q.), s $!s,,

BOUNDS

107

FOR EIGENVALLJES

We note that p,, E E(M, h, H) and consider the variation SF&,) of IQ,,) affected by the admissible variation of p, , Sp, = p0 - pO. Using the variational formula of Theorem 2, we obtain the identity

where &x) is greater than or equal to t, on S,, and less than or equal to t, on the complement Ssc. Since p, E E(A9, h, H), we have

so >,to[js H(x)--P&)h + js 44 - POW d-r] 0 UC (2.7)

1

=

t0

[S

o pa(x) dx -

~olpo(.~) d-x.1 > 0.

The inequality in (2.7) will be strict unless the functions p,, and p, are equal except on a set of measure zero. Thus any maximizing function for the functional F must have the form (2.3) and (2.4). For otherwise, there would exist a positive variation SF indicating a maximum had not been attained. This completes the proof of the first half of the theorem. To verify the second part we proceed in an analogous manner. We consider a function m = m(p) defined by m(t) = j

Q) S(f)

d.y + j

H(x) d.r. SC(f)

Then we take p to be a density pr at which F attains a minimum value over the class E(M, h, H). By choosing the value t, as the greatest value t for which m(t) is less than or equal to M, we find a set S, satisfying (2.6) and define the function pr given by (2.5). Forming the admissible variation Sp, = pr - p, , we find that the corresponding variation of F(p,) satisfies

WP,)G4 [js 1 h(s)

- pi(x) ds + J;- H(x) - p,(x) dx] = 1<

0.

The above inequality is strict unless p1 and p, agree almost everywhere. Hence we conclude the second assertion of the theorem is also valid. The Theorem 3 establishes the fact that the extreme points for the functional F over the class E(M, h, H) are on the boundary of the class. That is, a density p, at which the functional F assumes a maximum or a minimum value, may be perturbed outside of the class E(M, h, H) by an arbitrarily small local perturbation at any point x in [0, 11. We also note that Theorem 3 is a generalization of a result of Nehari [16].

3. EXISTENCE OF ESTRERIALS

In order to fully utilize the Theorem 3, which provides a characterization of those densities which yield the maximum or minimum value of a particular functional, it is necessary to know a priori that such extreme densities esist. Krein [15] considered the existence of extreme densities for the first eigenvalue of (1.2). TVe generalize Krein’s result to the consideration of functionals of the form (2.1) in the following: THEOREM 4. Let E(dI, h, H) denote the class of measurable density’ function p satisfying for constants M, h, H, 0
Proof. Because p E E(M, h, H) implies that 0 :iZ p(x) S: H, we have the inequality 4,/-V < A,(p) < AZ(p) < ... < X,,(p) -< z-*Hn’~M’. The upper bound for All(p) was obtained by Krein [ 151 and the lower bound is due to the fact that the lowest possible frequency of vibration of a string occurs when all the mass is concentrated at the center of the string. Consequently the set {fU( P) I P E E(N 11,H), ’ is bounded. Let p denote a point in the closure of {F(X(p)) 1p E E(A1, /z, H)j. Then there exists a sequence of densities {P”}%~ contained in E(M, A, H) such that the sequence {F(A(pz))j~zl converges to p. Since the sequence of vectors tA(pP)));7=is bounded, we may choose (p”) such that the sequence {A( converges to a vector (pI , p-LI!,..., ~L,J. By the continuity of F, we then have

The proof will be complete if we can show that there exists a density pa E E(ill, k, H) such that pj = A,(p”) for j = 1, 2,..., n. To accomplish this, we first show that the sequences of densities (pi). may be so chosen that the sequence converges to a density pa E E(M, h, H). Then, to show that the eigenvalues of (1.2) with p = po are given by the values pLi , following Krein, we argue that the original sequence {p’> can be selected such that the sequence of jth eigenfunctions (u~,~;., determined by the sequence {p”}, converges to a function ujo for each j = 1, ?,..., n. Finally, utilizing the convergence of the sequences {X,(p’)}, {u~,~} and (pi> we may claim that indeed the jth eigenvalue determined by pa is equal to pj . We first note that each density p? determines a nonnegative signed measure mj defined on the Bore1 subsets of [0, l] by mi(B) = jepi(x) dx. Since p’ E E(?l/, A, H) for each i, 11e have the sequence of measures {nl;l is uniformly

BOUNDS

109

FOR EIGENVALUES

bounded. Consequently there is a subsequence (vz~,~} of the sequence (mi} and a signed measure m, such that the subsequence converges to m, in the w*-sense; i.e., limj-, $if dmi,j I= sif dm, for each continuous function f defined on [0, 11. We may for convenience assume that our original sequence was so chosen that the sequence {mi} itself converges to m, . As m, is absolutely continuous there exists a function p” such that for each Bore1 subset of [0, I], B, m,(B) = jBpo(x) da. F rom this, the weak* convergence of the measures mi , and the integral characterization of the class E(ilI, R, H) we may conclude that the density p” E E(M, h, H). In order to examine the convergence of the eigenfunctions, uj,i = uj(p”), we convert the system (1.2) into the equivalent integral system

U(X) = h 1’ G&r, t) u(t) p(t) dt, where

is the Green function

(3.1)

0

associated with the second order differential

operator

L = d”/dx” on [0, l] with zero boundary conditions. We shall assume that the eigenfunctions are normalized by the condition J”: Z&(X) pi(x) dx = 1. For each j = 1, 2,..., n, we may conclude that the sequence {~~,~}~=r and {u:,~}+, are both uniformly bounded by utilizing the Cauchy-SchwarzBuniakowsky inequality in (3.1) and the fact that both G and aG/Ex are uniformly bounded. Hence, by Arzela’s Theorem, we may assert the existence of a continuous function uJ,o and a subsequence of our original sequence which converge uniformly to uj,o . We may thus assume that our original sequence of densities {p’)‘T=, is so chosen that the sequence of eigenfunctions {~~~~}~!r converges uniformly to uj,o for each j = 1, 2,..., n. We are now in a position to verify that pj = X,(p”) for j = 1, 2,..., 71.To this end we write the equivalent of the equation (3.1),

h,&&)z=z s’ G(s, t) z+(t)

dm,(t).

0

We now take the limit as icc in the equation (3.2). The limit of the left side of the equation is immediate. That we may pass to the limit under the integral of the right side of (3.2) is due to the uniform convergence of the sequence {~~.~>;=r to a continuous function uj,o and the weak* convergence of the measures m, (see Helmes [9, Lemmas 2.10, 2.111). Thus passing to the limit in (3.2) and using the representation, p”, of the measure m, , we obtain the equality

ZL~,~(.T) = pj 1’ G(r, t) qo(t) p”(t) dt. ‘0

(3.3)

GEN'I'RY AND BANKS

110

\Ye conclude from (3.3) that CL,is an eigenvalue associated with the density ~0. That p, is actually thejth eigenralue follows from the observation that the function uj,o must have precisely (j - 1) nodes in (0, 1) due to the uniform convergence involved and the fact that an eigenfunction cannot have multiple zeros. This completes the proof of Theorem 4.

4. THE ~~INIRIUM

0~ A,/h,

We are now in a position to search for the extreme values of particular functionals F. In light of Theorem 4, we may employ Theorem 3 to transform the search for the density or densities which yield a particular estremum to an examination of those densities which have the form given in Theorem 3. In the event that this subclass consists of a single element we will have guaranteed that this density uniquely yields the desired extremum. In this section, we use this approach to determine sharp lower bounds for the ratio of the second to the first eigenvalues of a vibrating string with fixed endpoints, a nonnegative density given by a measurable function bounded above by a constant H, and with total mass AZ. Such density functions constitute the class E(M, 0, H) with H d enoting the constant function. In particular, it follows from Theorem 4, that the infimum of AZ/A1over E(M, 0, H) is attained for some function in E(M, 0, H). M’e then use Theorem 3 to obtain the principal result of this section. THEOREM

5. For p in the class

let Al(p) and X,(p) be the first and second eigenzalue of a vibrating density function p. Then infimum

1% 1

string with

1p E E(113, 0, H)[ = #, 10

where p, E E(M, 0, H) is given 63

pa(x) =

H 0 I‘,H

and d = M/ZH.

if if if

0 < x < d, d
Furthermore,

there are no other local minima of h,(p)/h,(p) in w here z is the least positive root of the transcendental equation tan(z) = ~(1 - H/AZ).

WC 0, H) and UP~MP~)

= M77lW

111

BOUNDS FOR EIGENVALUES

The arguments to be presented in proving this theorem depend on the WA) (PN an d consequently on the shape of the graph of the corresponding function g(p) defined by (2.2). In this case, we find that

To make the appropriate analysis of the graph ofg(p), we rely on the following comparison theorem and some implications of it. COMPARISON

the d#erential

THEOREM

(Ince [13]).

Let u and v be nontrivia/

solutions of

equations, u”(X) -+ P(x) U(X) = 0, v”(Jc) + Q(x) v(x) = 0,

dejked on the interval [01,/3] such that either u satisfies the boundary condition U(U) = 0 or the conditions u(a) f 0, v(a) # 0, and u’(ol)/u(ol) 3 v’(c~)/v(o~) are satisjied. Let P and Q denote nonnegative measurable functions on [01,,B] with P(x) <

Q(x) for

3~E [a, PI.

(I) (Sturm) Then the zeros of the function v separate the zeros of the function u in the interval [a, p]. Th a t is, if x0 denotes the least zero of u in (ar, /3] there exists a zero y,, of v such that 01< y0 < x0 and ;f xi and xifl denote consecutive zeros of u in [01,/3] then there exists at least one zero y of v such that Xi < y < Xi+1 * (II)

If u and v have the same number of zeros in the interval

[cr, y) then

U’(Y) > V’(Y) u(y)‘-’ V(Y) Furthermore,

the equality in (II) may occur only if P EGQ = 0 on [01,y),

Although Ince required that P and Q not vanish on any subinterval of [01,/I], his proof utilizing the Picone identity is easily extended to this situation. Using the well known fact that the second eigenfunction of (2.1) has only one zero in (0, l), the above comparison theorem yields the COROLLARY. Let ui and us denote the jirst and second eigenfunctions respectively of (1.2) and denote the interior zero of up by 7. Then

u,‘(x) -iqq-

> l$‘(x) U*(X)

(4.2)

for all s E (0, T) and

for all sE(r,

I).

Proof. In the above comparison theorem, we take P = ,\ip and 0 = X,p. The inequality (4.2) then follows from part II of that theorem. To obtain (4.3), we apply the same argument to the reflected density p. Throughout the proof of Theorem 5, we use the convention and notation indicated by the following remarks: (a)

The first eigenfunction

ZQ is positive on (0, 1).

(b) The second eigenfunction (T, l), where ~~(7) = 0.

u2 is positive

on (0, T) and negative

(c) The set where ur attains its maximum Zr = [urt , ui2] with ori = (TV?if Zr is a single point.

value

is denoted

on by

(d) The sets where u., attains its maximum and minimum values are denoted bv S,a and 2: respectively. These may be intervals or isolated points. Furthermore, Zc C (0, uii) and ZZZ C (ui?, 1). Statements (a) and (b) are permissible since (2.1) is a homogeneous equation and Z+(X) # 0 in (0, 1) while uZ has only onzero in (0, l), see [7]. The set Z; is as indicated in (c) since p is nonnegative and (2.1) implies that ur is concave on (0, 1). A similar remark yields the first part of (d). The last part of (d) follows by noting that if -$’ contains a point 0 > (art then X, is the lowest eigenvalue of a fixed-free string of density P(X) for x E [0, U] while h, is the lowest eigenvalue of a fixed-free string of density p(x) for x E [0, urr]. But it is known that this would imply that A, 5; X, , a contradiction. il similar argument holds for the remainder of the statement. Proof of Theorem 5.

From Theorem S[X,(p)jh,(p)]

2, we find that

= 1’ g(p ; .y) Sp dx, ‘0

whereg(p) is defined by (4.1). For a minimizing densityp of X,/X, , Theorem 3 then yields the inequalities G(X) < t,(h,/h,) if x E SIC and G(X) 3 t,(h,/X,) if x E S, where G = ur2 - uEi?. Using the normalization of the eigenfunction with respect to p and (2.6), we obtain the inequality

Consequently,

t, must be a positive

number.

BOUNDS

113

FOR EIGENVALUES

We may take the set Src{O, I> to be open so that it is the union of a countable number of disjoint open intervals. Thus, Sc\{O, I} = lJiew (ai , bi) where w is a linearly ordered index set with the property that for i, j E w with i
(i) The behavior of or , zca and C = ui2 minimizing density is investigated.

ua2 on [b, a] when p is a

(ii) This beha vior . is then used to show the location of any interval [b, u] relative to the location of the extreme points ui and ua . (iii) The results of(i) and ( ii ) are used to show if ui attains its maximum on an interval of positive length then there is only one interval [b, a] and the minimizing density is of the form p, given in the statement of the theorem. (iv) Finally, it is shown that for any minimizing density, the corresponding first eigenfunction ui cannot attain its maximum at a single point.

Part (i). To investigate the nature of ui , ua and G == ui2 - ua2 on [b, u] for a minimizing density, we first note that ui and ue are linear functions there. From the inequalities of Theorem 3 and the continuity of G, we have G(u) == G(6) = t,(h,/h,). From this, we immediately LEnmA

(4.5)

obtain the following:

1. For a minimizing density of &/A, , ulz(u) = q*(b) if and only $

u**(u) = U*‘(b). It also follows from (4.5), that the graph of G on [b, a] is a parabola which is symmetric about the midpoint of [6, u]. Furthermore, from Theorem 3, (4.4), and (4.5), G(x) 3 tl(A,/A2) > 0 on [6, LZ] with G(a) == G(b) = tl(AJA2) so that G is concave there. It follows that the coefficient of the r-squared term of G is negative. The linearity of ui and II? on [b, u] then yields [u,‘(b)]”

= [q’(u)]”

< [u*‘(B)]2 = [U*‘(u)]*.

(4.6)

114

GENTRY

Xh-D

R.WKS

Equality cannot hold in (4.6) since ur and u., cannot be constant on [b, (11 by statement (d) at the beginning of this proof. From the symmetry of the graph of G about (u - b)!?, it also follows that G’(h) :- -G’(n) :- 0 which in turn vields

q’(b) u,(b) > u,‘(b) u,(b)

and

Ul’(U) u&)

< Us’

U’(U).

(4.7)

We complete this part of the proof by showing that the set of estremal points ZrZ and Za2 of ug , for a minimizing density, consist of isolated points. For suppose that Zrp is an interval [6, CZ] with z+‘(a) = ~~“(6). Then the lemma following (4.5) yields ~~‘(a) = or”. By statement (d), we must conclude that u,(b) + U?(U), a contradiction. A similar proof shows that Z2z cannot be an interval. Throughout the remainder of the proof, we denote the maximum and minimum points of Z12 and &a by ura and uZf respectively. Part (ii). We now show that the segments [b, a] can only be contained in certain subsets of [0, l] determined by the maximum set [or1 , ura] of ul and the extreme points ui4 and ui2 and the zero r of uq . If uit < urs , then the lemma of part (i) and the statement (d) imply that r = fr(uri + up2). If uii = up2 , either uri ::< T or urr > 7. Since the eigenvalues of a given string with fixed end points are the same as those of the reflected problem and since reflection would change the order of r and urr , we may assume without loss of generality that urr :< T. With this assumption, we have that either 0 < u1‘) < u11 < 7
<$

< 1,

when urr < u1;?or 0


< 1.

[b, a], a and b must satisfy 6 E (u1”, up)

(4.8)

a E (%I , 4

W)

6 E (u,g, T)

(4.10)

a E (%I ) u2’))

(4.11)

and

when urr < ula and and when

BOUNDS

115

FOR EIGENVALUES

We first assume that b E [0, ole]. Since uiu’ and uaua’ are both positive on (0, aia), the inequalities (4.6) and (4.7) imply that

u,‘(b) [U,‘W [~~‘W -= u,(b) u,‘(b)u,(b)< u,‘(b) u,(b)

[u,‘W

Uz’(b)

< up’(b) u,(b) - ~.ue(4

This contradicts the comparison inequality (4.2), so that b > cr12.The same argument applied to the reflected densityp shows that a < a,‘. We then have that ui2 < b < a < crZ2for any interval [b, a]. We also observe that by the conventions noted in remarks (a), (b) and (c), G’(x) = ui(x) u:(x) - ZL~(X)Q’(X) 2 0 in (a:, uli) while G’(x) < 0 for x E (ui2 , ai2) if ui2 > r or for x E (T, ui2) if uii = ui2 < 7. But (4.7) implies that G’(a) < 0 and G’(b) > 0. This and the results of the previous paragraph, then implies (4.8) and (4.9) or (4.10) and (4.11) for each of the respective cases. Part (iii). We now assume that for a minimizing density, the fundamental eigenfunction U, attains its maximum value on an interval [b, a]. We use (4.8) and (4.9) to show that in this case there can only be one interval in the set of intervals {(b, , ai+l)}, i.e., {(b, , ai+l)) = ((b, , a2)) and that 6, = ull density of ,\2/X, must have and a2 = u12 . It will follow that the minimizing value H on [0, uii] and [u12 , I] and value zero on (uii , u12). To verify the above assertion, we note that a,, must coincide with bi for some interval (bi , uifl). If this is the case for i > 1, then a2 < uii , contradicting (4.9). Similarly, if a, = ui2 for i < 7, then b, > ui2 contradicting (4.8). Hence conclude that a, == a,, . These two results together yield the desired conclusion. It is an immediate consequence that if ui is to attain its maximum value at every point of the interval [uii , ui2], then uli = 1 - ui2 = M/2H and the minimizing density must be given for p, as stated in the theorem. It is easily shown that in this case h, = rr2H/hIB and that X2 = 4z12H/M” where zi is the least positive root of tan z = ~(1 - H/M).

Part (iv). We now finish the proof of Theorem 5 by showing that if a function p is to minimize the ratio h,/h, , the associated first eigenfunction u1 cannot attain its maximum value at a single point. Assume p yields a minimum value of the ratio h,/h, and the maximum point of its first eigenfunction u1 , uil = ura is less than the zero 7 of its second eigenfunction. From the restrictions (4.10) and (4.11) we may conclude that the endpoint b, cannot lie in [0, uii]. For (4.10) prohibits its inclusion in [0, ui2] and ull being a single point, cannot lie in the interval [b, a]. Thus if b, were to lie in 2 uJ, we would have to also have aa in this interval in contradiction to ;::l’l). Thus th e f unction ui reaches its maximum in [0, b,]. We must then

have 6, > 1 - a,, for otherwise zlr would also reach a maximum interval [n,# , I] in Cen- of the fact that on [0, LJ,], we must have

in the

while on [n, , I], we have ul(s) = Cl, sin(X,H)lj’

(I - x).

By the same reasoning applied to the second eigenfunction ua , we conclude that 7 < a, , for if T > a,, we would also have to have a zero of ua in [0, b,]. By (4.1 l), we must have a, < ~a’) and by (4. lo), 6,-r must be less than 7. Hence 7 must lie in the interval [/j,,,,_,, a,]. We thus have

al2 < ull < 6, ::< b,-, < T < a, < uz2. That the function ur is decreasing on [un , l] and G(b,-,) = G(u,J then implies that u22(b,-1) > ~~‘(a,). Since uq is linear over [b,-, , u,]. We have the inequalities

T - b, > T - b,-, > a, - T.

(4.12)

We next consider the distance D from b, to the point of intersection of the tangent line to ua at the point (b, , u2(b1)) with the x-axis. Us is concave on [ale, T] and, consequently,

Dar--b U’e may actually

(4.13)

1'

compute the value D using the fact that u2(x) = cl2 sin((h,H)l!a

X)

for x E [0, b,].

We obtain the identity

D =

___ (A2&2

tan((X,H)l:’

b,).

In a similar way we may also compute the distance a, - 7 using the representation up(x) = c2nsin((A,H)r/”

(1 - x))

for x E [a, , I].

tan((haH)li2

(1 - a,)).

We obtain

uo- 7 = ~(h,Hj’,2

BOUNDS FOR EIGENVALUES

117

Using the fact that n/2 < (hzH)lla (1 - a,) < (&H)llf compare D and a,, - r to obtain

b, < =, we may

D
5. THE

RI~xrnnr~

OF A,+,/&,

In this section we consider the problem of determining which densities maximize the ratio X,+r(p)/A&) over E(M, 0, H). We utilized Theorem 3 to derive a general result, state a conjecture and then give a couple of specific results for the case n = 1. THEOREM

6. For p in the class E(M, 0, H) = ;p 10
< H, I1 p(x) d.r = 42; , ‘0

let h,(p) and X,+,(p) denote the nth and (n + 1)st eigenvalue of a vibrating string with density function p. Then supremum

&fib? t h,(p)

1p E E(M, 0, H);

exist and is attained by a density P,~ E E(M, 0, H) which vanishes on at least one and at most n + 1 subinteraals of (0, 1) an d assumes the constant H otherwise. Proqf. That the functional F,(p) = X,+l(p)/A,(p) assumes a maximum on E(M, 0, H) is assured by Theorem 4. From Theorem 3 it follows that a maximizing density, p, , must have the form (2.3) where the set So satisfies (2.4) and the function g(p,; X) given by (2.2) with p = p, must satisfy the inequalities

for sorne number

g(P,; 4 > to,

if

$,(.4

= H,

g(p,;r)
if

PTA4 = 0,

(5.1)

to . For the functional F,, we find that

where X, denotes the ith eigenvalue of (1.2) and lli its associated eigenfunction normalized with respect to the density p. Consequently the form of a masim’zing density will be somewhat determined bv the behavior of the function g(p; x). For convenience we scale the function g and consider the function G(x) .= (h,:‘h,+I)g(p;

x) :L u,‘(s)

- use+,.

union, Furthermore, the set s,, may be expressed as a countable (J [ui , bj], i E w, where w is a linearly ordered set with the property that if From the normalization of u, and u,+~, then b,
2 n+d-+l~(4 d-y=

js J-W4 dx > b(UL+,) M. 0

From this inequality, we may conclude that t, is negative and that the index set w has an initial element 1 and a terminal element 7 with a, = 0 and 6, = 1.

Notation. Throughout the remainder [6, u] any of the intervals [bi , ai+,] where i On an interval of zero mass [6, LZ] the (1.2), must be linear functions. It follows

of the proof, we shall denote by + 1 denotes the successor of i in W. eigenfunctions, being solutions of from (5.1) that the graph of

over an interval [b, a] must be a convex parabola symmetric point (b + a)/2 and that

about the mid-

G(a) = G(b) = (A~$L+~) t, < 0, G’(a) = -G’(b)

> 0.

(5.2)

Notation. The kth eigenfunction, uk , of (1.2) is characterized as that solution of (1.2) which vanishes exactly k - 1 times in the open interval (0, 1). We denote these nodal points by 7ik with 0 = To’i < T1k < ... < Tkh’ = 1. Between the points $, and rTik the function U, will assume an extreme value on a set which we denote by Zii”. In view of the fact that uk satisfies (1.2) an antinode Zik will either consist of a single point or an interval coinciding with an interval of zero mass. When we state inequalities involving ,Zi:i” it is meant that the inequality should hold for each member of ,&i”. Also, when Zik is used as a left or right end point of an interval, it is assumed to represent its own left or right end point respectively should .ZL’i consist of an interval.

BOUNDS

119

FOR EIGENVALUES

The nodes and antinodes of the eigenfunction ing properties:

u, and u,+r have the follow-

(i)

0 < 7T+i < 7iR < 7t+l < 7en < .** < 7z+l < 1.

(ii)

0 < ZT+’ < Zr” < Z:+l

(iii)

If p maximizes

A,+,/&

< ZZn < **. < Z,”

< ZiIt

< 0.

, Zin consists of one point.

(iv) If p maximizes A,+,/& , Z;+l consists of either a single point or Z?‘+’ z is an interval of zero mass, [b, a], and rin = (b + a)/2. Statement (i) follows from the conclusion I of the Comparison Theorem of Section 4 with P = x,p and Q = &+ip. Since uk’ = 0 on Zik the conclusion (ii) follows from a similar application of the conclusion II of the Comparison Theorem to the density p and its symmetric image F. The following lemma justifies the statement (iv) and, together with the observation that G(T~+‘) > 0 and from (5.2) that G(x) < 0 for x E [b, a], also verifies statement (iii). LEMMA 2. For a maximizing density of An+JAn , u,,*(u) = un2(b) ij and only if u’,+,(u) = u;+l(b) and uk(u) = u,(b) implies ul((a + b)/2) = 0 for k = n, I = 12+ 1 or k = 12+ I,1 = n.

The first part of Lemma 2 follows from (5.2) and the second part follows from the first, (ii) and the observation that uk can only be constant on an interval Zii”. To complete the proof of Theorem 6 we construct a partition of the interval [0, l] and show that an interval of zero mass, [b, a] may be contained in only one type of the possible subintervals, of which there are at most n + 1 in number. We then show that at most one interval [b, a] may be in any such interval and, noting that H > M implies the existence of at least one interval [b, u], obtain the conclusion of the theorem. The nodes and antinodes 7ik, Zik’, R = n, n + 1 partition the interval [0, l] into subintervals, each of which may be characterized as being one of seven possible types. The types are described by the following forms where K and I may be either n or n + 1 but k # 1: Ii. III.

[Tk:, TZ],

II.

[P, Z”],

[T”, q

IV.

[P, @I,

VI.

[F, +‘I,

v. [T’i,F], VI I. The

[,X:+1, Z,‘“].

interval

type VII

occurs if and only if an interval

[b, u] = ZF’l.

Thus, in the following discussion of the possible occurence of massless intervals in the first six interval types it will be assumed that the interval [6, LZ]does not coincide with an!- Zi+“. Since G(X) < 0 on [b, 01 we may assume that zl,r,+l(s) >- u,(s)

;- 0

(5.3)

on any interval [6, u]. The fact that the coefficient of the X” term of a conves parabola must be positive, combined with (5.2) ensure that either

on any interval [6, a]. From (5.4) we conclude that no interval [b, a] may intersect intervals of the type I or II since un’ . u:,+l < 0 on intervals of these types. The types IV and VI are merely the reflections of the types III and V respectively. It suffices to consider only one of the symmetric types for the same argument applies to the other when p is replaced by p. Using (5.4) we may conclude that for intervals of the type III to intersect an interval [b, a], necessarily both u,,’ and u’,+r would be poitive and furthermore that this would imply the relationship for 1 < i < k

where j = i when 1 = TI + 1, k = n and j = i - 1 when I= n, k = 11+ 1. The right inequality leads to a contradiction with the statement (ii). (When i = 0, [~a~, 7ik] is considered of type V since T,-,~= rl”.) Consequently no interval [b, a] may intersect intervals of the type III or IV. We next examine the intervals of type V and VI where k = n + 1, I = n. By utilizing the conclusion II of the Comparison Theorem, on the interval (&, T:+‘) we have

UT2- %

%+1

(-1’

%2+1

I %+I

%I

[ %z+1

%I

I

1<0.

Consequently, if the interval [Zlin, T;+‘] is part of our partition, on it we would have 0 > u,’ > uk+i and in view of (5.4) it could not intersect an interval [6, u]. The symmetric consideration yields the same conclusion for partition intervals of the form [~;+r, ,Y~i]. There remains then only the possibility of an interval [b, a] occuring in intervals of the type V and VI when k = n and I = n + 1. That at most one massless interval [b, a] may intersect an interval of the form [Tag, Zz,:‘] will follow from the

BOUNDS FOR EIGEYNVALUES

and G(q) < 0, G’(q) > 0 then G’(x) > 0

LEMMA 3. If xi E (Tag,ZTz) f0Y.q

<

x <

ri

=

121

min(fi , ZT$

where ti is the unique zero of G in (xi , T:::).

If the interval [7in, Z$t] contained an interval [bj , a,,,] then it could not contain the interval [bi+r , ai+J since we could take xi = ai+l in Lemma 3 and thus preclude the possibility that bj+r E [T~‘~,ZT++ii] since uj+l and bj+l must satisfy (5.2). Thus at most one interval [b, a] exist in any partition interval of the form [Tag,.ZZ~+:‘]or, by symmetry, of the form [Zf+:‘, ~~“1. Prooj of Lemma 3. We note that the conclusion II of the Comparison Theorem yields the inequality (6.5) We may assume u, and u,+r are positive on (T;~, pi”,:‘) and since G(x) < 0, G(T;+‘) > 0, G has at least one zero in (xi, T;“). If 3’ is such a zero then from (5.6) we have un’(y) > Us+,, i.e., the graph of the function u, must cross above the graph of u,+~ at x = 3’. This requirement could not be satisfied by two consecutive zeros and consequently we have the uniqueness of ti . By integrating f’ - hh’ one obtains the simple LEnrMA 4. Let f and h denote dz~eerentiuble functions over an interval and satisfy the two conditions: (i) f(a) > kh(a)

for

[LU,p]

some constant k 3 1,

and

(ii) Then

for

0 >f’(~) all

> h’(x)

x E [a, ,L3], f(x)

for >

all

x E [a, 81.

hh(x).

Since u, and u,+r satisfy (1.2) and (5.3) in (xi , ri) we may apply Lemma 4 with 01= xi , /3 = ri , f = u,’ and h = u’,+~ . Taking the constant R = %z+1(G44, we obtain the inequality un+1(4 24n+l(.2) u,‘(x) > ___ %dXi)

for X E (Xi , ri).

(5.7)

From (5.5) and (5.3) we also have the inequality %+1(Xi) > un(si)

%+1b9

44

> 1

for x E (xi, ri).

(5.8)

GENTKY AND BANtiS

122 Combining

(5.7), (5.8) and (5.3) yields the conclusion

G’(x) = 2[u,‘(s)

un(x) -

U’,+,(X) untt(x)]

: 0,

of Lemma

3:

.x E (Si ) Ti).

The proof of Theorem 6 is finally accomplished by noting that at tnost one interval of zero mass [b, a] may occur only in one of the at most IZ f 1 intervals associated with the nodes 7in, i = 0, I,..., n possible under the above partition; [Tag, Z$‘], or [ZFJ:,:‘, ~~‘~1when ZFA1 consist of one point or ,ZyA1 = [b, a] with T:)I = (6 + a)/2 when ZF:;’ consist of an interval. From the results of Krein [15] we see that A, is minimized when the total mass is symmetrically distributed as close as possible to the antinodes of the nth eigenfunction u, and that h,+r is maximized when the total mass is somewhat symmetrically distributed as close as possible to the nodes of the (n + 1)st eigenfunction u,,+r . Thus, it seems plausible that &+,/A, should be maximized when the total mass is somewhat symmetrically distributed avoiding as much as possible both the nodes T,~ and the antinodes Z,F+:‘. We see in Theorem 6 that indeed the only segments on which a maximizing density may vanish must lie within intervals determined by 7in and Zn+r. r+l While the maximizing density for h,+t is not unique, the minimizing density for A, is unique and symmetric and this combined with the restricted somewhat symmetric form of the maximizing density for h,+r suggest that the maximum An+JAn is uniquely attained by a symmetric function. Furthermore, the fact that each antinode .Zyfl where P,~ does not vanish would contribute to diminish A,,, in view of Theorem 6 and the representation: .l

h nt1

=

I u;?+~dx/

‘0

1’ p,,u:+, dx '0

together with the fact that the densities which maximizing mizing A, each vanish on n + 1 intervals lead us to

h,+t and mini-

Conjecture. A,+l(p)/A,(p) is uniquely maximized on E(M, 0, H) by a symmetric density, p, , which vanishes on exactly n + 1 subintervals of [0, l] and assumes the constant value H otherwise. For the case rz = 1 we have the following partial results where i!@W, 0, H) = {p E E(M, 0, H) j P = j?). COROLLARY

p, has the form

1. Ifp, yields the maximum {X&)/X,(p)

1p E E(M, 0, H)} then

123

BOUNDS FOR EIGENVALUES

where 0 < o+ < t511< CQ< /3, < 1 and

COROLLARY 2. The Supremum {A,(p)/&(p) attained by a density p E E(N, 0, H).

1p E E(M, 0, H)} is unique/y

Proof of Corollary 1. By Theorem 6 p, vanishes on at least one and at most two subintervals of (0, 1) and has the constant value H, otherwise. We assume that pr vanishes on only one interval, [b, a], and obtain a contradiction to arrive at the conclusion of Corollary 1. To this end we assume p, has the form p,(x) = H if 0 < x < b or a < x < 1 and P,(X) = 0 if b < .T <: a. Then the eigenfunctions of (1.2) with p = p, are given for i= 1,2 bv sin((&H)l/a u&y) = jCil 1Ci, sin((AiH)liz

if if

X) (1 - x))

O
and ui linear on (b, a) where the constants Ci, and Ci, are choosen so that ui and ui’ are continuous on [0, I] and ji ZQ’(X)pr(~) dx = 1. From the proof of Theorem 6 we may assume that [b, a] C (0, Zrz). Krein [15] has shown that for p E .E(M, 0, H), max X,(p) = T?H/M* and max A,(p) = 4?r2H/M”. Consequently, we have h,(p) > 4h,(p,) and the “periods” #or and #a of ur and u, satisfy & < #r/2. This last inequality implies that El1 < era, and, since ZQ is decreasing on (L’rl, l), that GY

+ (4;’ - a)) > u1(q2 + (~12 - 4).

Since or is symmetric (locally) about x = Zr’, function about x = 7r2, we have

and uq is (locally)

an odd

ul(u) = ul(41 + (4’ - a)) and -u2(u) Combining u12(~12

= Up(T12+ (T12 - a)).

the last three relations, +

(T)*

-

a))

-

u2’(T12

+

we obtain the inequality (71’

-

a))

<

u;‘(U)

-

u2’(U)

=

t,,(&/h&.

But this inequality violates a necessary requirement for the density p, to give a maximum value of the ratio X,/h,: G(x) >, (X,/h,) t, for all points x such that pr(.~) is positive. Proof of Corollary 2. Since all of the arguments used to prove Theorem 6 also apply when p is restricted to E(M, 0, H), the maximum is attained by a

density \\hich \ anishcs. ,111 t\‘:o sul~intcr\-als. C’onscquentl\- \\ c ileecl onI> look Such for a L;niquc maximizing function of Az(~)/Al(~~) nherl p lb ” so rcaictcd. functions form a one-parameter family E, ~ I/$[)] /I rI; [O. J/:?ZI]j. whrrr the function

Thus the set of values, (&&)/Al(p) 1p E E,j. may be simply determined as the range of the functionf defined on [0, dql2H] byf(b) = A,(p);X,(p) where p = p[b]. Ti le f unctionf is then continuous on [0, M/2H] and at any point b in this interval, the variation off determined by the change 66 of 6, is given by the first variation, 6(X,/h,), of the ratio X,/h, determined by the admissible variation Sp =: p[b + 661 - p[b]. W e note that the density corresponding to the value b = M/2H is none other than the density p, of Theorem 5, which we have shown provides the unique minimum for the ration X,/h, over the class E(M, 0, H). Consequently, the functionf does not assume its maximum at b = M/2H. That the point b == 0, also does not yield the maximum off over [0, M/2H] follows from the fact that the variation off at b = 0 will be positive for arbitrarily small positive variations 6b. This is clear from the observation that the function G corresponding to p = p[O] is decreasing we map conclude that the function on [0, (1 - 1lljH)/2]. Consequently, f attains its absolute maximum on the interval [0, ;V/2H] at an interior point. If /3 is any point of (0, :1fj2H) at which the variation off is zero, we must have, for the function G corresponding to the densityp = p[fl], G(P) = G(a), where in.= (1 - Llf/H)/2 + 13.Furthermore, the function G must necessarily be negative on the interval of zero mass [h, ~1. To verify this, we observe that G must be a symmetric parabola on the interval [p, a] and consequently, G’ must be an odd function about the point (p + a)/2. Since p is symmetric, we hare the maximum, ZI1, of the first eigenfunction coincides with the zero, TV’),of the second eigenfunction. Thus G’ is positive throughout the interval (rI”, ,YzJ) and therefore, by (5.2), the interval [/3, E] must be contained in the interval (0, ZIz). But, where the function G is positive in (0, ZIz), we may employ the inequality (4.2) to obtain the inequality UIUl = ulyul’Iul)

> Ul’(U,‘/U,) ;

u,yu,‘/u,)

= U&‘.

Hence, where G is positive on (0, El’), G ’ is also positive, and consequently can not be positive on [p, a],

G

125

BOUNDS FOR EIGENVALUES

Using Lemma 3 we may conclude that not only is G negative [p, a] but also

‘34 3 G(P)

if

.x-E [O, p] u [Lx, f]

G(x) < G(P)

if

.v E [/3, cd].

on the interval

and

Hence the density p = p[p] can yield a maximum but not a minimum of ha/h, . Thus the uniqueness of the maximizing density follows, since if fir and /?a were to yield distinct maximums, the continuity off would imply a minimum point /3 between them.

6. RE~LIARKS The methods of proof used for the Existence Theorem 4 generalizes to higher order equations in the linear case. But, in higher dimensional problems, difficulties are encountered even in the second order problem, since the kernel of the integral operator corresponding to (3.1) is not necessarily compact nor continuous. It is also apparent that the class of density functions utilized in Section 2 and 3 may be further restricted, provided the additional restraints imposed upon the densities are necessarily imposed on any density function obtained through convergence in the w*-sense (integral sense) of a TABLE Minimum

of X,(p)/X,(p),

I

p E E(M, 0, 1)

Mass

z”

Min h,/h, = [z/(~r/2)]?

0.1

1.64

1.089

0.2

1.71

1.185

0.3

1.80

1.313

0.4

1.91

1.478

0.5

2.03

1.670

0.6

2.17

1.908

0.7

2.35

2.238

0.8

2.57

2.616

0.9

2.87

3.338

a Jahnke and Emde [14].

126

LENTRY

AND

H.ViW

sequence of densities within the class. (For esamples of classes with this property, see Banks [l-4] and Barnes [Yj.) 1Vhile our method remains valid for any differentiable function of the eigenvalues, and quite easily provides the results of Krein [15], we encounter considerable difficulty when considering more complicated functions than those considered here. This is due to the fact that the nature of the variation 6F(h) = jg(s) 6p(x) dx’ is determined by the function g(p; s), which in turn is a function of the squares of the eigenfunctions uTLand the eigenvalues A, . We were able to utilize the Comparison Theorem to study the behavior of the function g in the cases presented. To handle more complicated functions, F, would necessitate having more general knowledge of the relation between two or more eigenfunctions than is presently available. Also, we did not discuss the ratio ha/A, for the fourth order equation associated with the Iibrating rod. This problem would lend itself to a similar attack, but presents considerable difficulties even on segments where a density function vanishes, for on these, the eigenfunctions are given by polynomials of degree 4. The extreme values of the ratio ha/A, depends on the ratio of the mass M to the height H. This constant M/H also determines intervals on which the extremizing densities p, and p, vanish. The Tables I and II indicate representative values where we have assumed the constant H = 1. TXBLE

Maximum of h,(p)/&(p), n1ass

II p E E(M, 0, 1) Max h,/h, .___

4 -

1.0

+

471’)

0.9

10.70134

48.2072

4.50478

0.8

11.76248

60.5347

5.14642

0.7

13.15053

0.6

15.02575

78.6865 106.892

4.0

5.98352 7.11395

0.5

17.67451

154.015

8.71399

0.4

21.66825

241.304

11.13630

0.3

28.33940

430.861

15.20350

0.2

41.68627

974.776

23.38362

0.1

81.70650

3923.036

48.01375

Note that the

BOUNDS

127

FOR EIGENVALUES

where p, gives the max of A, = 7r24HMe2 and p1 gives the max of A, = dHlW2. The density which gives the maximum A,/& vanishes on the intervals [q , PI], [l - j$ , 1 - a11 and is 1 otherwise where CX~ and /31 are given in Table III. TABLE

III

0.9

0.19266749

0.24266749

0.8

0.17477098

0.21477098

0.7

0.15602358

0.30602358

0.6

0.13637767

0.33637767

0.5

0.11580927

0.36580927

0.4

0.0943 1866

0.39431866

0.3

0.07193134

0.42193134

0.2

0.04869775

0.44869175

0.1

0.02469074

0.41469074

REFERENCES

1. D. 0. BANKS, Bounds for the eigenvalues of some vibrating systems, Pucz’fc J. Math. 10 (1960), 439-474. 2. D. 0. BANKS, Upper bounds for the eigenvalues of some vibrating systems, Pacific 1. Math. 11 (1961), 1183-1203. 3. D. 0. BANKS, Bounds for eigenvalues and generalized convexity, PaciJic J. Math. 12 (1963), 1031-1052. 4. D. 0. BANKS, Lower bounds for the eigenvalues of a vibrating string whose density satisfies a Lipschitz condition, Pacific 1. Math. 20 (1967), 393-410. 5. D. BARNES, Some isoperimetric inequalities for the eigenvalues of vibrating strings, Pacific J. Moth. 29 (1969), 43-61. 6. J. J. A. M. BRANDS, Bounds for the ratios of the first three eigenvalues, Arch. Rut. Mech. Anal. 16 (1964), 265-268. “Methods of Mathematical Physics,” Vol. 1, 7. R. COURANT AND D. HILBERT, Interscience Publishers, New York, 1966. 8. HANS DEVRIES, On the upper bound for the ratio of the first two membrane eigenvalues, 2. Nuturforsch. 22 (1967), 152-153. 9. L. L. HELMS, “Introduction to Potential Theory,” Wiley-Interscience, New York, 1969. 10. JOSEPH HERSCH, Characterisation variationelle dune some de valeurs propres consecutives: generalisation d’intgalites de Polya-Shiffer et de Weyl, C. R. Acud. Sci. 20 March (1961), 1714-1716. 409/5III-9

IntgalitCs pour des valeurs propres consCcutires de s+xnes ribrants inhomog&es, allant et sens inverse de celles de Polya-Shiffer et de Weyl, C. R. Acod. Sci. 24 April (1961), 2496-3498. JOSEPH HERKM, On symmetric membranes and conformal radius: Some complements to Polya’s and Szego’s inequalities, birch. Rot. Mech. And. 20 (1965), 378-390. E. L. IR’CE, “Ordinary Differential Equations,” Dover Publications Inc., Xen York, 1956. Dover Publications Inc , New E. JAHNKE AND F. EMDE, “Tables of Functions,” York, 1956. RI. G. KREIN, On certain problems on the maximum and minimum of characteristic values and on zones of stability, Amer. Math. Sot. Tmmsl. 1 (1955), 163-187. 2. NEH.ARI, Extremal problems for a class of functions defined on convex sets, Bull. .-lttler. Math. Sot. 73 (1967), 584-591. L. E. PAYNE, G. POLYA, .4ND H. F. WEINBERGER, On the ratios of consecutive eigenralues, /. of Math. ad Phys. 35 (1956), 289-298. HANS S.4G.4N, “Introduction to the Calculus of Variation,” hIcGran-Hill, New York, 1969. BENJAMIN SCHWARZ, Bounds for sums of reciprocal of eigenvalues, Bull. Res. Cotorc. of Israel SF (1959), 91-102. G. TEUPLB .~ND W. G. BICKW~, “Rayleigh’s Principle and its Applications to Engineering,” Dover Publications, New York, 1956. COLIN J. THOMPSON, On the ratio of consecutive eigenvalues in IV-dimensions, Stud. dppl. Math. 48 (1969), 281-283.

1 I. JOSEPH HERSCH,

12.

13. 14. 15. 16. 17. 18. 19. 20.

21.