CUBO A Mathematical Journal

Vol.11, No¯ 04, (59–71). September 2009

Estimates for solutions to nonlinear degenerate elliptic
equations

Fabrizio Cuccu,

Dipartimento di Matematica e Informatica,
Università di Cagliari,

Via Ospedale 72, 09124 Cagliari, Italy.
email: fcuccu@unica.it

Petar Popivanov

Institute of Mathematics and Informatics,
Bulgarian Academy of Sciences,

Sofia 1113, Bulgaria.
email: popivano@math.bas.bg

and

Giovanni Porru

Dipartimento di Matematica e Informatica,
Università di Cagliari,

Via Ospedale 72, 09124 Cagliari, Italy.
email: porru@unica.it

ABSTRACT

We find estimates of the L∞ norm of solutions to special nonlinear degenerate elliptic

partial differential equations in terms of norms of the data. We also discuss a spe-

cial isoperimetric inequality involved in the definition of the ellipticity of the above

equations.

RESUMEN

Encontramos estimaciones de la norma L∞ de las soluciones de ecuaciones diferenciales

parciales elípticas degeneradas nolineales en términos de la norma de los datos. Además,

discutimos una desigualdad isoperimétrica especial involucrada en la definición de la

elipticidad de las ecuaciones anteriormente descritas.



60 Fabrizio Cuccu, Petar Popivanov and Giovanni Porru CUBO
11, 4 (2009)

Key words and phrases: Degenerate elliptic equations; Isoperimetric inequality.

Math. Subj. Class.: 35B45; 35J70.

1 Introduction.

We are interested in the estimation of the L∞ norm of generalized solutions of a special class

of nonlinear second order degenerate elliptic partial differential equations in divergent form. The

Sobolev type spaces W
1,p
0

(ψ,D) to whom our solutions belong are defined below. Our main tools in

the proposed investigation are a variant Fubini’s theorem (see for example [7]) and a generalization

of the famous isoperimetric inequality (see [2]).

The condition (3) (see below) is inspired by the main assumption of [8]. Unfortunately, in [8]

there are no geometrical conditions leading to the fulfillment of such an assumption, and there are

not nontrivial examples satisfying it. We propose here the proof of (3) in the two dimensional case

and for functions φ(x) = λ|x|β, ψ(x) = Λ|x|γ with 0 < λ ≤ Λ and γ + 1 > β ≥ γ ≥ 0. Thus, the
complete study of the generalized isoperimetric inequality (3) is an open and, we think, a difficult

problem.

Our main result is the a priori estimate of the L∞ norm of the solution by means of appropriate

norms of ψ and fψ
1

s−1 , s > s0, and relies heavily on the parameter α of condition (3). Here s0 is

a special number which depends on p and α and it is sharp. In fact, as simple examples show, our

equation (1) even in the linear case can possess unbounded solutions for s = s0. Our results are

very precise when φ(x) and ψ(x) are constants, as in this case the classical isoperimetric inequality

holds. We propose a special example when the corresponding a priori estimate is sharp.

The paper is organized as follows: main results, special cases illustrating the main theorem,

proof of (3) in dimension two and for special radially symmetric functions φ and ψ.

2 Main results

Let D be a bounded smooth domain in RN and let 1 < p < N. Consider the following non linear

second order degenerate elliptic equation in divergent form

−
(
(aijuxiuxj )

p−2
2 aijuxi

)
xj

= f(x), x ∈ D. (1)

The summation convention from 1 to N over repeated indices is in effect. The matrix [aij] =

[aij(x,u,∇u)] is assumed to be symmetric and elliptic in the sense

0 ≤ φ(x)|ξ|p ≤
(
aijξiξj

)p
2 ≤ ψ(x)|ξ|p ∀ξ ∈ RN, (2)

with φ 6≡ 0 and ψ ∈ L1(D). The functions φ and ψ are subject to the following condition: there
exist two constants C > 0 and α ∈ (p−1

p
, 1) such that, for each Borel set E ⊂ D with smooth



CUBO
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Estimates for solutions to nonlinear degenerate elliptic equations 61

boundary ∂E we have ∫

∂E

φ(x)dσ ≥ C
(∫

E

ψ(x)dx

)α
. (3)

Note that (1) is the Euler equation of the functional

∫

D

[(
aijuxiuxj

)p
2 − p u f

]
dx.

We consider solutions u ∈ W 1,p
0

(ψ,D), where W
1,p
0

(ψ,D) is the completion of C1
0
(D) with respect

to the norm

‖u‖ =
(∫

D

(
|u|p + ψ(x)|∇u|p

)
dx

)1
p

.

We shall use the following formula ( [7] pag. 37). If g(x) ≥ 0 is measurable in the sense of
Borel in an open set Ω and if u ∈ C0,1(Ω) then

∫

Ω

g(x)|∇u|dx =
∫ ∞

0

dτ

∫

Fτ

g(x)dσ, (4)

where

Fτ = {x ∈ Ω : |u(x)| = τ},

and dσ stands for the (N − 1)-Hausdorff measure. The equality (4) has been extended to functions
u ∈ W 1,1loc (D) (see for example [6], Theorem 1.1). A detailed theory of Sobolev spaces including
W

1,1
loc (D) can be found in [7].

If u ∈ W 1,p
0

(ψ,D) is a solution of equation (1) we shall denote

Dt = {x ∈ D : u(x) > t}.

By putting Ω = Dt with t > 0, equality (4) yields

∫

Dt

g(x)|∇u|dx =
∫

sup u

t

dτ

∫

Fτ

g(x)dσ, (5)

with

sup u = sup
x∈D

u(x), Fτ = {x ∈ D : u(x) = τ}.

We note that ∂Dt ⊂ Ft.

Lemma 2.1. If condition (3) holds and if φ(x) ≤ H(x) ≤ ψ(x) then for almost all t > 0 we have
∫

Ft

H(x)|∇u|p−1dσ ≥ Cp (V (t))
αp

(
−V ′(t)

)p−1 ,

where

V (t) =

∫

Dt

ψ(x)dx.



62 Fabrizio Cuccu, Petar Popivanov and Giovanni Porru CUBO
11, 4 (2009)

Proof. By using the Hölder inequality we find

(∫

Ft

H(x)dσ

)p
=

(∫

Ft

(
H(x)|∇u|p−1

)1
p

(
H(x)

|∇u|
)p−1

p

dσ

)p

≤
∫

Ft

H(x)|∇u|p−1dσ
(∫

Ft

H(x)

|∇u| dσ
)p−1

.

It follows that
∫

Ft

H(x)|∇u|p−1dσ ≥

(∫
Ft
φ(x)dσ

)p

(∫
Ft

ψ(x)
|∇u|

dσ

)p−1 . (6)

Equality (5) with g(x) =
ψ(x)
|∇u|

yields

V (t) =

∫
sup u

t

dτ

∫

Fτ

ψ(x)

|∇u|dσ.

Hence, for almost all t > 0 we have

V
′
(t) = −

∫

Ft

ψ(x)

|∇u|dσ.

By using the latter equation and condition (3), from inequality (6) we get the statement of the

lemma.

Theorem 2.2. Assume conditions (2) and (3). Let s > 1
p(1−α)

, and let fψ−1+
1
s ∈ Ls(D). If

u ∈ W 1,p
0

(ψ,D) is a solution of equation (1) then we have

‖u‖L∞(D) ≤
1

C
p

p−1

s(p − 1)
ps(1 − α) − 1‖fψ

1
s
−1‖

1
p−1

Ls(D)

(∫

D

ψ(x)dx

)ps(1−α)−1
s(p−1)

.

Proof. Putting

H(x) =

(
aijuxiuxj

)p
2

|∇u|p ,

condition (2) implies

φ(x) ≤ H(x) ≤ ψ(x).
According to the definition of weak solution of (1) we have that for every v ∈ W 1,p

0
(ψ,D)

∫

D

(
aijuxiuxj

)p−2
2
aijuxivxj dx =

∫

D

f(x)v(x)dx.

Since u ∈ W 1,p
0

(ψ,D), for t > 0 we can take v = (u(x) − t)+. We find
∫

Dt

(
aijuxiuxj

)p
2
dx =

∫

Dt

f(x)(u(x) − t)dx,

where

Dt = {x ∈ D : u(x) > t}.



CUBO
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Estimates for solutions to nonlinear degenerate elliptic equations 63

Recalling our definition of H(x), by the latter equation we have

∫

Dt

H(x)|∇u|pdx =
∫

Dt

f(x)dx

∫ u

t

dτ =

∫
sup u

t

dτ

∫

Dτ

f(x)dx. (7)

As it concerns the left hand side, we apply (5) with g(x) = H(x)|∇u|p−1. We find
∫

Dt

H(x)|∇u|pdx =
∫

sup u

t

dτ

∫

Fτ

H(x)|∇u|p−1dσ.

The latter equation and (7) yield

∫
sup u

t

dτ

∫

Fτ

H(x)|∇u|p−1dσ =
∫

sup u

t

dτ

∫

Dτ

f(x)dx.

After a differentiation, for almost all t we get
∫

Ft

H(x)|∇u|p−1dσ =
∫

Dt

f(x)dx ≤
∫

Dt

|f(x)|dx. (8)

Applying Lemma 2.1 we find

C
p (V (t))

αp

(
−V ′(t)

)p−1 ≤
∫

Dt

|f(x)|ψ
1−s

s ψ
s−1

s dx ≤ ‖fψ 1s −1‖Ls(D)
(∫

Dt

ψ(x)dx

)s−1
s

.

Rearranging we have

1 ≤ 1
C

p
p−1

‖fψ 1s −1‖
1

p−1

Ls(D)
(V (t))

s(1−pα)−1
s(p−1) (−V ′(t)).

Finally, integrating over (0, sup u) we get

sup u ≤ 1
C

p
p−1

s(p − 1)
ps(1 − α) − 1‖fψ

1
s
−1‖

1
p−1

Ls(D)
(V (0))

ps(1−α)−1
s(p−1) .

Being V (0) ≤
∫
D
ψ(x)dx we find

sup u ≤ 1
C

p
p−1

s(p − 1)
ps(1 − α) − 1‖fψ

1
s
−1‖

1
p−1

Ls(D)

(∫

D

ψ(x)dx

)ps(1−α)−1
s(p−1)

. (9)

If u is a solution of equation (1) then −u is a solution the same equation with −f in place of
f. Therefore, the estimate (9) also holds for −u. The theorem follows.

3 Special cases illustrating Theorem 2.2

We shall begin this section with the case when φ(x) = λ and ψ(x) = Λ. Then condition (3) holds

with

α = 1 − 1
N
, C =

λ

Λ
N−1

N

Nω
1
N

N ,



64 Fabrizio Cuccu, Petar Popivanov and Giovanni Porru CUBO
11, 4 (2009)

where ωN is the measure of the unit ball in R
N. For 1 < p < N, s > N

p
, Theorem 2.2 yields

‖u‖L∞(D) ≤
Λ

(
Nλω

1
N

N

) p
p−1

Ns(p − 1)
ps − N ‖f‖

1
p−1

Ls(D)
|D|

ps−N
N s(p−1) . (10)

However, in this special case we can prove an inequality sharper than (10). Indeed, since λ ≤ H(x),
by the inequality (8) we find

λ

∫

Ft

|∇u|p−1dσ ≤ ‖f‖Ls(D)|Dt|
s−1

s . (11)

Instead of (6) we use the inequality

∫

Ft

|∇u|p−1dσ ≥

(∫
Ft
dσ

)p

(∫
Ft

1

|∇u|
dσ

)p−1 .

Putting µ(t) = |Dt| we know that

µ
′
(t) = −

∫

Ft

1

|∇u|dσ.

Using this equality and the familiar isoperimetric inequality (see, for example, [2])

∫

Ft

dσ ≥ Nω
1
N

N (µ(t))
N−1

N

we find ∫

Ft

|∇u|p−1dσ ≥ N
p
ω

p
N

N (µ(t))
p(N−1)

N

(−µ′(t))p−1 .

Hence, by (11) we get

λN
p
ω

p
N

N (µ(t))
p(N−1)

N

(−µ′(t))p−1 ≤ ‖f‖L
s(D)(µ(t))

s−1
s ,

and

1 ≤ µ
−1+ 1

p−1
(

p
N

− 1
s
)
(t)

(
λNpω

p
N

N

) 1
p−1

(−µ′(t))‖f‖
1

p−1

Ls(D)
. (12)

Integration over (0, sup u) yields

sup u ≤ 1
(Nλ)

1
p−1 ω

p
N(p−1)

N

s(p − 1)
ps − N ‖f‖

1
p−1

Ls(D)
|D|

ps−N
N s(p−1) .

The same estimate can be found for −u. Therefore,

‖u‖L∞(D) ≤
1

(Nλ)
1

p−1 ω

p
N(p−1)

N

s(p − 1)
ps − N ‖f‖

1
p−1

Ls(D)
|D|

ps−N
N s(p−1) .

The latter inequality improves (10) by the factor λ/Λ.



CUBO
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Estimates for solutions to nonlinear degenerate elliptic equations 65

In case f is bounded in D then we can take s → ∞ and we find

‖u‖L∞(D) ≤
1

(Nλ)
1

p−1 ω

p
N(p−1)

N

p − 1
p

(sup
x∈D

|f(x)|)
1

p−1 |D|
p

N(p−1) . (13)

In the case λ = Λ, f is a constant and D is a ball, the inequality (13) is sharp. Indeed, if R

is the radius of the ball and f = A > 0, with u(x) = u(r) for |x| = r, the equation reads as

λ
(
r
N−1|u′|p−1

)′
= r

N−1
A.

Integrating we find

−u′ = r
1

p−1

(Nλ)
1

p−1

A
1

p−1 ,

u(r) =
(p − 1)A

1
p−1

p(Nλ)
1

p−1

[
R

p
p−1 − r

p
p−1
]
,

and

‖u‖L∞(D) = u(0) =
(p − 1)A

1
p−1

p(Nλ)
1

p−1

R
p

p−1 . (14)

Equation (14) yields (13) with the equality sign.

We shall consider now the case when aij = |x|βδij, β ≥ 0, δij being the Kronecker symbol.
In this case condition (2) holds with φ(x) = ψ(x) = |x| βp2 . If we look for condition (3) when E are
balls centered in the origin we find α = 1 − 1

N+β
. We think that this value of α is correct for all

Borel sets E, but we can prove this fact in case of N = 2 only (see the next section).

Let us show that the conclusion of Theorem 2.2 is generally false if fψ
1
s
−1 ∈ Ls(D) with

s =
N+β
p

and p ≥ 2.
We have

(
(aijuxiuxj )

p−2
2 aijuxi

)
xj

=
(
(|x|β|∇u|2)

p−2
2 |x|βuxi

)
xi
.

If u(x) is a radial function and u(x) = v(r) for |x| = r, we have uxi = v′ xir and

(
(|x|β|∇u|2)

p−2
2 |x|βuxi

)
xi

= r
1−N

(
r
N−1+

βp
2 |v′|p−2v′

)′
.

Consider problem (1) when Ω is a ball B centered in the origin and f = f(r) is a radial function.

Then the solution v is radial and satisfies the equation

−r1−N
(
r
N−1+

βp
2 |v′|p−2v′

)′
= f(r). (15)

Let B be the ball with radius 1/e. Consider the unbounded function

v(r) = log(− log r).



66 Fabrizio Cuccu, Petar Popivanov and Giovanni Porru CUBO
11, 4 (2009)

We find

v
′
=

1

r log r

and

−r1−N
(
r
N−1+

βp
2 |v′|p−2v′

)′
= r

1−N
(
r
N+

βp
2
−p

(− log r)1−p
)′

= r
βp
2
−p

(− log r)1−p
[
N +

βp

2
− p + (1 − p) 1

log r

]
.

Hence, our function v satisfies equation (15) with

f(r) = C(r)r
βp
2
−p

(− log r)1−p,

where C(r) is a bounded function for r < 1
e
.

If s = N+β
p

we have

(
f(r)ψ

(
1
s
−1)
)s

= C̃(r)r
βp
2
−N−β

(− log r)(1−p)
N+β

p .

If β > 0 it is easy to see that f(r)ψ(
1
s
−1) ∈ Ls(B) for s ≤ N+β

p
and p ≥ 2.

The same computations with β = 0 show that when f ∈ Ls(D) with s = N
p

and p > N
N−1

we

may have unbounded solutions of equation (1).

4 Proof of the isoperimetric inequality in dimension two and

for radially symmetric functions φ and ψ

We shall deal with φ(x) = λ|x|β, ψ(x) = Λ|x|γ, 0 < λ ≤ Λ, γ + 1 > β ≥ γ ≥ 0 in this section.
We use polar coordinates (ρ,θ). If E is a given set, we define a new set E′ according to the

following rule. For every ρ > 0, if Fρ = {x ∈ R2 : |x| = ρ} we replace E ∩ Fρ with the arc with
radius ρ, having the same 1-dimensional measure as E ∩ Fρ, centered in (ρ, 0). The sets E′ are
situated symmetrically with respect to θ = 0. We have

∫

E

|x|γdx =
∫

E′
|x|γdx.

Indeed, if we integrate from ρ to ρ + dρ we find ργldρ, where l is the 1-dimensional measure of

E ∩ Fρ (which is equal to the 1-dimensional measure of E′ ∩ Fρ). Moreover, we have
∫

∂E

|x|βds ≥
∫

∂E′
|x|βds′.

Indeed, if β = 0 this is the classical isoperimetric inequality (see, for example, [5]). If β > 0 we

can apply this inequality to the region of E enclosed between ρ and ρ + dρ. The boundary integral

of this elementary part of E is ρβ(ds + 2l) (l has the same meaning as before). Similarly, the



CUBO
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Estimates for solutions to nonlinear degenerate elliptic equations 67

boundary integral of the part of E′ enclosed between ρ and ρ + dρ is ρβ(ds′ + 2l). Hence, ds ≥ ds′
for each value of ρ. The inequality follows.

Therefore, in order to prove condition (3) we can confine ourselves to sets E having the

representation

E = {r ≤ ρ ≤ R, −h(ρ) ≤ θ ≤ h(ρ)}

with r ≥ 0. Of course, 0 ≤ h(ρ) ≤ π.
Consider first the case h(ρ) = π for 0 ≤ ρ ≤ R, that is the disc centered in the origin and of

radius R. We find
∫

∂E

λ|x|βds = 2πλRβ+1,
∫

E

Λ|x|γdx = 2πΛ
γ + 2

R
γ+2

.

Therefore, with

α =
β + 1

γ + 2
(20)

we have ∫
∂E

λ|x|βds
(∫

E
Λ|x|γdx

)α =
2πλ

(
2πΛ
γ+2

)α .

Let h(ρ) < π for 0 ≤ ρ ≤ R. Define the new set

Dτ = {0 ≤ ρ ≤ R, −τ ≤ θ ≤ τ},

with τ such that ∫

E

|x|γdx =
∫

Dτ

|x|γdx.

This value of τ ∈ (0,π) exists because Dπ is the disc with radius R and D0 is the segment (0,R),
E ⊂ Dπ but E 6= Dπ and

F(τ) =

∫

Dτ

|x|γdx

is a continuous monotonically increasing function for 0 < τ < π. Let us show that

∫

∂E

|x|βds ≥ 2
∫

L

|x|βds′ = 2
β + 1

R
β+1

, (21)

where L is the segment θ = τ, 0 ≤ ρ ≤ R. Indeed, if ds is the length of the part of the arc ∂E
between ρ and ρ + dρ, and if ds′ is the length of the part of the segment L between ρ and ρ + dρ,

we have ds ≥ 2ds′. We notice that the segment L is situated at θ = τ, whereas ∂E has one part
situated at θ ≥ 0, and the symmetric part situated at θ ≤ 0.

Easy computations yield ∫

Dτ

|x|γdx = 2τ
γ + 2

R
γ+2

.



68 Fabrizio Cuccu, Petar Popivanov and Giovanni Porru CUBO
11, 4 (2009)

Therefore, with α as in (20) we have

∫
∂E

λ|x|βds
(∫

E
Λ|x|γdx

)α ≥
2λ
β+1(
2τΛ
γ+2

)α >
2λ
β+1(
2πΛ
γ+2

)α .

Now we consider r > 0. If h(r) = π for r ≤ ρ ≤ R we can replace E by the ball h(r) = π for
0 ≤ ρ ≤ R. The integral of Λ|x|γ over the ball is greater than the integral over E, whereas, the
integral of λ|x|β over the boundary of the ball is smaller than the integral over ∂E. Hence, in this
situation we find ∫

∂E
λ|x|βds

(∫
E

Λ|x|γdx
)α ≥

2πλ
(

2πΛ
γ+2

)α .

Let h(r) < π for r ≤ ρ ≤ R. Define the set

Gτ = {r ≤ ρ ≤ R, −τ ≤ θ ≤ τ},

with τ such that ∫

E

|x|γdx =
∫

Gτ

|x|γdx.

We have 0 < τ < π. Arguing as in the proof of (21), now we find
∫

∂E

|x|βds ≥ 2
∫

L

|x|βds′ = 2
β + 1

(R
β+1 − rβ+1), (22)

where L is the segment θ = τ, r ≤ ρ ≤ R.
Let us show that we also have

∫

∂E

|x|βds ≥ 2
∫

Γ

|x|βds′ = 2τ rβ+1, (23)

where Γ is the arc ρ = r, 0 < θ < τ. Indeed, if ds is the length of the part of the arc ∂E between

θ and θ + dθ, θ > 0, and if ds′ is the length of the part of the arc Γ between θ and θ + dθ, we have

ds ≥ ds′. Recall that ∂E is symmetric with respect to θ = 0.
If we add (22) and (23) we get

∫

∂E

|x|βds ≥ 1
β + 1

(R
β+1 − rβ+1) + τ rβ+1.

On the other side, direct computation yields

∫

E

|x|γdx =
∫

Gτ

|x|γdx = 2τ
∫ R

r

ρ
γ+1

dρ =
2τ

γ + 2
(R

γ+2 − rγ+2).

Therefore, with α as in (20) we find

∫
∂E

λ|x|βds
(∫

E
Λ|x|γdx

)α ≥
λ
β+1

(R
β+1 − rβ+1) + τ λ rβ+1

(
2τΛ
γ+2

)α
(Rγ+2 − rγ+2)α

.



CUBO
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Estimates for solutions to nonlinear degenerate elliptic equations 69

If τ ≥ 1
β+1

we get

λ
β+1

(R
β+1 − rβ+1) + τ λ rβ+1

(
2τΛ
γ+2

)α
(Rγ+2 − rγ+2)α

≥
λ
β+1

R
β+1

(
2πΛ
γ+2

)α
(Rγ+2 − rγ+2)α

≥
λ
β+1(
2πΛ
γ+2

)α .

To discuss the case τ < 1
β+1

we consider the function

F(r,t) =
R
β+1 − rβ+1 + t rβ+1
(
t(Rγ+2 − rγ+2)

)α

for 0 < t < 1 and 0 < r < R. It is easy to see that ∂F
∂t

= 0 for

t =
α

1 − α
R
β+1 − rβ+1
rβ+1

.

With t ∈ (0, 1), the function F(r,t) is positive, strictly decreasing for t < t and strictly increasing
for t > t. Then two cases are considered.

a) 0 ≤ r ≤ Rα
1

β+1 . Then we have t ≥ 1. Therefore, recalling that α is given by equation (20) we
find

F(r,t) ≥ F(r, 1) = R
β+1

(
Rγ+2 − rγ+2

)α ≥ 1, ∀t ∈ (0, 1).

b) Rα
1

β+1 < r < R.

(i) From γ + 1 > β ≥ γ ≥ 0 we find

1

2
≤ γ + 1
γ + 2

≤ α = β + 1
γ + 2

< 1,

which implies 2α − 1 ≥ 0 and α(γ + 1) − β(1 − α) > 0.
(ii) Since α

rβ+1
<

1

Rβ+1
we have

F(r,t) ≥ F(r,t) = 1
1 − α

R
β+1 − rβ+1

(
α

1−α
Rβ+1−rβ+1

rβ+1
(Rγ+2 − rγ+2)

)α

≥ 1
1 − α

R
β+1 − rβ+1

(
1

1−α
Rβ+1−rβ+1

Rβ+1
(Rγ+2 − rγ+2)

)α

= (1 − α)α−1 (R
β+1 − rβ+1)1−αRα(β+1)

(Rγ+2 − rγ+2)α .

(iii) As we know from the theory of homogeneous functions,

R
β+1 − rβ+1 ≥ Cβ(R − r)(R + r)β



70 Fabrizio Cuccu, Petar Popivanov and Giovanni Porru CUBO
11, 4 (2009)

and

R
γ+2 − rγ+2 ≤ Cγ(R − r)(R + r)γ+1,

for suitable positive constants Cβ and Cγ. This implies

(R
β+1 − rβ+1)1−αRα(β+1)

(Rγ+2 − rγ+2)α ≥
C

1−α
β

Cαγ

R
α(β+1)

(R − r)2α−1(R + r)α(γ+1)−β(1−α)

≥
C

1−α
β

Cαγ

R
α(β+1)

R2α−1(2R)α(γ+1)−β(1−α)
=

C
1−α
β

Cαγ 2
α(γ+1)−β(1−α)

.

Therefore, for this kind of sets E there is Cβ,γ > 0 such that
∫
∂E

λ|x|βds
(∫

E
Λ|x|γdx

)α ≥ Cβ,γ
λ

Λα
.

Let h(ρ) = π for 0 ≤ ρ ≤ r and h(ρ) < π for r < ρ ≤ R. If there is a set Gτ = {r ≤ ρ ≤
R, −τ ≤ θ ≤ τ}, with τ < π such that

∫

E

|x|γdx = 2
∫

Gτ

|x|γdx,

then we can argue as in the previous case and find
∫
∂E

λ|x|βds
(∫

E
Λ|x|γdx

)α ≥ Cβ,γ
λ

(2Λ)α
.

Otherwise we must have ∫

E

|x|γdx ≥ 2
∫

Gπ

|x|γdx.

Since
∫
BR

|x|γdx ≥
∫
E
|x|γdx and Gπ = BR \ Br, this implies that

∫

BR

|x|γdx ≥ 2
∫

BR

|x|γdx − 2
∫

Br

|x|γdx,

from which we get (
r

R

)γ+2
≥ 1

2
. (24)

On the other side, since Br ⊂ E ⊂ BR we have
∫

E

|x|γdx ≤
∫

BR

|x|γdx,
∫

∂E

λ|x|βds ≥
∫

∂Br

λ|x|βds.

Hence, ∫
∂E

λ|x|βds
(∫

E
Λ|x|γdx

)α ≥
∫
∂Br

λ|x|βds
(∫

BR
Λ|x|γdx

)α =
2πλ
β+1(
2πΛ
γ+2

)α
(
r

R

)β+1
. (25)



CUBO
11, 4 (2009)

Estimates for solutions to nonlinear degenerate elliptic equations 71

Estimates (24) and (25) yield ∫
∂E

λ|x|βds
(∫

E
Λ|x|γdx

)α ≥
2πλ
β+1(
4πΛ
γ+2

)α .

The case E has the representation h(ρ) ≤ π for 0 ≤ ρ ≤ R, can be reduced to one of the
previous cases. Indeed, let b = sup{ρ : h(ρ) = π}. If b = R we replace E with the ball Ẽ with
radius R. If b < R we replace E with the set Ẽ having the representation h̃(ρ) = π for 0 ≤ ρ ≤ b,
and h̃(ρ) = h(ρ) for b < ρ ≤ R. The integral of Λ|x|γ over Ẽ is greater than the corresponding
integral over E, whereas, the integral of λ|x|β over ∂Ẽ is smaller than the corresponding integral
over ∂E.

The case E has the representation h(ρ) ≤ π for r ≤ ρ ≤ R can be treated similarly. If b is as
before and b = R we replace E with the ball Ẽ with radius R. If b < R we replace E with the set

Ẽ having the representation h̃(ρ) = π for 0 ≤ ρ ≤ b, and h̃(ρ) = h(ρ) for b < ρ ≤ R.
This way we have completed the proof of (3) in our special case.

Received: May 2008. Revised: September 2008.

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[2] Evans, L. C. and Gariepy, R. F., Measure theory and fine properties of functions, CRC

Press 1992.

[3] Federer, H., Curvature measures Transaction AMS, Vol. 93 (1959), 418–451.

[4] Gilbarg, D. and Trudinger, N. S., Elliptic Partial Differential Equations of Second Order,

Springer Verlag, Berlin, 1977.

[5] Kawohl, B., Rearrangements and Convexity of Level Sets in PDE, Lectures Notes in Math-

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[6] Malý, J. Swanson, D. and Ziemer, W. P., The coarea formula for Sobolev mappings,

arXiv:math.CA/0112008 v1, 1 Dec. 2001, 1–16.

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