Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 6, Issue 2, Article 56, 2005
SHARP CONSTANTS FOR SOME INEQUALITIES CONNECTED TO THE
p-LAPLACE OPERATOR
JOHAN BYSTRÖM
L ULEÅ U NIVERSITY OF T ECHNOLOGY
SE-97187 L ULEÅ , S WEDEN
johanb@math.ltu.se
Received 14 May, 2005; accepted 19 May, 2005
Communicated by L.-E. Persson
A BSTRACT. In this paper we investigate a set of structure conditions used in the existence theory
of differential equations. More specific, we find best constants for the corresponding inequalities
in the special case when the differential operator is the p-Laplace operator.
Key words and phrases: p-Poisson, p-Laplace, Inequalities, Sharp constants, Structure conditions.
2000 Mathematics Subject Classification. 26D20, 35A05, 35J60.
1. I NTRODUCTION
When dealing with certain nonlinear boundary value problems of the kind
(
− div (A (x, ∇u)) = f on Ω ⊂ Rn ,
u ∈ H01,p (Ω) , 1 < p < ∞,
it is common to assume that the function A : Ω × Rn → Rn satisfies suitable continuity and
monotonicity conditions in order to prove existence and uniqueness of solutions, see e.g. the
books [6], [9], [11] and [12]. For C1∗ and C2∗ finite and positive constants, a popular set of such
structure conditions are the following:
|A (x, ξ1 ) − A (x, ξ2 )| ≤ C1∗ (|ξ1 | + |ξ2 |)p−1−α |ξ1 − ξ2 |α ,
hA (x, ξ1 ) − A (x, ξ2 ) , ξ1 − ξ2 i ≥ C2∗ (|ξ1 | + |ξ2 |)p−β |ξ1 − ξ2 |β ,
where 0 ≤ α ≤ min (1, p − 1) and max (p, 2) ≤ β < ∞. See for instance the articles [1], [2],
[3], [4], [7], [8] and [10], where these conditions (or related variants) are used in the theory of
homogenization. It is well known that the corresponding function
A (x, ∇u) = |∇u|p−2 ∇u
ISSN (electronic): 1443-5756
c 2005 Victoria University. All rights reserved.
158-05
2
J OHAN B YSTRÖM
for the p-Poisson equation satisfies these conditions, see e.g. [12], but the best possible constants
C1∗ and C2∗ are in general not known. In this article we prove that the best constants C1 and C2
for the inequalities
p−2
|ξ1 | ξ1 − |ξ2 |p−2 ξ2 ≤ C1 (|ξ1 | + |ξ2 |)p−1−α |ξ1 − ξ2 |α ,
p−2
|ξ1 | ξ1 − |ξ2 |p−2 ξ2 , ξ1 − ξ2 ≥ C2 (|ξ1 | + |ξ2 |)p−β |ξ1 − ξ2 |β ,
are
C1 = max 1, 22−p , (p − 1) 22−p ,
C2 = min 22−p , (p − 1) 22−p ,
see Figure 1.1.
2
1.5
C1
1
C2
0.5
0
1
2
3
4
p
5
6
7
8
Figure 1.1: The constants C1 and C2 plotted for different values of p.
2. M AIN R ESULTS
Let h·, ·i denote the Euclidean scalar product on Rn and let p be a real constant, 1 < p < ∞.
Moreover, we will assume that |ξ1 | ≥ |ξ2 | > 0, which poses no restriction due to symmetry
reasons. The main results of this paper are collected in the following two theorems:
Theorem 2.1. Let ξ1 , ξ2 ∈ Rn and assume that the constant α satisfies
0 ≤ α ≤ min (1, p − 1) .
Then it holds that
p−2
|ξ1 | ξ1 − |ξ2 |p−2 ξ2 ≤ C1 (|ξ1 | + |ξ2 |)p−1−α |ξ1 − ξ2 |α ,
J. Inequal. Pure and Appl. Math., 6(2) Art. 56, 2005
http://jipam.vu.edu.au/
S HARP C ONSTANTS FOR S OME I NEQUALITIES C ONNECTED TO THE p-L APLACE O PERATOR
with equality if and only if

ξ1 = −ξ2 ,






∀ξ1 , ξ2 ∈ Rn ,



ξ1 = ξ2 ,




ξ1 = kξ2 , 1 ≤ k < ∞,





ξ1 = kξ2 when k → ∞,
3
for 1 < p < 2,
for p = 2,
for 2 < p < 3 and α = 1,
for p = 3,
for 3 < p < ∞.
The constant C1 is sharp and given by
C1 = max 22−p , (p − 1) 22−p , 1 .
Theorem 2.2. Let ξ1 , ξ2 ∈ Rn and assume that the constant β satisfies
max (p, 2) ≤ β < ∞.
Then it holds that
p−2
|ξ1 | ξ1 − |ξ2 |p−2 ξ2 , ξ1 − ξ2 ≥ C2 (|ξ1 | + |ξ2 |)p−β |ξ1 − ξ2 |β ,
with equality if and only if

ξ1 = ξ2 ,
for 1 < p < 2 and β = 2,



∀ξ1 , ξ2 ∈ Rn , for p = 2,



ξ1 = −ξ2 ,
for 2 < p < ∞.
The constant C2 is sharp and given by
C2 = min 22−p , (p − 1) 22−p .
3. S OME AUXILIARY L EMMAS
In this section we will prove the four inequalities
p−2
|ξ1 | ξ1 − |ξ2 |p−2 ξ2 ≤ c1 |ξ1 − ξ2 |p−1 , 1 < p ≤ 2,
p−2
|ξ1 | ξ1 − |ξ2 |p−2 ξ2 , ξ1 − ξ2 ≥ c2 (|ξ1 | + |ξ2 |)p−2 |ξ1 − ξ2 |2 , 1 < p ≤ 2,
p−2
|ξ1 | ξ1 − |ξ2 |p−2 ξ2 ≤ c1 (|ξ1 | + |ξ2 |)p−2 |ξ1 − ξ2 | , 2 ≤ p < ∞,
p−2
|ξ1 | ξ1 − |ξ2 |p−2 ξ2 , ξ1 − ξ2 ≥ c2 |ξ1 − ξ2 |p , 2 ≤ p < ∞.
Note that, by symmetry, we can assume that |ξ1 | ≥ |ξ2 | > 0. By putting
η1 =
ξ1
,
|ξ1 |
|η1 | = 1,
η2 =
γ = hη1 , η2 i , − 1 ≤ γ ≤ 1, k =
ξ2
,
|ξ2 |
|ξ1 |
|ξ2 |
|η2 | = 1,
≥ 1,
we see that the four inequalities above are in turn equivalent with
p−1
k η1 − η2 ≤ c1 |kη1 − η2 |p−1 , 1 < p ≤ 2,
(3.1)
p−1
(3.2)
k η1 − η2 , kη1 − η2 ≥ c2 (k + 1)p−2 |kη1 − η2 |2 , 1 < p ≤ 2,
p−1
k η1 − η2 ≤ c1 (k + 1)p−2 |kη1 − η2 | , 2 ≤ p < ∞,
(3.3)
p−1
(3.4)
k η1 − η2 , kη1 − η2 ≥ c2 |kη1 − η2 |p , 2 ≤ p < ∞.
Before proving these inequalities, we need one lemma.
J. Inequal. Pure and Appl. Math., 6(2) Art. 56, 2005
http://jipam.vu.edu.au/
4
J OHAN B YSTRÖM
Lemma 3.1. Let k ≥ 1 and p > 1. Then the function
h (k) = (3 − p) 1 − k p−1 + (p − 1) k − k p−2
satisfies h (1) = 0. When k > 1, h (k) is positive and strictly increasing for p ∈ (1, 2) ∪ (3, ∞) ,
and negative and strictly decreasing for p ∈ (2, 3) . Moreover, h (k) ≡ 0 for p = 2 or p = 3.
Proof. We easily see that h (1) = 0. Two differentiations yield
h0 (k) = (p − 1) (p − 3) k p−2 + 1 − (p − 2) k p−3 ,
1
00
p−3
h (k) = (p − 1) (p − 2) (p − 3) k
1−
,
k
with h0 (1) = 0 and h00 (1) = 0. When p ∈ (1, 2) ∪ (3, ∞) , we have that h00 (k) > 0 for k > 1
which implies that h0 (k) > 0 for k > 1, which in turn implies that h (k) > 0 for k > 1. When
p ∈ (2, 3) , a similar reasoning gives that h0 (k) < 0 and h (k) < 0 for k > 1. Finally, the lemma
is proved by observing that h (k) ≡ 0 for p = 2 or p = 3.
Remark 3.2. The special case p = 2 is trivial, with equality (c1 = c2 ≡ 1) for all ξi ∈ Rn in all
four inequalities (3.1) – (3.4). Hence this case will be omitted in all the proofs below.
Lemma 3.3. Let 1 < p < 2 and ξ1 , ξ2 ∈ Rn . Then
p−2
|ξ1 | ξ1 − |ξ2 |p−2 ξ2 ≤ c1 |ξ1 − ξ2 |p−1 ,
with equality if and only if ξ1 = −ξ2 . The constant c1 = 22−p is sharp.
Proof. We want to prove (3.1) for k ≥ 1. By squaring and putting γ = hη1 , η2 i , we see that this
is equivalent with proving
p−1
k 2(p−1) + 1 − 2k p−1 γ ≤ c21 k 2 + 1 − 2kγ
,
where −1 ≤ γ ≤ 1. Now construct
2
(k p−1 − 1) + 2k p−1 (1 − γ)
k 2(p−1) + 1 − 2k p−1 γ
=
f1 (k, γ) =
p−1 .
(k 2 + 1 − 2kγ)p−1
(k − 1)2 + 2k (1 − γ)
Then
f1 (k, γ) < ∞.
Moreover,
2k (1 − k
∂f1
=−
∂γ
p−2
p
) (k − 1) + (2 − p) 2k
p−1
(1 − γ) + (k
p−1
p
(k 2 + 1 − 2kγ)
2
− 1)
< 0.
p
Hence we attain the maximum for f1 (k, γ) (and thus also for f1 (k, γ)) on the border γ = −1.
We therefore examine
p
k p−1 + 1
.
g1 (k) = f1 (k, −1) =
(k + 1)p−1
We have
p−1
p−2
g10 (k) = −
≤ 0,
p 1−k
(k + 1)
with equality if and only if k = 1. The smallest possible constant c1 for which inequality (3.1)
will always hold is the maximum value of g1 (k) , which is thus attained for k = 1. Hence
c1 = g1 (1) = 22−p .
This constant is attained for k = 1 and γ = −1, that is, when ξ1 = −ξ2 .
J. Inequal. Pure and Appl. Math., 6(2) Art. 56, 2005
http://jipam.vu.edu.au/
S HARP C ONSTANTS FOR S OME I NEQUALITIES C ONNECTED TO THE p-L APLACE O PERATOR
5
Lemma 3.4. Let 1 < p < 2 and ξ1 , ξ2 ∈ Rn . Then
p−2
|ξ1 | ξ1 − |ξ2 |p−2 ξ2 , ξ1 − ξ2 ≥ c2 (|ξ1 | + |ξ2 |)p−2 |ξ1 − ξ2 |2 ,
with equality if and only if ξ1 = ξ2 . The constant c2 = (p − 1) 22−p is sharp.
Proof. We want to prove (3.2) for k ≥ 1. By putting γ = hη1 , η2 i , we see that this is equivalent
with proving
k p + 1 − k p−2 + 1 kγ ≥ c2 (k + 1)p−2 k 2 + 1 − 2kγ ,
where −1 ≤ γ ≤ 1. Now construct
f2 (k, γ) =
(k p−1 − 1) (k − 1) + (k p−1 + k) (1 − γ)
k p + 1 − (k p−2 + 1) kγ
.
=
(k + 1)p−2 (k 2 + 1 − 2kγ)
(k + 1)p−2 (k − 1)2 + 2k (1 − γ)
Then
f2 (k, γ) > 0.
Moreover,
∂f2
k (1 − k p−2 ) (k 2 − 1)
=−
≤ 0,
∂γ
(k + 1)p−2 (k 2 + 1 − 2kγ)2
with equality for k = 1. Hence we attain the minimum for f2 (k, γ) on the border γ = 1. We
therefore examine
k p−1 − 1
g2 (k) = f2 (k, 1) =
.
(k − 1) (k + 1)p−2
By Lemma 3.1 we have that
(3 − p) (1 − k p−1 ) + (p − 1) (k − k p−2 )
≥ 0,
(k − 1)2 (k + 1)p−1
with equality if and only if k = 1. The largest possible constant c2 for which inequality (3.2)
always will hold is the minimum value of g2 (k) , which thus is attained for k = 1. Hence
g20 (k) =
k p−1 − 1
= (p − 1) 22−p .
k→1
k→1 (k − 1) (k + 1)p−2
This constant is attained for k = 1 and γ = 1, that is, when ξ1 = ξ2 .
c2 = lim g2 (k) = lim
Lemma 3.5. Let 2 < p < ∞ and ξ1 , ξ2 ∈ Rn . Then
p−2
|ξ1 | ξ1 − |ξ2 |p−2 ξ2 ≤ c1 (|ξ1 | + |ξ2 |)p−2 |ξ1 − ξ2 | ,
with equality if and only if

ξ1 = ξ2 ,
for 2 < p < 3,



ξ1 = kξ2 when 1 ≤ k < ∞, for p = 3,



ξ1 = kξ2 when k → ∞,
for 3 < p < ∞.
The constant c1 is sharp, where c1 = (p − 1) 22−p for 2 < p < 3 and c1 = 1 for 3 ≤ p < ∞.
Proof. We want to prove (3.3) for k ≥ 1. By squaring and putting γ = hη1 , η2 i , we see that this
is equivalent with proving
k 2(p−1) + 1 − 2k p−1 γ ≤ c21 (k + 1)2(p−2) k 2 + 1 − 2kγ ,
where −1 ≤ γ ≤ 1. Now construct
f3 (k, γ) =
k 2(p−1) + 1 − 2k p−1 γ
(k + 1)2(p−2) (k 2 + 1 − 2kγ)
J. Inequal. Pure and Appl. Math., 6(2) Art. 56, 2005
2
=
(k p−1 − 1) + 2k p−1 (1 − γ)
.
(k + 1)2(p−2) (k − 1)2 + 2k (1 − γ)
http://jipam.vu.edu.au/
6
J OHAN B YSTRÖM
Then
f3 (k, γ) < ∞.
Moreover,
∂f3
2k (k p−2 − 1) (k p − 1)
=
≥ 0,
∂γ
(k + 1)2(p−2) (k 2 + 1 − 2kγ)
with equality for k = 1. Hence we attain the maximum for f3 (k, γ) (and thus also for
on the border γ = 1. We therefore examine
p
k p−1 − 1
g3 (k) = f3 (k, 1) =
.
(k − 1) (k + 1)p−2
p
f3 (k, γ))
First we note that
g3 (k) ≡ 1
when p = 3, implying that c1 = 1 with equality for all ξ1 = kξ2 , 1 ≤ k < ∞. Moreover, we
have that
(3 − p) (1 − k p−1 ) + (p − 1) (k − k p−2 )
g30 (k) =
.
(k − 1)2 (k + 1)p−1
By Lemma 3.1 it follows that g3 (k) ≤ 0 for 2 < p < 3 with equality if and only if k = 1. The
smallest possible constant c1 for which inequality (3.3) will always hold is the maximum value
of g3 (k) , which thus is attained for k = 1. Hence
k p−1 − 1
= (p − 1) 22−p , for 2 < p < 3.
p−2
k→1 (k − 1) (k + 1)
c1 = lim g3 (k) = lim
k→1
This constant is attained for k = 1 and γ = 1, that is, when ξ1 = ξ2 .
Again using Lemma 3.1, we see that g3 (k) ≥ 0 for 3 < p < ∞, with equality if and only
if k = 1. The smallest possible constant c1 for which inequality (3.3) will always hold is the
maximum value of g3 (k) , which thus is attained when k → ∞. Hence
k p−1 − 1
= 1, for 3 < p < ∞.
k→∞ (k − 1) (k + 1)p−2
c1 = lim g3 (k) = lim
k→∞
This constant is attained when k → ∞ and γ = 1, that is, when ξ1 = kξ2 , k → ∞.
Lemma 3.6. Let 2 < p < ∞ and ξ1 , ξ2 ∈ Rn . Then
p−2
|ξ1 | ξ1 − |ξ2 |p−2 ξ2 , ξ1 − ξ2 ≥ c2 |ξ1 − ξ2 |p ,
with equality if and only if ξ1 = −ξ2 . The constant c2 = 22−p is sharp.
Proof. We want to prove (3.4) for k ≥ 1. By squaring and putting γ = hη1 , η2 i , we see that this
is equivalent to proving
2
p
k p + 1 − k p−2 + 1 kγ ≥ c22 k 2 + 1 − 2kγ ,
where −1 ≤ γ ≤ 1. Now construct
2
(k p + 1 − (k p−2 + 1) kγ)
f4 (k, γ) =
(k 2 + 1 − 2kγ)p
2
((k p−1 − 1) (k − 1) + (k p−1 + k) (1 − γ))
=
.
p
(k − 1)2 + 2k (1 − γ)
Then
f4 (k, γ) > 0.
J. Inequal. Pure and Appl. Math., 6(2) Art. 56, 2005
http://jipam.vu.edu.au/
S HARP C ONSTANTS FOR S OME I NEQUALITIES C ONNECTED TO THE p-L APLACE O PERATOR
7
Moreover,
∂f4
2k ((p − 2) A (k) + B (k)) A (k)
=
> 0,
∂γ
(k 2 + 1 − 2kγ)p+1
where
A (k) = k p−1 − 1 (k − 1) + k p−1 + k (1 − γ) ,
B (k) = k p−2 − 1 k 2 − 1 .
p
Hence we attain the minimum for f4 (k, γ) (and thus also for f4 (k, γ)) on the border γ = −1.
We therefore examine
p
k p−1 + 1
g4 (k) = f4 (k, −1) =
.
(k + 1)p−1
We have
g40 (k) =
(p − 1) (k p−2 − 1)
≥ 0,
(k + 1)p
with equality if and only if k = 1. The largest possible constant c2 for which inequality (3.4)
will always hold is the minimum value of g4 (k) , which thus is attained for k = 1. Hence
c2 = g4 (1) = 22−p .
This constant is attained for k = 1 and γ = −1, that is, when ξ1 = −ξ2 .
4. P ROOF OF THE M AIN T HEOREMS
Proof of Theorem 2.1. Let 1 < p < 2. Then the condition 0 ≤ α ≤ min (1, p − 1) = p − 1
implies that p − 1 − α ≥ 0. From Lemma 3.3 it follows that
p−2
|ξ1 | ξ1 − |ξ2 |p−2 ξ2 ≤ c1 |ξ1 − ξ2 |p−1−α |ξ1 − ξ2 |α
≤ c1 (|ξ1 | + |ξ2 |)p−1−α |ξ1 − ξ2 |α ,
with c1 = 22−p , and we have equality when ξ1 = −ξ2 . Now let 2 < p < ∞. Then 0 ≤ α ≤
min (1, p − 1) = 1 implies that 1 − α ≥ 0. From Lemma 3.5 it follows that
p−1−α
p−2
|ξ1 | ξ1 − |ξ2 |p−2 ξ2 ≤ c1 (|ξ1 | + |ξ2 |)
|ξ1 − ξ2 |
(|ξ1 | + |ξ2 |)1−α
≤ c1 (|ξ1 | + |ξ2 |)p−1−α |ξ1 − ξ2 |α ,
with
a) c1 = (p − 1) 22−p , equality for ξ1 = ξ2 when α = 1 for 2 < p < 3,
b) c1 = 1, equality for ξ1 = kξ2 when k → ∞ for 3 < p < ∞.
The case p = 2 is trivial and the case p = 3 has equality for ξ1 = kξ2 , 1 ≤ k < ∞, both
cases with constant c1 = 1. The theorem follows by taking these two inequalities together. Proof of Theorem 2.2. Let 1 < p < 2. Then the condition 2 = max (p, 2) ≤ β < ∞ implies
that β − 2 ≥ 0. From Lemma 3.4 it follows that
p−2
|ξ1 | ξ1 − |ξ2 |p−2 ξ2 , ξ1 − ξ2 ≥ c2 (|ξ1 | + |ξ2 |)p−β (|ξ1 | + |ξ2 |)β−2 |ξ1 − ξ2 |2
≥ c2 (|ξ1 | + |ξ2 |)p−β |ξ1 − ξ2 |β ,
J. Inequal. Pure and Appl. Math., 6(2) Art. 56, 2005
http://jipam.vu.edu.au/
8
J OHAN B YSTRÖM
with c2 = (p − 1) 22−p and equality for ξ1 = ξ2 when β = 2. Now let 2 < p < ∞. Then
p = max (p, 2) ≤ β < ∞ implies that β − p ≥ 0. From Lemma 3.6 it follows that
p−2
|ξ1 | ξ1 − |ξ2 |p−2 ξ2 , ξ1 − ξ2 ≥ c2 |ξ1 − ξ2 |p−β |ξ1 − ξ2 |β
≥ c2 (|ξ1 | + |ξ2 |)p−β |ξ1 − ξ2 |β ,
with c2 = 22−p and equality for ξ1 = −ξ2 . The case p = 2 is trivial, with constant c2 = 1. The
theorem is proven by taking these two inequalities together.
R EFERENCES
[1] A. BRAIDES, V. CHIADÒ PIAT AND A. DEFRANCESCHI, Homogenization of almost periodic
monotone operators, Ann. Inst. Henri Poincaré, 9(4) (1992), 399–432.
[2] J. BYSTRÖM, Correctors for some nonlinear monotone operators, J. Nonlinear Math. Phys., 8(1)
(2001), 8–30.
[3] J. BYSTRÖM, J. ENGSTRÖM AND P. WALL, Reiterated homogenization of degenerate nonlinear
elliptic equations, Chin. Ann. Math. Ser. B, 23(3) (2002), 325–334.
[4] V. CHIADÒ PIAT AND A. DEFRANCESCHI, Homogenization of quasi-linear equations with natural growth terms, Manuscripta Math., 68(3) (1990), 229–247.
[5] G. DAL MASO AND A. DEFRANCESCHI, Correctors for the homogenization of monotone operators, Differential Integral Equations, 3(6) (1990), 1151–1166.
[6] P. DRÁBEK, A. KUFNER AND F. NICOLOSI, Quasilinear Elliptic Equations with Degenerations
and Singularities, De Gruyter, Berlin, 1997.
[7] N. FUSCO AND G. MOSCARIELLO, On the homogenization of quasilinear divergence structure
operators, Ann. Mat. Pura. Appl., 146 (1987), 1–13.
[8] N. FUSCO AND G. MOSCARIELLO, Further results on the homogenization of quasilinear operators, Ricerche Mat., 35(2) (1986), 231–246.
[9] S. FUČIK
1980.
AND
A. KUFNER, Nonlinear Differential Equations, Elsevier Scientific, New York,
[10] J.L. LIONS, D. LUKKASSEN, L.-E. PERSSON AND P. WALL, Reiterated homogenization of
nonlinear monotone operators, Chin. Ann. Math. Ser. B, 22(1) (2001), 1–12.
[11] A. PANKOV, G-Convergence and Homogenization of Nonlinear Partial Differential Operators.
Kluwer, Dordrecht, 1997.
[12] E. ZEIDLER, Nonlinear Functional Analysis and its Applications II/B. Springer Verlag, Berlin,
1980.
J. Inequal. Pure and Appl. Math., 6(2) Art. 56, 2005
http://jipam.vu.edu.au/