- Email: [email protected]
Bornes supérieures isopérimètriques pour les valeurs propres du type de Sturm-Liouville
C. R. Acad. Analyse
Sci. Paris, t. 325,
SCrie I, p. 835-840,
1997
Analysis
mathCmatique/Mathematical
Isoperimetric upper bounds for the eigenvalues of the Sturm-Liouville type Samir
KARAA
Laboratoire des Mathematiques pour 1’Industrie et la Physique, CNRS UMR 5640, Universiti: Paul-Sabatier, 118, route de Narbonne 31062, Toulouse, France. E-mail : [email protected]
Abstract.
We study the problem of maximizing the eigenvalues of the differential equation (q(z)y’)’ + Xp(z)y = 0 defined on a finite interval. The problem is solved by means of sufficient conditions of optimality.
Bornes
supkrieures isop6rim6triques pour les valeurs propres
du type de Sturm-LiouviUe RCSUlnC.
On
Ptudie le problhe de maximisation des valeurs propres de l’kquation (q(z)y’)’ + Xp(x)y = 0 d@nie sur un intervalle borne’ de R. Le probkme est t&oh ir l’aide de
conditions
Version
francaise
On considere
sujfisantes
d ‘optimalit&.
abrbgke
un couple (4, fi) E U, x V tel que X1(q, p) < designe la premiere valeur propre du probleme (l)-(2) et U, et V sont les ensembles definis par (3) et (4) respectivement. Posons iw? = u”u,pvh(cL PI. X1(4,@)
V(q,p)
le probleme
E U,
(P)
: Trouver
x V, oti X1(4, p)
a
Alors, on a le theoreme suivant : THBORGME. - Soit Q > 1 un nombre Gel et posons p = 2a/(a fonction
propre
du problkme
non-linkaire
(Ally’Ij;-ply’lp-2y’)’ 02 jj est donnke par (6). Posons optimal, et on a :
t3 &ant la fonction
ensuite
- 1). De%gnons par jj la premibre
:
+ X/5y = 0, Q(z)
y(O) = y(e) = 0,
= AlI~‘II~-“l~‘(2)lp-2.
Alors
le couple
(@,p) est
d’Euler.
On generalise ensuite ces resultats a toutes les valeurs propres du probleme (l)-( 2). Note prbsentbe par Haim BR~ZIS.
0764~4442/97/03250835 0 Academic des Sciences/Elsevier. Paris
835
S. Karaa
We shall be concerned
with the eigenvalue
problem
M4Y’)’ + Jv(X)Y = 0,
(1) (2)
Y(O)
= Y(l) = 0,
e being a positive real number, and we denote by Xi (q, p) its first eigenvalue. Let H, A, and B be positive numbers such that HC > I?. Define the sets of constraints U, and V by:
P
(4)
V
=
p
10 5 p(x) 2 H,
E L"(O,l)
J
0 p(x)dx
where N is a number
’
= B
2 1. The aim of this Note is to study the problem:
(PI
maximize
Xl(q, p)
In [4], estimates of Xi(q, p) are obtained are subjected to the constraints
subject to
(q, p) E U, x V.
when the functions
J qN(x)dx =l, I
0
q(x)
and p(x)
are nonnegative,
and
Y
PO(X)dx
= 1,
J0
where Q and p are non-zero real numbers. It is proven that Xl(q, p) cannot be estimated from above when a: 2 1 and ,/3 = 1. For other extremal problems concerning eigenvalues, see [l]-[3], [5], [6], and the references therein. Put now
where the sup is taken in the class of all pairs (q, p) E U, x V. It is clear that standard compactness arguments do not enable us to establish the existence of solutions to problem (P). We proceed then to exploit the sufficient conditions given by: PROPOSITION.- Let (ij, 6) be a couple
of functions
from
U, x V and $ be any first eigenfunction
of the problem: (i(x)y’)’
(5)
Je-
P(x)$(x)~dx
0
J
+ X@(x)y
e
<
= 0, y(0)
J
836
= 0.
e
~(x)lJ(x)~
dx,
0
(5) is in fact the reason for choosing
J P
q(x)$(x)’
dx 5
0
for every couple (q, p) E U, x V, then (@,c) is a solution Condition
= y(l)
~(x)$(x)~
0
to problem
the optimal
function
(P). b given below.
dx
lsoperimetric
1.
Optimal
upper
bounds for the eigenvalues
of the Sturm-Liouville
type
solutions
Let fi be the function
defined
by: H p(z) = 0 i H
(6)
The real a is choosen
so that p(z)
if 0 5 :C 5 n, ifa
E V. That is a = (2H)-‘B.
THEOREM1. - Let a > 1 and set p = 2a/(a
We establish
- 1). Let fj be the$rst
now our main result:
eigenfunction
of the nonlinear
problem: (AIIY’~~;-~IY’I~-~Y’~’
+ XPY
where p is given by (6). Put q = A](ij’J]~-P]fj’]P-2. (P), and we have
where B is Euler’s
=
0, Y(O) = de) = 0,
Then th e couple
(4, p) is a solution
to problem
Beta function.
To prove this theorem,
we will use the following
LEMMA 1. - Let p be a number
lemmas:
> 1. Let m. = inf, G[y], where
G[yl = (J; IY’IP w2’p .d @Y2 dx and p is given by (6). The inf above Then G has a nonnegative minimizer (i) jj is concave and symmetric (ii) 1 = {x E [o,e]; g(x) = max (iii) jj’ is everywhere continuous
is taken in the class of all non-zero functions y from W,‘“(O, e). fj in W,‘“(O, 1) with the following properties: about x = e/2. I$[} is exactly the interval [a,e - a]; a = (2H)-lB. and jj satisjies the equation (IIY’II~-PIy’IP-2y’)’
at every point of the interval
+
m/?y= 0
(0, e).
(iv> m = 2J!T?!&$-2’p(~)1-2’pB~(;,~
(v)
_ 9.
We have
G(x) = G(a)+ cl(a - x) p’(p-l)[l y’(x)
= cp(a - x) l’(p--l)[1
G(x) = QX[l + o(l)] where cl, ~2, and c3 are non-zero
+ o(l)]
+ o(l)]
as x + a-, as z + a-, as z + O+,
numbers. 837
S. Karaa
t? and ij be us in Lemma 1. Put q(x) = 1IQ’1Isep ~fj’(~)~p-“.
LEMMA 2. - Let p > 2 and let thefunctions Let
inf
P= Then /L is attained
?I-;
.Ib”4~” dx
(08) j-i ,5y2 d:l: ’
on ji and u = m.
Proof of Theorem
1. - Let cy > 1 and set p = ~cx/(Q: - l), p > 2. Put t(x)
where $ is indicated Holder’s inequality,
in Lemma we obtain
= A~I~‘II~-P(~J’(x)~P-~,
1. Then we have s, G(x)” dx = A”, which implies for all q E U,
s
that @ E U,. By
e
0
q(x)G’(x)’dx 5
(sil~~(x)l~dx)‘;“(i’q(x)-dx)“l
= A(1
On the other hand, it is easily
verified
jQ’(x)[pdx)2”
= ~q(x)1/‘(x)2dx.
that for each x E [0, !I,
and hence Ji @(x)$(x)~ dx < 1: p(x)$(x)” d x f or every function p E V. Besides, since the differential equation (ll~‘ll~-pl~‘lp-2~‘)’ + 7n@j = 0 holds everywhere
in (0, a), Proposition
1 and Lemma 2 imply that (4, ,6) is a solution
COROLLARY 1. - Let q and p be two functions p E L”(0, e), where Q: > 1. Then
where C(a)
not identically
to problem
zero such that q E
L"(0, 1) and
= I& for the case A = B = H = 1.
THEOREM 2. - If (II = 1, then for all (q, p) E VI x V &(q,p) The equality
is attained
< 12AH2B-“.
if p equals the function
i(x) =
( MIH(u’ 0
fi defined by (6), and - x2)/2
t @(e- x) where Ml
= 12AH2B-”
if 0 2 x 5 a, if u < x 5 e/2, if e/2 5 2 5 e,
and a = (2H)-lB.
Proof. - The key of the proof is the appropriate
choice of the test function if O
838
(P).
jj defined by:
lsoperimetric
for which we have for all (q,p)
2. Estimates
upper
bounds for the eigenvalues
of the Sturm-Liouville
type
E U1 x V,
of all the eigenvalues
We now give an optimal upper bound for the n-th eigenvalue of problem (l)-(2). The main result presented here is: THEOREM3. - Let X,(q,
functions
p) be the n-th eigenvalue of problem (l)-(2). Suppose that the coefjicient q and p are not identically zero and satisfy: q E L”(0, e) and p E L” (0, !), where cu > 1. Then
where C?(a) = C(a) g’tven by Corollary 1 if o! > 1 and c(1) = 12. In addition, the equality is attained by two periodicfinctions q and p with a period equal to e/n and such that q(x) = q( nrc), p(x) = /?(nz) for all x E (0, e/n), where q and p are the optimal functions indicated in Theorems 1 and 2.
To prove this theorem, we shall use the following lemma: LEMMA 3. - Let r, s be two real numbers such that rs < 0 and r + s 2 1. Denote by E the set of all vectors X = (x1,x2, . ...2,) E (IO, l[)” satisfying cy=“=,xi < 1, and define the function F:ExE--+Rby:
F(X,
Y) = xc;‘y; + x;y;
+(1 - Xl - x2 - . . . - xn)r(l Then
+ . . . + xLy;
- y1 - y2 - . . . - 7&&y.
F attains its minimum value Fmin = (n + l)l-(‘+“) when X 1, . . , 1). A4oreover, this minimum point is unique in the case r + s > 1.
the function
(n + l)(-‘)(l,
=
Y
=
Proof of Theorem 3. - We begin by the case n = 2. The general case follows by using similar arguments. Let q and p be two measurable functions such that ]]q]]ol= 1, Q! > 1, ]]p]ll = 1, and 0 5 p(x) 5 H for all x E [0, C]. Let ya (resp. X2(q, p)) be the second eigenfunction (resp. eigenvalue) of the problem
(464d)~+ XP(X)Y
= 0,
Y(O) = 0)
= 0.
As is well known, the function y2 admits precisely one zero (call it x1) in the interval (0, e). Furthermore, the number X2(q, p) is in fact the first eigenvalue of the two problems: (d4Y’)
+ XP(X)Y = 0,
(q(x)y’)’ + xp(x)y = 0,
Y(0) = Y(Q) Y(X~ =
= 0,
de) = 0.
According to Corollary 1 and Theorem 2, we have:
(7)
X2(q, p) 5 C(o)H1+l’”
(~z’q(x)-dx)1’~(~z’p(r)dx)-(2+1’~’,
839
S. Karaa
where c(a)
(8) Similarly,
is given by Corollary
1 for o > 1 and c(1)
(I~‘~(~)~di)-1’~(JC~11i(X)iii)2+1’~
= 12. Inequality
(7) can be written
as:
5 C(~)H1+l/“[X2(4,P),-l.
we have:
(9)
Now put b = St1 q(x)a dz and c = ST1 p(z) dz. By summing (10)
b-1/ac2+1/a
+ (1 - b)-I/“(1
- c)~++
(8) and (9), we obtain:
< C(n)H1+l’”
[X,(9, p)]-‘.
By Lemma 3 (applied with n = l), the left side of (10) is not less than 2-l c are both in the interval (0,l). Therefore l/2 which shows that &(q,p)
< 2 C(o) H 1+1’a
< 4C(cr)H’+““.
[XzkJ,
since the reals b and
/w,
If ljqllcy = A and llplli = B, then we obtain
X2(4, p) 2 4 C(o)
H1+‘la
A B-(2+1’“).
To prove the general case n > 2, we argue as above and take into account the fact that the n-th eigenfunction of problem (l)-( 2) h as exactly (n - 1) zeros between 0 and e. Note remise le le’ avril 1997, acceptee le 28 juillet 1997.
References [l] Bandle C., 1987. Extremal problems for eigenvalues of the Sturn-Liouville type, Internat.Ser. Numer. Math , General inequalities,5, Birkhauser, 80, pp. 319-336. [2] Egorov Yu. V. and Karaa S., 1994. Optimisation de la premiere valeur propre de l’optrateur de Sturm-Liouville, C. R. Acad. Sci. Paris, 319, SCrie I, pp. 793-798. [3] Egorov Yu. V. and Karaa S., 1996. SW la forme optimale d’une colonne en compression, C. R. Acad. Sci. Paris, 322, Strie I, pp. 519-524. [4] Egorov Yu. V. and Kondratiev Yu. V., 1991. On an estimate of the principal eigenvalue of the Sturm-Liouville operator, Vestnik Mosk. (In-ta Math. Mech., 6, pp. 5-l 1. [5] Karaa S., 1996. Extremal eigenvalues and their associated nonlinear equations, Boll. Un. Math. Ital., IO-B, pp. 625-649. [6] Talenti G., 1984. Estimates of eigenvalues of Sturm-Liouville problems, Internat. Ser. Numer. Math., General inequalities, 4, Birkhluser, pp. 341-350.
840
Sci. Paris, t. 325,
SCrie I, p. 835-840,
1997
Analysis
mathCmatique/Mathematical
Isoperimetric upper bounds for the eigenvalues of the Sturm-Liouville type Samir
KARAA
Laboratoire des Mathematiques pour 1’Industrie et la Physique, CNRS UMR 5640, Universiti: Paul-Sabatier, 118, route de Narbonne 31062, Toulouse, France. E-mail : [email protected]
Abstract.
We study the problem of maximizing the eigenvalues of the differential equation (q(z)y’)’ + Xp(z)y = 0 defined on a finite interval. The problem is solved by means of sufficient conditions of optimality.
Bornes
supkrieures isop6rim6triques pour les valeurs propres
du type de Sturm-LiouviUe RCSUlnC.
On
Ptudie le problhe de maximisation des valeurs propres de l’kquation (q(z)y’)’ + Xp(x)y = 0 d@nie sur un intervalle borne’ de R. Le probkme est t&oh ir l’aide de
conditions
Version
francaise
On considere
sujfisantes
d ‘optimalit&.
abrbgke
un couple (4, fi) E U, x V tel que X1(q, p) < designe la premiere valeur propre du probleme (l)-(2) et U, et V sont les ensembles definis par (3) et (4) respectivement. Posons iw? = u”u,pvh(cL PI. X1(4,@)
V(q,p)
le probleme
E U,
(P)
: Trouver
x V, oti X1(4, p)
a
Alors, on a le theoreme suivant : THBORGME. - Soit Q > 1 un nombre Gel et posons p = 2a/(a fonction
propre
du problkme
non-linkaire
(Ally’Ij;-ply’lp-2y’)’ 02 jj est donnke par (6). Posons optimal, et on a :
t3 &ant la fonction
ensuite
- 1). De%gnons par jj la premibre
:
+ X/5y = 0, Q(z)
y(O) = y(e) = 0,
= AlI~‘II~-“l~‘(2)lp-2.
Alors
le couple
(@,p) est
d’Euler.
On generalise ensuite ces resultats a toutes les valeurs propres du probleme (l)-( 2). Note prbsentbe par Haim BR~ZIS.
0764~4442/97/03250835 0 Academic des Sciences/Elsevier. Paris
835
S. Karaa
We shall be concerned
with the eigenvalue
problem
M4Y’)’ + Jv(X)Y = 0,
(1) (2)
Y(O)
= Y(l) = 0,
e being a positive real number, and we denote by Xi (q, p) its first eigenvalue. Let H, A, and B be positive numbers such that HC > I?. Define the sets of constraints U, and V by:
P
(4)
V
=
p
10 5 p(x) 2 H,
E L"(O,l)
J
0 p(x)dx
where N is a number
’
= B
2 1. The aim of this Note is to study the problem:
(PI
maximize
Xl(q, p)
In [4], estimates of Xi(q, p) are obtained are subjected to the constraints
subject to
(q, p) E U, x V.
when the functions
J qN(x)dx =l, I
0
q(x)
and p(x)
are nonnegative,
and
Y
PO(X)dx
= 1,
J0
where Q and p are non-zero real numbers. It is proven that Xl(q, p) cannot be estimated from above when a: 2 1 and ,/3 = 1. For other extremal problems concerning eigenvalues, see [l]-[3], [5], [6], and the references therein. Put now
where the sup is taken in the class of all pairs (q, p) E U, x V. It is clear that standard compactness arguments do not enable us to establish the existence of solutions to problem (P). We proceed then to exploit the sufficient conditions given by: PROPOSITION.- Let (ij, 6) be a couple
of functions
from
U, x V and $ be any first eigenfunction
of the problem: (i(x)y’)’
(5)
Je-
P(x)$(x)~dx
0
J
+ X@(x)y
e
<
= 0, y(0)
J
836
= 0.
e
~(x)lJ(x)~
dx,
0
(5) is in fact the reason for choosing
J P
q(x)$(x)’
dx 5
0
for every couple (q, p) E U, x V, then (@,c) is a solution Condition
= y(l)
~(x)$(x)~
0
to problem
the optimal
function
(P). b given below.
dx
lsoperimetric
1.
Optimal
upper
bounds for the eigenvalues
of the Sturm-Liouville
type
solutions
Let fi be the function
defined
by: H p(z) = 0 i H
(6)
The real a is choosen
so that p(z)
if 0 5 :C 5 n, ifa
E V. That is a = (2H)-‘B.
THEOREM1. - Let a > 1 and set p = 2a/(a
We establish
- 1). Let fj be the$rst
now our main result:
eigenfunction
of the nonlinear
problem: (AIIY’~~;-~IY’I~-~Y’~’
+ XPY
where p is given by (6). Put q = A](ij’J]~-P]fj’]P-2. (P), and we have
where B is Euler’s
=
0, Y(O) = de) = 0,
Then th e couple
(4, p) is a solution
to problem
Beta function.
To prove this theorem,
we will use the following
LEMMA 1. - Let p be a number
lemmas:
> 1. Let m. = inf, G[y], where
G[yl = (J; IY’IP w2’p .d @Y2 dx and p is given by (6). The inf above Then G has a nonnegative minimizer (i) jj is concave and symmetric (ii) 1 = {x E [o,e]; g(x) = max (iii) jj’ is everywhere continuous
is taken in the class of all non-zero functions y from W,‘“(O, e). fj in W,‘“(O, 1) with the following properties: about x = e/2. I$[} is exactly the interval [a,e - a]; a = (2H)-lB. and jj satisjies the equation (IIY’II~-PIy’IP-2y’)’
at every point of the interval
+
m/?y= 0
(0, e).
(iv> m = 2J!T?!&$-2’p(~)1-2’pB~(;,~
(v)
_ 9.
We have
G(x) = G(a)+ cl(a - x) p’(p-l)[l y’(x)
= cp(a - x) l’(p--l)[1
G(x) = QX[l + o(l)] where cl, ~2, and c3 are non-zero
+ o(l)]
+ o(l)]
as x + a-, as z + a-, as z + O+,
numbers. 837
S. Karaa
t? and ij be us in Lemma 1. Put q(x) = 1IQ’1Isep ~fj’(~)~p-“.
LEMMA 2. - Let p > 2 and let thefunctions Let
inf
P= Then /L is attained
?I-;
.Ib”4~” dx
(08) j-i ,5y2 d:l: ’
on ji and u = m.
Proof of Theorem
1. - Let cy > 1 and set p = ~cx/(Q: - l), p > 2. Put t(x)
where $ is indicated Holder’s inequality,
in Lemma we obtain
= A~I~‘II~-P(~J’(x)~P-~,
1. Then we have s, G(x)” dx = A”, which implies for all q E U,
s
that @ E U,. By
e
0
q(x)G’(x)’dx 5
(sil~~(x)l~dx)‘;“(i’q(x)-dx)“l
= A(1
On the other hand, it is easily
verified
jQ’(x)[pdx)2”
= ~q(x)1/‘(x)2dx.
that for each x E [0, !I,
and hence Ji @(x)$(x)~ dx < 1: p(x)$(x)” d x f or every function p E V. Besides, since the differential equation (ll~‘ll~-pl~‘lp-2~‘)’ + 7n@j = 0 holds everywhere
in (0, a), Proposition
1 and Lemma 2 imply that (4, ,6) is a solution
COROLLARY 1. - Let q and p be two functions p E L”(0, e), where Q: > 1. Then
where C(a)
not identically
to problem
zero such that q E
L"(0, 1) and
= I& for the case A = B = H = 1.
THEOREM 2. - If (II = 1, then for all (q, p) E VI x V &(q,p) The equality
is attained
< 12AH2B-“.
if p equals the function
i(x) =
( MIH(u’ 0
fi defined by (6), and - x2)/2
t @(e- x) where Ml
= 12AH2B-”
if 0 2 x 5 a, if u < x 5 e/2, if e/2 5 2 5 e,
and a = (2H)-lB.
Proof. - The key of the proof is the appropriate
choice of the test function if O
838
(P).
jj defined by:
lsoperimetric
for which we have for all (q,p)
2. Estimates
upper
bounds for the eigenvalues
of the Sturm-Liouville
type
E U1 x V,
of all the eigenvalues
We now give an optimal upper bound for the n-th eigenvalue of problem (l)-(2). The main result presented here is: THEOREM3. - Let X,(q,
functions
p) be the n-th eigenvalue of problem (l)-(2). Suppose that the coefjicient q and p are not identically zero and satisfy: q E L”(0, e) and p E L” (0, !), where cu > 1. Then
where C?(a) = C(a) g’tven by Corollary 1 if o! > 1 and c(1) = 12. In addition, the equality is attained by two periodicfinctions q and p with a period equal to e/n and such that q(x) = q( nrc), p(x) = /?(nz) for all x E (0, e/n), where q and p are the optimal functions indicated in Theorems 1 and 2.
To prove this theorem, we shall use the following lemma: LEMMA 3. - Let r, s be two real numbers such that rs < 0 and r + s 2 1. Denote by E the set of all vectors X = (x1,x2, . ...2,) E (IO, l[)” satisfying cy=“=,xi < 1, and define the function F:ExE--+Rby:
F(X,
Y) = xc;‘y; + x;y;
+(1 - Xl - x2 - . . . - xn)r(l Then
+ . . . + xLy;
- y1 - y2 - . . . - 7&&y.
F attains its minimum value Fmin = (n + l)l-(‘+“) when X 1, . . , 1). A4oreover, this minimum point is unique in the case r + s > 1.
the function
(n + l)(-‘)(l,
=
Y
=
Proof of Theorem 3. - We begin by the case n = 2. The general case follows by using similar arguments. Let q and p be two measurable functions such that ]]q]]ol= 1, Q! > 1, ]]p]ll = 1, and 0 5 p(x) 5 H for all x E [0, C]. Let ya (resp. X2(q, p)) be the second eigenfunction (resp. eigenvalue) of the problem
(464d)~+ XP(X)Y
= 0,
Y(O) = 0)
= 0.
As is well known, the function y2 admits precisely one zero (call it x1) in the interval (0, e). Furthermore, the number X2(q, p) is in fact the first eigenvalue of the two problems: (d4Y’)
+ XP(X)Y = 0,
(q(x)y’)’ + xp(x)y = 0,
Y(0) = Y(Q) Y(X~ =
= 0,
de) = 0.
According to Corollary 1 and Theorem 2, we have:
(7)
X2(q, p) 5 C(o)H1+l’”
(~z’q(x)-dx)1’~(~z’p(r)dx)-(2+1’~’,
839
S. Karaa
where c(a)
(8) Similarly,
is given by Corollary
1 for o > 1 and c(1)
(I~‘~(~)~di)-1’~(JC~11i(X)iii)2+1’~
= 12. Inequality
(7) can be written
as:
5 C(~)H1+l/“[X2(4,P),-l.
we have:
(9)
Now put b = St1 q(x)a dz and c = ST1 p(z) dz. By summing (10)
b-1/ac2+1/a
+ (1 - b)-I/“(1
- c)~++
(8) and (9), we obtain:
< C(n)H1+l’”
[X,(9, p)]-‘.
By Lemma 3 (applied with n = l), the left side of (10) is not less than 2-l c are both in the interval (0,l). Therefore l/2 which shows that &(q,p)
< 2 C(o) H 1+1’a
< 4C(cr)H’+““.
[XzkJ,
since the reals b and
/w,
If ljqllcy = A and llplli = B, then we obtain
X2(4, p) 2 4 C(o)
H1+‘la
A B-(2+1’“).
To prove the general case n > 2, we argue as above and take into account the fact that the n-th eigenfunction of problem (l)-( 2) h as exactly (n - 1) zeros between 0 and e. Note remise le le’ avril 1997, acceptee le 28 juillet 1997.
References [l] Bandle C., 1987. Extremal problems for eigenvalues of the Sturn-Liouville type, Internat.Ser. Numer. Math , General inequalities,5, Birkhauser, 80, pp. 319-336. [2] Egorov Yu. V. and Karaa S., 1994. Optimisation de la premiere valeur propre de l’optrateur de Sturm-Liouville, C. R. Acad. Sci. Paris, 319, SCrie I, pp. 793-798. [3] Egorov Yu. V. and Karaa S., 1996. SW la forme optimale d’une colonne en compression, C. R. Acad. Sci. Paris, 322, Strie I, pp. 519-524. [4] Egorov Yu. V. and Kondratiev Yu. V., 1991. On an estimate of the principal eigenvalue of the Sturm-Liouville operator, Vestnik Mosk. (In-ta Math. Mech., 6, pp. 5-l 1. [5] Karaa S., 1996. Extremal eigenvalues and their associated nonlinear equations, Boll. Un. Math. Ital., IO-B, pp. 625-649. [6] Talenti G., 1984. Estimates of eigenvalues of Sturm-Liouville problems, Internat. Ser. Numer. Math., General inequalities, 4, Birkhluser, pp. 341-350.
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