J. of Inequal, & Appl., 1997, Vol. 1, pp. 165-170
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Extension of Two Inequalities of Payne
R. SPERB
Seminar for Angewandte Mathematik, ETH ZOrich, Switzerland
(Received 22 August 1996)
In this note isoperimetric bounds are derived for the maximum of the solution to the Poisson
problem for a plane domain. This extends previous bounds of Payne valid for the torsion problem.
Keywords: Isoperimetric inequalities; Poisson problem.
1 INTRODUCTION
The boundary value problem
in
on
Ar+I=0
p
0
cIR2
(1.1)
is usually called the torsion problem because of its mechanical interpretation.
Another interpretation relates (1.1) to a laminar flow in a pipe of cross-section
f2. Then, p is proportional to the flow velocity. A third important possibility
is a stationary heat flow problem with measuring the temperature.
An important quantity in all these contexts is
s
f lvrl2 f
dx
dx
(dx
areaelement).
In the mechanical interpretation of (1.1) S is called the torsional rigidity. A
second quantity of interest is
1/rm
max lr(x)
165
R. SPERB
166
Many bounds for lpm and S are known, see e.g. 1, 2, 4]. In particular P61ya
and Szeg6 proved that
1/rrn <
A
(1.2)
with A denoting the area of f2 and furthermore that
2
s< A
.
(.3)
Later on Payne [3] proved the sharper inequality
S 1/2
(1.4)
and also gave the lower bound
4re lpm >_. A
(A 2
87/’S) 1/2
(1.5)
In all these inequalities the equality sign holds if is a disk.
In this note the primary concern is to give an extension of Payne’s
inequalities (1.4), (1.5.) to the Poisson problem in the plane, i.e. the boundary
value problem
Au+p(x)=0 in g2
(1.6)
on 02
u=0
where p(x) is a smooth, strictly positive function satisfying
A (log p)
< K
2p
(1.7)
for some constant K. If K > 0 an additional requirement is that
K
f
p dx < 4zr
(1.8)
Remark Problem (1.6) is equivalent to problem (1.1) for a domain on
a surface of Gaussian curvature K
2p A(log p) (see [1, 4] for more
details).
2 EXTENSION OF PAYNE’S INEQUALITIES
The analogue of inequalities (1.4) and (1.5) can be stated as
EXTENSION OF TWO INEQUALITIES OF PAYNE
THEOREM
167
Suppose p(x) satisfies (1.7) and (1.8) in the simply connected
plane domain fa and set
Then one has for Um
max u the inequalities
1
Um
and
Um-+-
(1- e -Kum) <
A
1
(e Kum
K)
K.S
(2.1)
4:re
1) >
K.S
4re
Equality holds in (2.2), (2.3) if fa is a disk and p is of the form
(1
4rg
(2.2)
q- r2) 2’
positive number
distance from the center of the disk.
c
r
Remark For K --+ 0 inequalities (2.1) and (2.2) reduce to the inequalities
(1.4) and (1.5) of Payne as a Taylor expansion with respect to K shows.
Proof of the Theorem: Let 1-’t be the level-line where u
domain enclosed by Ft. We set for v 6 (0, urn)
S())=fUm(t,u, ds)
and fat the
dt,
(2.3)
p dx =: a(v)
(2.4)
Then
IVul ds
dv
where we have used Green’s identity and defined the quantity a(v) such that
a(O)
f
p dx
=- A
and a(um)
O
Next we make use of the fact that (see 1], p. 53)
da
dv
fr
ds
P
IVul
a.e. in (0, urn)
(2.5)
R. SPERB
168
By Schwarz’s inequality one has
I’ulds"
P
>
IVul
(2.6)
ds
At this point we can use Bol’s inequality (see 1], p. 36) which states that if
p(x) satisfies (1.7) and (1.8) then
( fr /-ds)Z > a(v)(4yr
Ka(v))
(2.7)
Combining now (2.4), (2.6) and (2.7) we e led to the inequality
d2S
K"
dv 2
dS
dv
(2.8)
4
or in equivalent form as
> 4 e_
(_Kv dS)
dv
d
e
dv
v0 to v =Um gives after some
Integration of (2.9) from a value v
reangement
dS
4
>
(1
dv vo- K
since
dS
For v0
(2.9)
e -K(um-v))
(2.10)
O.
Um
0 (2.10) reads
-
4
A>(1- e_Kum)
K
or equivalently
Um <
1
log
(2.11)
4zr
( 4zr-KA
)
as noted by Bandle (see [1]).
If we now integrate (2.10) one more time from v0
led to
IXZul ds
S(0)
>
4r[
which is inequality (2.1).
1
Um
(1
0 to v
IVul2dx
dt
e -Kum)
(2.12)
]
Um we are
S
(2.13)
EXTENSION OF TWO INEQUALITIES OF PAYNE
169
(For the second equality sign in (2.13), see e.g. [4], p. 190). Inequality
(2.2) is obtained in a completely analogous manner: the first integration of
(2.9) is now from v 0 to v v0 and the second is from v0 0 to v Um
as before.
3 REMARKS
--
(a) It was shown by Bandle (see [1]) that
S<
47/"
log
A
K
4zr
4r
KA
If we write (2.13) as
S-+-
4zr
(1- e -Kum)
>"
4r
Um
and use (2.11).and (2.12), we see that the upper bound for Um given
in (2.1) is sharper than the bound (2.12), but it requires the knowledge
of S or a close upper bound for S.
(b) There are other types of bounds that can be obtained from the
differential inequality (2.8). For example if we write it in terms of
a(v) as
da
> 4re Ka(v)
drand then change the independent variable and writing u in the place
of v it becomes
1
du
<
(.3.2)
da
4re Ka
This inequality can be integrated in many ways. As an example we perform
a double integration as follows:
foA[fsA(_dlg__, in f.
.a/ da ds
Setting
f
are
)
log
(
fo
u 2pdx<-
4r-Ka
un p dx <
foA[fs
A
da
4r
Ka
In
ds
(3.3)
upper bound for/gm one has e.g.
A
2-1
27rf
2(
-K
K
4zr
K f
)
(3.4)
R. SPERB
170
If instead of the double integral A
f0 fsA
S >_
A’um d-
1
"-
(47r
we select
KA).
f
f0a f
A
K
then we obtain
(3.5)
(c) A number of other types of bounds for problems (1.1) and (1.6) can
be found in 1, 2, 4].
References
C. Bandle, Isoperimetric inequalities and applications. Pitman (1980).
[2] L.E. Payne, Isoperimetric inequalities and their applications. SlAM Rev., 9 (1967), 453-488.
[3] Some isoperimetric inequalities in the torsion problem for multiply connected regions.
Studies in Math. Anal. and Related Topics: essays in honor of G. P61ya, Stanford Univ.
Press (1962), 270-280.
[4] R.P. Sperb, Maximum principles and their applications. Academic Press (1981).