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Some isoperimetric inequalities of the kind of Hersch-Monkewitz
MECH. RES. COMM.
SOME
Vol.l,
ISOPERIMETRIC
335-340,
1974.
INEQUALITIES
OF
G@rard Philippin Department of Civil Engineering, University of Waterloo, Waterloo, (Received
i.
7 October
1974;
accepted
Pergamon Press.
THE
KIND
OF
Printed in USA.
HERSCH-MONKEWITZ
Solid Mechanics Division Ontario, Canada
as r e a d y for p r i n t
14 O c t o b e r
1974)
Introduction
Let 0 < Ii < ~2 ~ 13 ~ "'" denote the eigenvalues of a vibrating homogeneous membrane in a domain G
z
of area A of the complex z-plane, fixed along its
boundary ~G : Z
A~ + I~ = 0 in Gz,
(1.1)
~ = 0 on ~G z.
Let 0 = ~i < ~2 ! ~3 ! "'" denote the eigenvalues corresponding to the homogeneous membrane, free on 3Gz: Au + ~u = 0
in Gz,
3u Dn _ 0 on 3G z .
G. Szeg~ and H.F. Weinberger
(1.2)
[1,2] have shown that
(~2 - 1 + ~3- 1) A- i = Min f o r the c i r c l e . J. Hersch and M. Monkewitz [3] improved t h i s (cX1 1+~2 i+~3 1 ) A - I = Min f o r the c i r c l e The purpose of t h i s paper i s to e s t a b l i s h the same k i n d . 2.
Let us consider the unit circle G
W
(1.3) last result: f o r any c i n [0, 1 . 2 5 8 8 ] . ( 1 . 4 )
some i s o p e r i m e t r i c
inequalities
of
centred at the origin of the w-plane,
and a conformal mapping w(z) of G z onto Gw, which maps the point zo £ G z into the centre w = 0 of the circle Gw. eigenfunctions of (i .i) , resp.
Let 5i(w) and ui(w), i = 1,2,3 ..... be the
(1.2) for the circle Gw-
With the help of con-
formal transplantation, we construct the functions Ui(z) = ui(w(z)) U. (z) = ui(w(z)), i = 1,2,3 .... defined on G z, and admissible for the variai n -i tional characterization of Z ~. (see [4,5]): i=l i Sci en t i fi c Comm uni ca t i on
335
336
G.
nZ 171 > i:l
1
--
z
R- [Ui] for any positive integer n, with i=l z
[~i]
z
No.S/6
(2.1)
IG f grad2UidAz
fG "fU~dAz
_
z
fG fU~dAz
Z
(DGz(Ui)
Voi.l,
G1
nE
DG (Ui) RG
PHILIPPIN
(Rayleigh quotient).
Z
is the Dirichlet integral of 0i(z ) in Gz, dAz = dxdy is the element
of a r e a ) .
G. Szeg5 [1] has p r o v e d the e x i s t e n c e
the transplanted
functions
z ° E Gz such that
of a point
U2, U3 ( o f the second and t h i r d
eigenfunctions
of
(1.2)) have both the mean value zero in Gz: IG fU2dAz : IG /U3dAz = O. Z
Z
Hence, ~2 - 1
+ ~3 - 1 >-- RGI[u2] + RGI[U3]" Z
By combining t h e i n e q u a l i t i e s obtain:
3 z-l_ - 1 -i C g 1 +>2 +k13 > i=l
(2.2)
Z
(2.1) where we choose n = 3, and ( 2 . 2 ) ,
we
C{klfo f~21dAz+k2f G I(U2+U3)dAz}+IG ~2 ~2 2 Az I(U22+ U3)d z
z
z
(z .5)'-.-"
DG (u 2) W
DG (u 2) This relationship is valid for all positive c, with k I : DG (u 2) w ~ 0.3397515 k 2 = DG (~2) w
W
= 0.259542,
DGw(~1)
Comment - In the preceding calculation we have used the fact that the Dirichlet integral remains invariant under conformal transplantation. The numerator on the right of (2.3) can be written the following way: 1
fo(C[k 1j2o(Jor)+k2Jl(Jlr)]+Jl(Pl 2 2
2~ dz 2
r)
}o~o = dw
d@ rdr,
(2.4)
where Jo(X), Jl(X) are the Bessel functions of first, resp. second order, Jo' resp. Jl are their first zero, and where Pl denotes the smallest value of x such that Jl(X) is a maximum.
(r,@) denote polar coordinates in the w-plane.
2 z(w) being an analytic function, d)_~,z is a sub-harmonic function, thus 2 g(r) = ~" ,,d~. ~, d@ is a monotonic non decreasing function of r in 0 ~ r ~ 1 r=const (see [5,6]). We shall now apply a lemma proved in [3]: Lemma - Let g(r) be a non decreasing function of r in [O,1], and f(r) satisfy the inequalities:
VoI.I,
No.5/6
ISOPERIMETRIC
2__ f~ f'(r)rdr < f = F2 o -
2f~
337
INEQUALITIES
f(r)rdr for any ~ £ (0 I) ' "
(2.5)
Then : fl f(r).g(r)rdr > ~.fl g(r)rdr. O
--
(2.6)
O
Let us define: 2 f(r) = C(klJ~(jo r) + k2J~(Jlr)) + Jl(Plr).
(2.7)
With c = 0, f(r) is an increasing function, and the inequality course satisfied. condition yields c
(2.6) is of
Let us now determine the largest value of c such that the
(2.5) remains satisfied. ~ 0.7985.
A numerical estimate with the computer
Thus, for 0 < c < 0.7985 we can use (2.6) for estimating
max
--
--
(2.4) from below: dz 2 d@ rdr > 2f~{C[klJ~(Jor ) +k2Jl(3 2 .1r)]+Jl2(Plr))rdr.AG f~f(r) ~2v ~-~ 0=0 z AG
=
2 ~z fG f{C[klJ2(Jor)+k2J21(Jlr)]+Jl(Plr)}dAw" w
From (2.3) and the above inequality, we thus obtain: 3 AG fG fJo2 (Jo r) dAw fG fJ21(jlr)dAw IG fJ21(plr)dAw c Z ,-i -i -i z w w .}, i=izi +~2 +~3 --> ~r" {c[ DG (~1) + ]+ wDG (u 2) DG (a2) W
W
W
whence :
3 ,-i -i -i~ 1 {c Z A i +~2 +P3 ).A- = Min for the circle for any c in [0, 0.7985].(2.8) i=l
Remark - The same combination of (2.1) and (2.2),
but with
n=5 in (2.1) gives
the following result: 5 {Ci=l Z ~[11+~2-i+~3 I)'A-I = Min for the circle,
(2.9)
where the constant c is to be chosen such that the function: D G (u 2) D G (u2) D G (u 2) G j2 w 2 w 2 }+j~ f(r) = c{D (~I) o(Jo r)+ DG (~2) Jl(Jl r)+ D G (d4) J2(J2 r) (Pl r) W
satisfies the inequalities
W
W
(2.5).
order, and J2 is its first zero.
(2. i0)
J2(x) is the Bessel function of second A numerical estimate yields Cma x
This fact ensures the validity of the isoperimetric inequality
0 .6412.
(2.9) for any
c in [0, 0.6412]. 3.
Let us now consider a domain G
Z
symmetric of order q = 2.
(A domain G
Z
is
called symmetric of order q if there exists a point 0 such that G z is invariant under a rotation of angle 2--~ around 0 (see [7]). q
If we choose a conformal
338
G.
PHILIPPIN
mapping of G onto the u n i t c i r c l e
v,: I]
GW, such t h a t
.
1
~
~J<-
.
)
the c e n t r e o f svlmnetrv ~I oi ~
G i s mapped i n t o t h e c e n t r e of t h e c i r c l e , then we have (hi a d d i t i o n to the z inequalities (2.1) and (2.2)), the f o l l o w i n g i n e q u a l i t y (see [5] p. 383): fG f (U22+U~)dAz 1+;~ 1 >
DG (fi2)
(3.1
W
Combining
(S.1)
and ( 2 . 2 ) ,
. . . . 1} {c(k21+XS1)+p21+~S
we e a s i l y .A- 1
o b t a i n the i s o p e r i m e t r i c
inequality:
= Min f o r t h e c i r c l e ,
(5.2)
where the constant c is to be chosen such that the function:
f(r)
DG (u 2) w -
= c
2
(o 2) 'J1 (j
DC
1
2
r)
+
(is. s)
Ol (pl r)
W
satisfies the inequalities
This f a c t e n s u r e s Remark - If G
(2.5).
the v a l i d i t y
A numerical estimate yields c x 1.8120. max
o f (3.2)
f o r any c in IO, 1 . 8 1 2 0 ] .
is symmetric of order q = 3, there exists two linear combinaZ
tions 0(z) = ~02(z)+BU3(z), U(z) = ~U2(z)+BU3(z), such that 2 fG fJ21(Jlr) l ~ >R
_
1
z)]
z
dAw
w
:
2D(; (~9)
'
(S. 4)
W
dz 2
fG /'J~(Pl r) dw -1 GI[u w P3 >- R z ( z ) ] = 2DG (ug)~
dAw (5.5)
,
W
(see [5] p. 388 for the proof). These two inequalities yields the following isoperimetric inequalities: -i -1 .A-l {ck3 +P3 } = Min for the circle.
(3.6)
This last result is valid for any c in [0, l.gl20], like in the case of the inequality (3.2) 4.
Let us finally consider the case of a domain G
symmetric of order q = 4. Z
In addition to the inequalities following inequalities
(2.1), (2.2), and (3.1), we then have the
(see [5]):
/G /(042 + U~)diz -
D G
(a4) w
'
Vol .1 , No. 5/6
-l
U4
ISOPERIMETRIC
-I + pS >
339
INEQUALITIES
Z
DG (u4)
(4.2)
w
With the help of a l l these i n e q u a l i t i e s ,
i t would be possible to find some
theorems of the kind: ClXll + c2(~21 + ~3 I) + c3(~41 + ~ l ) + c4(~21 + P3 I) + -I -i P4 + P5 = Min for the circle, where the last relationship would be valid for (ci,c2,c3,c4) in some regic,- ~ of R 4+. However, the interest of such a research seems to be too limited and does not justify the numerical calculus (even with the computer) necessary for the determination of g.
We shall only give some
particular results where only one parameter appears: {cAil+p41+p51).A -I
[0,0.4574]
(4.3)
= Min for the circle
Vc
= Min for the circle
Vc ~ [0,0.5095]
(4.4)
= Min for the circle
Vc ~ [0,0.4179]
(4.S)
= Min for the circle
Vc E [0,1.1811]
(4.6)
{C(~41+~51)'U41.DSI}.A-I
= Min for the circle Yc ~ [ 0 , 2 . 1 2 2 0 [
(4.7)
{c(~41+~51)+U21+pS1}.A-1
= Hin for the circle Vc ~ [0,2.0885]
(4.8)
5 1 {c~il+ ~ p i l } . A -
= Min for the circle
(4.9)
3 {c ~ xil+p41+p5-I}.A-I i=l 5 {c X Ai-l +P4-i +PS-1} "A-I i=l {c(A:I+~:I)+H]I+M:I}.A-I g
J
q
b
Vc e [0,i.7161]
i=2 3
5
{c z x:l+ ~ uT1}.A -1 i=l 5
l
i:2 5
{c ~ ~1+ Z u i l } . A -1 i=l
= Min for the circle V c E [0,1.3188]
(4.10)
= Min for the circle
Vc E [0,1.0590]
(4.11)
-- Min for the circle V c I~ [0,2.9928]
(4.12)
= Min for the circle V c ~ [0,3.7626]
(4.13)
1
i=2 5
{c(~21+~31)+ g p~I}.A -1 i=2 S 1
{c(~1+~51)+ Z UT1}.Ai=2 i
5.
Illustration - Let us consider a square of area A = i.
For every preceding
inequality, (excepted for (3.6) which can be applied only for domains symmetric of order q = 3), let us compare the lower bound given by the circle, with the exact value.
We choose of course c = c
max
in every inequality:
340
G. PHILIPPIN
inequalities
(2.sj (2.9)
(3.2) (4.3) (4.4)
(4.5) (4.6) (4.7)
(4.8) (4.9) (4.1o) (4.ii) (4.12)
(4.13)
syn~netry order q ] 1 2 4 4 4 4 4 4 4 4 4 4 4
Cmax
Vo] ,]
[,b).:,,'
exact values for values for t h e the square ( A : ] ) u n i t c i r c l e (A=I)
0.7985 0.6412 1.8120 0.4574 0.5095 0.4179 ].1811 2.1220 2.0885 1.7161 1.3188 1.0590 2.9928 3.7626
0.2968 0.2909 0.2962 0.09711 0.1206 0.1217 0.1212 0.1219 0.2703 0.3856 0.4185 0.4186 0.4174 0.3819
0. 2791 0.2791 0.2788 0.09438 0.1194 0. 1205 0.1203 O. 1207 0.2509 0.3643 0. 3996 0. 3997 0. 3991 0. 3607
Acknowledgement
This research was carried out under the support of the National Research Council of Canada under Grant No. A 7297.
References
I. 2. 3.
4.
5.
6. 7.
G. Szeg~, Inequalities for certain eigenvalues of a membrane of given area, J. Rat. Mech. Analysis, 3, pp. 343-356, 1954. H.F. Weinberger, An isoperimetric inequality of the N-dimensional free membrane problem, J. Rat. Mech. Analysis, 5, pp. 633-636, 1956. J. Hersch, M. Monkewitz, Une in~galit6 isop6rim6trique renfor~ant celle de Szeg~-Weinberger sur les membranes fibres, C.R. Acad. Sci. Paris, 273, pp. 62-64, 1971. J. ;lersch, Caract6risation variationnelle d'une somme de valeurs propres cons6cutives; gen~ralisation d'in~galit6s de P61ya-Schiffer et de Weyl, C.R. Acad. Sci. Paris, 252, pp. 1714-1716, 1961. J. Hersch, On Symmetric Membranes and Conformal Radius: Some Complements to P61ya's and SzegS's Inequalities, J. Rat. Mech. Analysis, 20, No. 5, pp. 378-390, 1965. G. P61ya, G. SzegS, Aufgaben und Lehrs~tze aus der Analysis, Springer, 1954. Vol. I. G. P61ya, On the characteristic frequencies of a symmetric membrane, Hath Zeitschr., 63, pp. 331-337, 1955.
SOME
Vol.l,
ISOPERIMETRIC
335-340,
1974.
INEQUALITIES
OF
G@rard Philippin Department of Civil Engineering, University of Waterloo, Waterloo, (Received
i.
7 October
1974;
accepted
Pergamon Press.
THE
KIND
OF
Printed in USA.
HERSCH-MONKEWITZ
Solid Mechanics Division Ontario, Canada
as r e a d y for p r i n t
14 O c t o b e r
1974)
Introduction
Let 0 < Ii < ~2 ~ 13 ~ "'" denote the eigenvalues of a vibrating homogeneous membrane in a domain G
z
of area A of the complex z-plane, fixed along its
boundary ~G : Z
A~ + I~ = 0 in Gz,
(1.1)
~ = 0 on ~G z.
Let 0 = ~i < ~2 ! ~3 ! "'" denote the eigenvalues corresponding to the homogeneous membrane, free on 3Gz: Au + ~u = 0
in Gz,
3u Dn _ 0 on 3G z .
G. Szeg~ and H.F. Weinberger
(1.2)
[1,2] have shown that
(~2 - 1 + ~3- 1) A- i = Min f o r the c i r c l e . J. Hersch and M. Monkewitz [3] improved t h i s (cX1 1+~2 i+~3 1 ) A - I = Min f o r the c i r c l e The purpose of t h i s paper i s to e s t a b l i s h the same k i n d . 2.
Let us consider the unit circle G
W
(1.3) last result: f o r any c i n [0, 1 . 2 5 8 8 ] . ( 1 . 4 )
some i s o p e r i m e t r i c
inequalities
of
centred at the origin of the w-plane,
and a conformal mapping w(z) of G z onto Gw, which maps the point zo £ G z into the centre w = 0 of the circle Gw. eigenfunctions of (i .i) , resp.
Let 5i(w) and ui(w), i = 1,2,3 ..... be the
(1.2) for the circle Gw-
With the help of con-
formal transplantation, we construct the functions Ui(z) = ui(w(z)) U. (z) = ui(w(z)), i = 1,2,3 .... defined on G z, and admissible for the variai n -i tional characterization of Z ~. (see [4,5]): i=l i Sci en t i fi c Comm uni ca t i on
335
336
G.
nZ 171 > i:l
1
--
z
R- [Ui] for any positive integer n, with i=l z
[~i]
z
No.S/6
(2.1)
IG f grad2UidAz
fG "fU~dAz
_
z
fG fU~dAz
Z
(DGz(Ui)
Voi.l,
G1
nE
DG (Ui) RG
PHILIPPIN
(Rayleigh quotient).
Z
is the Dirichlet integral of 0i(z ) in Gz, dAz = dxdy is the element
of a r e a ) .
G. Szeg5 [1] has p r o v e d the e x i s t e n c e
the transplanted
functions
z ° E Gz such that
of a point
U2, U3 ( o f the second and t h i r d
eigenfunctions
of
(1.2)) have both the mean value zero in Gz: IG fU2dAz : IG /U3dAz = O. Z
Z
Hence, ~2 - 1
+ ~3 - 1 >-- RGI[u2] + RGI[U3]" Z
By combining t h e i n e q u a l i t i e s obtain:
3 z-l_ - 1 -i C g 1 +>2 +k13 > i=l
(2.2)
Z
(2.1) where we choose n = 3, and ( 2 . 2 ) ,
we
C{klfo f~21dAz+k2f G I(U2+U3)dAz}+IG ~2 ~2 2 Az I(U22+ U3)d z
z
z
(z .5)'-.-"
DG (u 2) W
DG (u 2) This relationship is valid for all positive c, with k I : DG (u 2) w ~ 0.3397515 k 2 = DG (~2) w
W
= 0.259542,
DGw(~1)
Comment - In the preceding calculation we have used the fact that the Dirichlet integral remains invariant under conformal transplantation. The numerator on the right of (2.3) can be written the following way: 1
fo(C[k 1j2o(Jor)+k2Jl(Jlr)]+Jl(Pl 2 2
2~ dz 2
r)
}o~o = dw
d@ rdr,
(2.4)
where Jo(X), Jl(X) are the Bessel functions of first, resp. second order, Jo' resp. Jl are their first zero, and where Pl denotes the smallest value of x such that Jl(X) is a maximum.
(r,@) denote polar coordinates in the w-plane.
2 z(w) being an analytic function, d)_~,z is a sub-harmonic function, thus 2 g(r) = ~" ,,d~. ~, d@ is a monotonic non decreasing function of r in 0 ~ r ~ 1 r=const (see [5,6]). We shall now apply a lemma proved in [3]: Lemma - Let g(r) be a non decreasing function of r in [O,1], and f(r) satisfy the inequalities:
VoI.I,
No.5/6
ISOPERIMETRIC
2__ f~ f'(r)rdr < f = F2 o -
2f~
337
INEQUALITIES
f(r)rdr for any ~ £ (0 I) ' "
(2.5)
Then : fl f(r).g(r)rdr > ~.fl g(r)rdr. O
--
(2.6)
O
Let us define: 2 f(r) = C(klJ~(jo r) + k2J~(Jlr)) + Jl(Plr).
(2.7)
With c = 0, f(r) is an increasing function, and the inequality course satisfied. condition yields c
(2.6) is of
Let us now determine the largest value of c such that the
(2.5) remains satisfied. ~ 0.7985.
A numerical estimate with the computer
Thus, for 0 < c < 0.7985 we can use (2.6) for estimating
max
--
--
(2.4) from below: dz 2 d@ rdr > 2f~{C[klJ~(Jor ) +k2Jl(3 2 .1r)]+Jl2(Plr))rdr.AG f~f(r) ~2v ~-~ 0=0 z AG
=
2 ~z fG f{C[klJ2(Jor)+k2J21(Jlr)]+Jl(Plr)}dAw" w
From (2.3) and the above inequality, we thus obtain: 3 AG fG fJo2 (Jo r) dAw fG fJ21(jlr)dAw IG fJ21(plr)dAw c Z ,-i -i -i z w w .}, i=izi +~2 +~3 --> ~r" {c[ DG (~1) + ]+ wDG (u 2) DG (a2) W
W
W
whence :
3 ,-i -i -i~ 1 {c Z A i +~2 +P3 ).A- = Min for the circle for any c in [0, 0.7985].(2.8) i=l
Remark - The same combination of (2.1) and (2.2),
but with
n=5 in (2.1) gives
the following result: 5 {Ci=l Z ~[11+~2-i+~3 I)'A-I = Min for the circle,
(2.9)
where the constant c is to be chosen such that the function: D G (u 2) D G (u2) D G (u 2) G j2 w 2 w 2 }+j~ f(r) = c{D (~I) o(Jo r)+ DG (~2) Jl(Jl r)+ D G (d4) J2(J2 r) (Pl r) W
satisfies the inequalities
W
W
(2.5).
order, and J2 is its first zero.
(2. i0)
J2(x) is the Bessel function of second A numerical estimate yields Cma x
This fact ensures the validity of the isoperimetric inequality
0 .6412.
(2.9) for any
c in [0, 0.6412]. 3.
Let us now consider a domain G
Z
symmetric of order q = 2.
(A domain G
Z
is
called symmetric of order q if there exists a point 0 such that G z is invariant under a rotation of angle 2--~ around 0 (see [7]). q
If we choose a conformal
338
G.
PHILIPPIN
mapping of G onto the u n i t c i r c l e
v,: I]
GW, such t h a t
.
1
~
~J<-
.
)
the c e n t r e o f svlmnetrv ~I oi ~
G i s mapped i n t o t h e c e n t r e of t h e c i r c l e , then we have (hi a d d i t i o n to the z inequalities (2.1) and (2.2)), the f o l l o w i n g i n e q u a l i t y (see [5] p. 383): fG f (U22+U~)dAz 1+;~ 1 >
DG (fi2)
(3.1
W
Combining
(S.1)
and ( 2 . 2 ) ,
. . . . 1} {c(k21+XS1)+p21+~S
we e a s i l y .A- 1
o b t a i n the i s o p e r i m e t r i c
inequality:
= Min f o r t h e c i r c l e ,
(5.2)
where the constant c is to be chosen such that the function:
f(r)
DG (u 2) w -
= c
2
(o 2) 'J1 (j
DC
1
2
r)
+
(is. s)
Ol (pl r)
W
satisfies the inequalities
This f a c t e n s u r e s Remark - If G
(2.5).
the v a l i d i t y
A numerical estimate yields c x 1.8120. max
o f (3.2)
f o r any c in IO, 1 . 8 1 2 0 ] .
is symmetric of order q = 3, there exists two linear combinaZ
tions 0(z) = ~02(z)+BU3(z), U(z) = ~U2(z)+BU3(z), such that 2 fG fJ21(Jlr) l ~ >R
_
1
z)]
z
dAw
w
:
2D(; (~9)
'
(S. 4)
W
dz 2
fG /'J~(Pl r) dw -1 GI[u w P3 >- R z ( z ) ] = 2DG (ug)~
dAw (5.5)
,
W
(see [5] p. 388 for the proof). These two inequalities yields the following isoperimetric inequalities: -i -1 .A-l {ck3 +P3 } = Min for the circle.
(3.6)
This last result is valid for any c in [0, l.gl20], like in the case of the inequality (3.2) 4.
Let us finally consider the case of a domain G
symmetric of order q = 4. Z
In addition to the inequalities following inequalities
(2.1), (2.2), and (3.1), we then have the
(see [5]):
/G /(042 + U~)diz -
D G
(a4) w
'
Vol .1 , No. 5/6
-l
U4
ISOPERIMETRIC
-I + pS >
339
INEQUALITIES
Z
DG (u4)
(4.2)
w
With the help of a l l these i n e q u a l i t i e s ,
i t would be possible to find some
theorems of the kind: ClXll + c2(~21 + ~3 I) + c3(~41 + ~ l ) + c4(~21 + P3 I) + -I -i P4 + P5 = Min for the circle, where the last relationship would be valid for (ci,c2,c3,c4) in some regic,- ~ of R 4+. However, the interest of such a research seems to be too limited and does not justify the numerical calculus (even with the computer) necessary for the determination of g.
We shall only give some
particular results where only one parameter appears: {cAil+p41+p51).A -I
[0,0.4574]
(4.3)
= Min for the circle
Vc
= Min for the circle
Vc ~ [0,0.5095]
(4.4)
= Min for the circle
Vc ~ [0,0.4179]
(4.S)
= Min for the circle
Vc E [0,1.1811]
(4.6)
{C(~41+~51)'U41.DSI}.A-I
= Min for the circle Yc ~ [ 0 , 2 . 1 2 2 0 [
(4.7)
{c(~41+~51)+U21+pS1}.A-1
= Hin for the circle Vc ~ [0,2.0885]
(4.8)
5 1 {c~il+ ~ p i l } . A -
= Min for the circle
(4.9)
3 {c ~ xil+p41+p5-I}.A-I i=l 5 {c X Ai-l +P4-i +PS-1} "A-I i=l {c(A:I+~:I)+H]I+M:I}.A-I g
J
q
b
Vc e [0,i.7161]
i=2 3
5
{c z x:l+ ~ uT1}.A -1 i=l 5
l
i:2 5
{c ~ ~1+ Z u i l } . A -1 i=l
= Min for the circle V c E [0,1.3188]
(4.10)
= Min for the circle
Vc E [0,1.0590]
(4.11)
-- Min for the circle V c I~ [0,2.9928]
(4.12)
= Min for the circle V c ~ [0,3.7626]
(4.13)
1
i=2 5
{c(~21+~31)+ g p~I}.A -1 i=2 S 1
{c(~1+~51)+ Z UT1}.Ai=2 i
5.
Illustration - Let us consider a square of area A = i.
For every preceding
inequality, (excepted for (3.6) which can be applied only for domains symmetric of order q = 3), let us compare the lower bound given by the circle, with the exact value.
We choose of course c = c
max
in every inequality:
340
G. PHILIPPIN
inequalities
(2.sj (2.9)
(3.2) (4.3) (4.4)
(4.5) (4.6) (4.7)
(4.8) (4.9) (4.1o) (4.ii) (4.12)
(4.13)
syn~netry order q ] 1 2 4 4 4 4 4 4 4 4 4 4 4
Cmax
Vo] ,]
[,b).:,,'
exact values for values for t h e the square ( A : ] ) u n i t c i r c l e (A=I)
0.7985 0.6412 1.8120 0.4574 0.5095 0.4179 ].1811 2.1220 2.0885 1.7161 1.3188 1.0590 2.9928 3.7626
0.2968 0.2909 0.2962 0.09711 0.1206 0.1217 0.1212 0.1219 0.2703 0.3856 0.4185 0.4186 0.4174 0.3819
0. 2791 0.2791 0.2788 0.09438 0.1194 0. 1205 0.1203 O. 1207 0.2509 0.3643 0. 3996 0. 3997 0. 3991 0. 3607
Acknowledgement
This research was carried out under the support of the National Research Council of Canada under Grant No. A 7297.
References
I. 2. 3.
4.
5.
6. 7.
G. Szeg~, Inequalities for certain eigenvalues of a membrane of given area, J. Rat. Mech. Analysis, 3, pp. 343-356, 1954. H.F. Weinberger, An isoperimetric inequality of the N-dimensional free membrane problem, J. Rat. Mech. Analysis, 5, pp. 633-636, 1956. J. Hersch, M. Monkewitz, Une in~galit6 isop6rim6trique renfor~ant celle de Szeg~-Weinberger sur les membranes fibres, C.R. Acad. Sci. Paris, 273, pp. 62-64, 1971. J. ;lersch, Caract6risation variationnelle d'une somme de valeurs propres cons6cutives; gen~ralisation d'in~galit6s de P61ya-Schiffer et de Weyl, C.R. Acad. Sci. Paris, 252, pp. 1714-1716, 1961. J. Hersch, On Symmetric Membranes and Conformal Radius: Some Complements to P61ya's and SzegS's Inequalities, J. Rat. Mech. Analysis, 20, No. 5, pp. 378-390, 1965. G. P61ya, G. SzegS, Aufgaben und Lehrs~tze aus der Analysis, Springer, 1954. Vol. I. G. P61ya, On the characteristic frequencies of a symmetric membrane, Hath Zeitschr., 63, pp. 331-337, 1955.