Directory UMM :Journals:Journal_of_mathematics:EJQTDE:

Electronic Journal of Qualitative Theory of Differential Equations
Proc. 7th Coll. QTDE, 2004, No. 4 1-12;
http://www.math.u-szeged.hu/ejqtde/

ISOPERIMETRIC INEQUALITIES FOR SOME NONLINEAR
EIGENVALUE PROBLEMS
´
GABRIELLA BOGNAR

Abstract. In this paper we intend to review many of the known inequalities for eigenvalues of the Laplacian in Euclidean plane. Our aim is to show
that we can generalize some results for the pseudo-Laplacian. We focus on
isoperimetric inequalities for the first eigenvalue of the Dirichlet eigenvalue
problem.

1. The nonlinear eigenvalue problem
We seek the eigenfunctions uj and corresponding eigenvalues λj
of the following nonlinear eigenvalue problem
−Qp = λ |u|p−1 u

(D)

(j = 1, 2, ...)

in Ω,

where the nonlinear operator Qp is the pseudo-Laplacian defined by
!
!
 p−1
 p−1
∂  ∂u 
∂u
∂u
∂  ∂u 
+
for 0 < p < ∞.
Qp =
∂x  ∂x 
∂x
∂y  ∂y 
∂y

The boundary condition corresponding to the problem is
u|∂Ω = 0.

Here Ω denotes a two-dimensional body with boundary ∂Ω. The boundary value
problem in which equation (D) is to be solved is called Dirichlet problem.
For p = 1 the problem (D) describes the vibration of an elastic membrane with
fixed boundary:
(L)

∆u + λu = 0

in Ω,

u = 0 on ∂Ω.
The function u ∈ W01,p+1 (Ω) is a generalized or weak solution of (D) if for every
v ∈ W01,p+1 (Ω)


 p−1

 p−1
Z
Z
 ∂u 
 ∂u 
∂u
∂u
∂v
∂v
 




+
dx = λ vu |u|p−1 dx.
∂x ∂x  ∂x 
∂y ∂y  ∂y 
Ω

Ω

It is known (see [3]) that there exist countably many number of distinct normalized eigenfunctions in W01,p+1 (Ω) with associated eigenvalues to the eigenvalue
problem (D) . For the eigenvalues λk (p) of the Dirichlet eigenvalue problem (D)
1991 Mathematics Subject Classification. 35J20, 35P15.
Key words and phrases. eigenvalue, pseudo-Laplace operator, isoperimetric inequality, FaberKrahn inequality.
This paper is in final form and no version of it will be submitted for publication elsewhere.

EJQTDE, Proc. 7th Coll. QTDE, 2004 No. 4, p. 1

the relation λk (p) → ∞ holds when k → ∞. Every eigenvalue is positive. Here, the
first eigenfunction has also many special properties. The first eigenfunction does
not change sign and the corresponding first eigenvalue λ1 is simple [19].
The main objective of the paper is to give lower bounds for the first eigenvalue
to the eigenvalue problem (D) .
2. The classical isoperimetric inequality
The classical isoperimetric inequality after which all such inequalities are named
states that the circle encloses the largest area of all plane curves of given perimeter.
Our aim is to show that many Euclidean results remain valid in Minkowski plane
1/(p+1)


p+1
p+1
with the norm kxkp := |x1 |
+ |x2 |
. Of course, for p = 1 it is the

usual Euclidean norm on R2 .
Let Ω be a simply connected convex domain. We denote the Minkowski arc
length of ∂Ω by Lp , the usual area of Ω by A, and the radius of the greatest
inscribed curve c̺ of Ω by ̺, when domain Ω is bounded by curve c̺
1

1

1

|x| p +1 + |y| p +1 = ̺ p +1 ,

(c̺ )

̺ ∈ R+ .

Curve c̺ is significant because many Euclidean results are preserved if the unitsphere is replaced by the solution of the isoperimetric problem which is called
isoperimetrix. The Minkowski length of curve c̺ is
v
p+1
Z̺ q
Z̺ u
u
|x| p
p+1
p+1
p+1
t
′
(2.1) Lp (c̺ ) = 4
1 + |y | dx = 4
1 + p+1
p+1 dx = 2P ̺,

p
p
−
|x|
ρ
x=0
x=0
(see [25]) and the area of the domain bounded by this curve is
Z̺
p+1
p+1
p
(2.2)
A(c̺ ) = 4
[̺ p − |x| p ] p+1 dx = P ̺2 ,
x=0

where
(2.3)

p
P =2
B
p+1



p
p
,
p+1 p+1




p
p
is a Beta function (see [13]). If p = 1, then P = π.
,
p+1 p+1

G. D. Chakerian [8] proved and applied the Bonnesen inequality

and B

(2.4)



(Lp ) ̺ ≥ A + P ̺2

in the Minkowski plane for any convex n-gon (and consequently for any convex
curve). This inequality was proved by L. Fejes T´
oth [12] for nonconvex curves
in the Euclidean plane. This proof can be generalized without difficulty to such
Minkowski geometry where the ”circle” is any centrally symmetric convex curve. So
the Bonnesen inequality (2.4) is valid for non-convex curves in Minkowski geometry.
In the case p = 1 the inequality (2.4) is reduced to the Bonnesen inequality valid
on the Euclidean plane.
EJQTDE, Proc. 7th Coll. QTDE, 2004 No. 4, p. 2

From the Bonnesen inequality for a simply connected convex domain G. D.
Chakerian [8] showed that the isoperimetric inequality in the Minkowski metric for
a simply connected convex domain Ω has the form
(2.5)

2

(Lp ) − 4P A ≥ 0.

In (2.5) equality holds if and only if domain Ω is bounded by curve c̺ .
3. The isoperimetric inequality in ”broader sense”
There are several interesting and important geometrical and physical quantities
depending on the shape and size of a curve: e.g., the length of its perimeter, the
area included, the moment of inertia, with respect to the centroid, of a homogeneous plate bounded by the curve, the torsional rigidity of an elastic beam the
cross-section of which is bounded by the given curve, the principal frequency of a
membrane of which the given curve is the rim, the electrostatic capacity of a plate
of the same shape and size, and several other quantities.
By the help of the isoperimetric inequalities we can estimate physical quantities
on the basis of easily accessible geometrical data.
3.1. Bounds for eigenvalues. Isoperimetric inequalities are also useful in the

derivation of explicit a priori inequalities employed in the determination of a priori
bounds in various types of initial or boundary value problems.
In the linear case (L) many papers and books were published on the estimation of
the first eigenvalue (see [22],.[1], [23] or [18]). Such bounds are based on geometrical
data of the domain.
For the case of (L) Lord Rayleigh conjectured that for all membranes with given
area A the circle yields the minimum value of λ1 . This property can be expressed
by the inequality
πj02
A
with equality only for the circle and where j0 is the first positive zero of the Bessel
function of the first kind J0 (x). G.Faber [11] and E.Krahn [16] found independently
essentially the same proof of Rayleigh’s conjecture. Another proof was given by
G.P´
olya and G.Szeg˜
o [22] by using Steiner symmetrization.
We know that for the domain Ω with sufficiently smooth boundary ∂Ω the first
eigenvalue in the fixed membrane problem (D) admits the following characterization:

R
p+1
p+1
dx
|ux |
+ |uy |
Ω
λ1 =
min
.
R p+1
u∈W01,p+1 (Ω)
|u|
dx
(3.1)

λ1 ≥

Ω

This characterization gives us an upper bound for λ1 , i.e., for any v ∈ W01,p+1 (Ω)

R
p+1
p+1
|vx |
+ |vy |
dx
λ1 ≤ Ω
.
R p+1
|v|
dx
Ω

The equality sign will always hold for some choice of v.
EJQTDE, Proc. 7th Coll. QTDE, 2004 No. 4, p. 3

3.2. A lower bound for the first eigenvalue. As it was shown in the introduction many papers were published on the estimation of the first eigenvalue for the
linear eigenvalue problem.
Let the domain Ω be a simply connected convex domain. We define the curve
(cρ0 ) by
(3.2)

1

1

1

|x − x0 | p +1 + |y − y0 | p +1 = ρ0p

+1

,

ρ0 ∈ R + ,

(x0 , y0 ) ∈ Ω.

In this part let us denote the radius of the greatest inscribed curve (cρ0 ) of Ω by ρ.
We shall consider (x0 , y0 ) as the origin of the new coordinate system.
Theorem 1. Let λ1 be the smallest eigenvalue of the eigenvalue problem (D) such
that u1 ≥ 0 in the simply connected convex domain Ω∈ R2 . Then the inequality

p+1
A+σ
(3.3)
λ1 ≥
(p + 1)ρA
holds, where A is the area of Ω , ρ is the radius of the greatest inscribed isoperimetrix
cρ of Ω , and σ = P ̺2 is the area of the isoperimetrix of radius ρ.

Proof. As in [22] and [15] we shall reconstruct the first eigenfunction u 1 (x, y) from
its level sets Ω(τ ) = {(x, y) ∈ Ω| u1 (x, y) ≥ c}, where the area of Ω(τ ) is τ , so
0 ≤ τ ≤ A and Ω(A) = Ω (see [5]). The domain Ω(τ ) may be disconnected but
each of its components is simply connected because u1 does not have local minima
in Ω. Instead of coordinates x, y we introduce the new coordinates τ and s in the
domain Ω, where s is the arc length in Minkowski metric of the level line which
bounds Ω(τ ). Therefore 0 ≤ s ≤ L(τ ), where L(τ ) is the total length of the level
line.
Let the function ϕ(τ ) be defined as
ϕ(τ ) = u1 (x, y)

on ∂Ω(τ ).

Evidently ϕ(A) = 0 and ϕ(τ ) is monotonically decreasing when 0 ≤ τ ≤ A. We
have for the derivatives
∂ϕ
∂u1 ∂x ∂u1 ∂y
∂u1
=
= ϕ′ (τ ) =
+
,
∂τ
∂τ
∂x ∂τ
∂y ∂τ
∂u1
∂ϕ
∂u1 ∂x ∂u1 ∂y
=
=0=
+
.
∂s
∂s
∂x ∂s
∂y ∂s
Since

therefore we get
(3.4)

 p+1  p+1
 ∂y 
 ∂x 
 
+  
= 1.
 ∂s 
∂s





 ∂u1 p+1  ∂u1 p+1
|ϕ′ (τ )|p+1




+
=
,
 ∂y 
 ∂x 
|△|p+1

where the Jacobian △ is



 ∂x ∂y 


 ∂τ ∂τ 
△ =  ∂x

∂y




∂s ∂s
EJQTDE, Proc. 7th Coll. QTDE, 2004 No. 4, p. 4

(similarly as in [17] for the linear case p = 1). The Rayleigh quotient can be expressed by the new coordinates τ and s:


 !
Z 
#
"
 ∂u1 p+1  ∂u1 p+1
L(τ
R ) ds
RA




′
p+1
dx
dy
+
 ∂x 
 ∂y 
dτ
|ϕ (τ )|
p
s=0 |△|
τ =0
Ω
R
"
#
.
(3.5) λ1 =
=
L(τ)
|u1 |p+1 dx dy
R
RA
|ϕ(τ )|p+1
|△|ds dτ
Ω
s=0

τ =0

Since

τ=

Z

Ω(τ )

Z
Zτ L(t)
dx dy =
|△| ds dt
t=0 s=0

we obtain
L(τ)
Z

(3.6)

|△| ds = 1.

s=0

Applying the H¨
older inequality we get
p
1
 p+1

 p+1

L(τ
L(τ)
Z )
Z
ds 



|△| ds
≥ L(τ ).



|△|p
s=0

s=0

Hence by (3.6)

L(τ
Z )

s=0

ds
≥ [L(τ )]p+1 .
|△|p

Now from (3.5) we have

λ1 ≥

RA

τ =0

|ϕ′ (τ )L(τ )|p+1 dτ
RA

τ =0

.

|ϕ(τ )|p+1 dτ

Let us denote by ρ(τ ) the radius of the greatest inscribed curve (cρ0 ) of Ω(τ ). Now
we have the inequality ρ(τ 1 ) ≤ ρ(τ 2 ) when τ 1 < τ 2 . We shall use the isoperimetric
inequality (2.5)


√
p
p
p
L(τ ) ≥ 4P τ , P = 2
B
,
p+1
p+1 p+1
and the Bonnesen inequality (2.4)
L(τ ) ≥
For σ ≤ τ ≤ A where

τ
+ P ̺(τ ).
̺(τ )

σ = P ̺2
EJQTDE, Proc. 7th Coll. QTDE, 2004 No. 4, p. 5

we show that L(τ ) ≥
t > 1 and

τ
+ P ̺. Because the function f (t) = t +
̺
√
√
√
τ
τ
σ
√
=1
≥ √
≥ √
P ̺(τ )
P̺
P̺

then
L(τ ) ≥



√
√ 
τ
τ

+ P ̺(τ ) = P τ  √
+
̺(τ )
 P ̺(τ )



√
√  τ
1 

≥ Pτ 
 √P ̺ + √τ  =
√
P̺
We define the function X(τ ) as follows
( √
4P τ
if
τ
X(τ ) =
+ Pρ
if
ρ
and we have the relation
(3.7)
Applying (3.7) we obtain

1
t

is increasing if




1

√
≥
τ 
√
P ̺(τ )

τ
+ P ̺.
̺

0 ≤ τ < σ,
σ ≤ τ ≤ A,

X(τ ) ≤ L(τ ) (0 ≤ τ ≤ A)

λ1 ≥

RA

τ =0

|ϕ′ (τ )X(τ )|p+1 dτ
RA

τ =0

For the function X(τ ) we find

.

|ϕ(τ )|p+1 dτ

lim X(τ ) = lim X(τ ) = 2P ̺,

τ →σ−0

τ →σ+0

1
̺
therefore X(τ ) ∈ C 1 (0, A). Now we introduce the function
lim X ′ (τ ) = lim X ′ (τ ) =

τ →σ−0

τ →σ+0

Y (τ ) =

A+σ
τ
A̺

for which Y (0) = X(0), Y (A) = X(A). For 0 ≤ τ < σ we get
s
s
r
r

2
√
A+σ
σ 2 τ 2
τ
τ
τ
=
τ.
1+
4P τ ≥ 4 P ρ2 2 = 4 σ 2 ≥ 4 2 ≥
ρ
ρ
ρ
A
ρ2
A̺
If σ ≤ τ ≤ A then
τ
τ
Pρ
τ
σ
A+σ
+P ̺ ≥ +
τ= +
τ=
τ,
̺
ρ
A
ρ Aρ
A̺

therefore Y (τ ) ≤ X(τ ) when 0 ≤ τ ≤ A.
EJQTDE, Proc. 7th Coll. QTDE, 2004 No. 4, p. 6

Using the function Y (τ ) we get


A+σ
λ1 ≥
A̺

p+1

RA

τ =0
RA

|τ ϕ′ (τ )|p+1 dτ

τ =0

For reals X, Y and p > 0 inequality
p+1

|X|

(3.8)

+ p |Y |

p+1

.

|ϕ(τ )|p+1 dτ

− (p + 1)X Y |Y |

p−1

≥0

holds, with equality if and only if X = Y.
Making use of inequality (3.8) with X = −(p + 1)τ ϕ′ and Y = ϕ we can write
∗

|(p + 1)τ ϕ′ |p+1 + p|ϕ|p+1 + (p + 1)2 τ ϕ′ ϕ p ≥ 0,

and therefore

∗

ϕ p = |ϕ|p sgn ϕ,

|(p + 1)τ ϕ′ |p+1 − |ϕ|p+1 = |(p + 1)τ ϕ′ |p+1 + p|ϕ|p+1 − (p + 1)|ϕ|p+1 ≥
∗

≥ −(p + 1)[(p + 1)τ ϕ′ ϕ p + |ϕ|p+1 ].
Integrating by parts we have
ZA

τ =0

[|(p + 1)τ ϕ′ |p+1 − |ϕ|p+1 ]dτ ≥

≥ −(p + 1)

ZA

τ =0

since ϕ(A) = 0. Hence

∗

[(p + 1)τ ϕ′ ϕ p + |ϕ|p+1 ]dτ =

= −(p + 1) [|ϕ|p+1 τ ]A
τ =0 = 0
RA

τ =0
RA

|τ ϕ′ (τ )|p+1 dτ

τ =0

|ϕ(τ )|p+1 dτ

≥

1
,
(p + 1)p+1

thus for the first eigenvalue λ1 we obtain the inequality (3.3).
In the case p = 1 the lower bound for the smallest eigenvalue is reduced to the
lower bound for λ1 given by Makai in [17].
3.3. Application of symmetrization. For the nonlinear eigenvalue problem (D)
let us consider the bounded domain Ω and the continuous function u(x, y) ∈
W01,p+1 (Ω) on Ω satisfying u|∂Ω = 0. We introduce the level set Ωc of u for c ∈ R
by
Ωc = {(x, y) ∈ Ω : u(x, y) ≥ c}.
The most frequently used type of symmetrization is the Schwarz symmetrization
(0)
which is centrally symmetric. We define the Schwarz symmetrization Ωc of Ωc ⊂
2
R by
EJQTDE, Proc. 7th Coll. QTDE, 2004 No. 4, p. 7

Ω(0)
c :=



bounded by curve (cρ ) with the same area as Ωc if Ωc 6= ∅,
∅
if Ωc = ∅.

For u the level sets Ωc are replaced by concentric curves (cρ ) centered at zero.
Therefore the Schwarz symmetrization of u is defined by
o
n
.
u(0) := sup c ∈ R| (x, y) ∈ Ω(0)
c

Let us consider the bounded domain Ω and the continuous function u(x, y) on
Ω satisfying u|∂Ω = 0. We introduce the level set Ωc of u for c ∈ R by
Clearly

and

¯ : u(x, y) ≥ c}.
Ωc = {(x, y) ∈ Ω
Ωc′ ⊃ Ωc′′

if

umin ≤ c′ < c′′ ≤ umax ,

Ωc = Ω if

c < umin ,

Ωc = ∅

c > umax .

if

By these properties we can replace function u by a related function u (0) which has
some desired properties (see [15] p.21).
Theorem 2. Let λ1 be the smallest eigenvalue and u1 be the corresponding eigen(0)
function, respectively, to the eigenvalue problem (D) on the domain Ω and λ1 be
(0)
the smallest eigenvalue with the corresponding eigenfunction u 1 on the domain
(0)
Ω(0) . Then λ1 > λ1 unless ∂Ω is a curve (c̺ ).
Proof. Our proof is based on the proof given by G.Faber [11] and E. Krahn [16]
in the linear case (L). First of all we give the differences between the linear and
nonlinear cases. We consider the level sets of the first eigenfunction u 1
¯:
Ωc = {(x, y) ∈ Ω

u1 (x, y) ≥ c},

c ∈ R.

Instead of coordinates x, y we introduce the new coordinates w, and s. The
intersection of the plane w = c with the surface of w = u1 gives the level sets
Ωc . Therefore we get 0 ≤ w ≤ a, where a is the maximum value of u1 . Let the
coordinate s be the arc length of the level line from 0 the total length L (w) of ∂Ω c
We have the following connections:
∂u1
= 1,
∂w

∂u1
=0
∂s

that is
∂u1 ∂y
∂u1 ∂x
+
= 1,
∂x ∂w
∂y ∂w
∂u1 ∂x ∂u1 ∂y
+
= 0,
∂x ∂s
∂y ∂s
EJQTDE, Proc. 7th Coll. QTDE, 2004 No. 4, p. 8

and

 p+1  p+1
 ∂y 
 ∂x 
 
+  
= 1.
 ∂s 
∂s
For the new coordinates we have the Jacobian


 ∂x ∂x 



∂s  .
∆ =  ∂w
 ∂y ∂y 


∂w ∂s
Hence we obtain

p+1
1


p+1
|(u1 )x |
+ (u1 )y 
=
p+1
|∆|
so
Z
Za L(w)
Z 

p+1 
1


p+1
dx dy =
|(u1 )x |
+ (u1 )y 
p ds dw.
|∆|
w=0 s=0

Ω

We denote by A (w) the area of Ωc that is
A (w) =

Z

dx = −

Dc

Zw

L(
Zw)

w=a s=0

and A (0) =mes Ω, A (a) = 0. Therefore we have
′

A (w) = −

(3.9)

L(w)
Z
s=0

|∆| ds dw
e

|∆| ds,

moreover by the H¨
older inequality and by inequality (2.5) we obtain

 p1
L(w)
L(w)
Z
Z
p+1
1
1


p +1
(3.10)
|∆| ds 
≥ [4P A(w)] 2p .
p ds ≥ |L(w)|
|∆|
s=0

s=0

By equation (3.9) we get
L(w)
Z
s=0

p+1
1
−p−1
′
[−A′ (w)] [4P A(w)] 2
p ds ≥ |−A (w)|
|∆|

therefore

Z 

(3.11)

Ω

(4P )

p+1
2

p+1

|(u1 )x |

Za

w=0


p+1 


+ (u1 )y 
dx dy ≥

|−A′ (w)|

−p−1

[−A′ (w)] [A(w)]

p+1
2

dw.

EJQTDE, Proc. 7th Coll. QTDE, 2004 No. 4, p. 9

(0)

Since u1 is symmetric function we can write
(0)

u1 (x, y) = v (ρ) .
We obtain
p+1 #
Z  p
Za L(w)
 dv 

  ds dw =

dx dy =
 dρ 
y


Z " 

 
 (0) p+1  (0)
+  u1

 u1
x

Ω

(3.12)

=

Za

w=0
2

Since A(w) = P ρ ,

w=0 s=0



 p
Za
 dw p
 dv 
p+1
 dw.
ρp+1 
2P ρ   dw = 2 P
dρ
d (ρ2 ) 
w=0

ρ=
thus



A(w)
P

 12



 dw 
P


 d (ρ2 )  = − A′ (w) ,

and from (3.12)


Z "
 

 (0) p+1  (0)
+  u1
 u1


(3.13)

p+1 #


dx dy =
y

x

Ω

= (4P )

p+1
2

Za

w=0

|−A′ (w)|

−p−1

[−A′ (w)] [A(w)]

p+1
2

dw.

Comparing with (3.11) we obtain

Z 
Z 

p+1 

 

 (0) p+1  (0)

p+1
+  u1
|(u1 )x |
+ (u1 )y 
dx dy ≥  u1

x

Ω

Ω

Using the property ii.) from [15] we get
Z 

 (0) p+1
p+1
dx dy.
(3.14)
ku1 kLp+1 =  u1 

p+1

 dx dy.
y

Ω

Therefore (3.14) yields the estimate
R

Ω

λ1 =




p+1 


dx dy
|(u1 )x |p+1 + (u1 )y 
p+1

ku1 kLp+1

"

R  (0)  p+1  (0) 
+  u1

 u1
Ω

≥

x

p+1 #


dx dy
y

R  (0) p+1
 u1  dx dy

≥

Ω

EJQTDE, Proc. 7th Coll. QTDE, 2004 No. 4, p. 10

≥

inf

v∈F (Ω(0) )

R
Ω

p+1

|vx |

R

Ω

p+1

+ |vy |

|v|

p+1



dx dy

dx dy
(0)

= λ1 .

From the isoperimetric inequality (2.5) it follows that in (3.11) we have equality
(0)
(also in (3.10), respectively) if the level lines are curves (c̺ ). Thus λ1 > λ1 unless
∂Ω is a curve (c̺ ).
Corollary 1. In the case when the domain Ω is bounded by the curve (c̺ ), then
the first eigenvalue of (D) can be given by
 p+1
p+1
h0
P 2 hp+1
(0)
0
λ1 =
,
=
p+1
ρ
A 2
where h0 is the first positive zero of the generalized nonlinear Bessel function H 0 (x)
satisfying the nonlinear ordinary differential equation
"
#
p−1

p−1
dH0
dH0
d  dH0 
1  dH0 
p−1
+ λ |H0 |
H0 = 0
(3.15)
+
dx  dx 
dx
x  dx 
dx

with conditions
(3.16)

H0 (1) = 0

and

dH0
(0) = 0,
dx 

p
p
p
λ = 1 and P = 2
(see [6]). Hence for any domain on the
B
,
p+1
p+1 p+1
plane with area A we get a lower bound for the first eigenvalue:
(3.17)

λ1 ≥

P

p+1
2

A

hp+1
0

p+1
2

.

Inequality (3.17) is the generalization of the Faber-Krahn inequality (3.1) to the
nonlinear problem (D).
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(Received September 30, 2003)
´ ros, HunMathematical Institute, University of Miskolc, H-3515 Miskolc-Egyetemva
gary

EJQTDE, Proc. 7th Coll. QTDE, 2004 No. 4, p. 12