Journal of Inequalities in Pure and
Applied Mathematics
SHARP CONSTANTS FOR SOME INEQUALITIES CONNECTED
TO THE P -LAPLACE OPERATOR
volume 6, issue 2, article 56,
2005.
JOHAN BYSTRÖM
Luleå University of Technology
SE-97187 Luleå, Sweden
Received 14 May, 2005;
accepted 19 May, 2005.
Communicated by: L.-E. Persson
EMail: johanb@math.ltu.se
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
158-05
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Abstract
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.
2000 Mathematics Subject Classification: 26D20, 35A05, 35J60.
Key words: p-Poisson, p-Laplace, Inequalities, Sharp constants, Structure conditions.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3
Some Auxiliary Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4
Proof of the Main Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
References
Sharp Constants for Some
Inequalities Connected to the
p-Laplace Operator
Johan Byström
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1.
Introduction
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
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 ,
Sharp Constants for Some
Inequalities Connected to the
p-Laplace Operator
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are
C1 = max 1, 22−p , (p − 1) 22−p ,
C2 = min 22−p , (p − 1) 22−p ,
see Figure 1.
2
Sharp Constants for Some
Inequalities Connected to the
p-Laplace Operator
1.5
Johan Byström
C1
1
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C2
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Figure 1: The constants C1 and C2 plotted for different values of p.
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2.
Main Results
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 ,
with equality if and only if

ξ1 = −ξ2 ,






∀ξ1 , ξ2 ∈ Rn ,



ξ1 = ξ2 ,





ξ1 = kξ2 , 1 ≤ k < ∞,




ξ1 = kξ2 when k → ∞,
Sharp Constants for Some
Inequalities Connected to the
p-Laplace Operator
Johan Byström
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for 1 < p < 2,
for p = 2,
for 2 < p < 3 and α = 1,
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for p = 3,
for 3 < p < ∞.
The constant C1 is sharp and given by
C1 = max 22−p , (p − 1) 22−p , 1 .
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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 < ∞.
Sharp Constants for Some
Inequalities Connected to the
p-Laplace Operator
Johan Byström
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The constant C2 is sharp and given by
2−p
C2 = min 2
Contents
2−p
, (p − 1) 2
.
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3.
Some Auxiliary Lemmas
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,
γ = hη1 , η2 i , − 1 ≤ γ ≤ 1,
η2 =
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.
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
Sharp Constants for Some
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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
,
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where −1 ≤ γ ≤ 1. Now construct
2
k 2(p−1) + 1 − 2k p−1 γ
(k p−1 − 1) + 2k p−1 (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
< 0.
p−2
p
) (k − 1) + (2 − p) 2k
p−1
(1 − γ) + (k
p−1
2
− 1)
(k 2 + 1 − 2kγ)p
Sharp Constants for Some
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p-Laplace Operator
Johan Byström
Hence we attain the maximum for f1 (k, γ) (and thus also for
the border γ = −1. We therefore examine
p
k p−1 + 1
.
g1 (k) = f1 (k, −1) =
(k + 1)p−1
p
f1 (k, γ)) on
We have
p−1
p−2
≤ 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
g10 (k) = −
c1 = g1 (1) = 22−p .
This constant is attained for k = 1 and γ = −1, that is, when ξ1 = −ξ2 .
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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
k p + 1 − (k p−2 + 1) kγ
(k + 1)p−2 (k 2 + 1 − 2kγ)
(k p−1 − 1) (k − 1) + (k p−1 + k) (1 − γ)
.
=
(k + 1)p−2 (k − 1)2 + 2k (1 − γ)
f2 (k, γ) =
Then
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f2 (k, γ) > 0.
Moreover,
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II
I
∂f2
k (1 − k p−2 ) (k 2 − 1)
=−
≤ 0,
∂γ
(k + 1)p−2 (k 2 + 1 − 2kγ)2
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with equality for k = 1. Hence we attain the minimum for f2 (k, γ) on the
border γ = 1. We therefore examine
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p−1
g2 (k) = f2 (k, 1) =
k
−1
.
(k − 1) (k + 1)p−2
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By Lemma 3.1 we have that
g20 (k) =
(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
k p−1 − 1
= (p − 1) 22−p .
p−2
k→1 (k − 1) (k + 1)
c2 = lim g2 (k) = lim
k→1
This constant is attained for k = 1 and γ = 1, that is, when ξ1 = ξ2 .
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 ,



ξ1 = kξ2 when 1 ≤ k < ∞,



ξ1 = kξ2 when k → ∞,
for 2 < p < 3,
for p = 3,
Sharp Constants for Some
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for 3 < p < ∞.
2−p
The constant c1 is sharp, where c1 = (p − 1) 2
3 ≤ p < ∞.
for 2 < p < 3 and c1 = 1 for
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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)
(k
p−1
(k + 1)
2
− 1) + 2k
2(p−2)
p−1
(1 − γ)
.
(k − 1) + 2k (1 − γ)
Sharp Constants for Some
Inequalities Connected to the
p-Laplace Operator
2
Johan Byström
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
p for k = 1. Hence we attain the maximum for f3 (k, γ) (and thus
also for f3 (k, γ)) on the border γ = 1. We therefore examine
g3 (k) =
p
f3 (k, 1) =
k p−1 − 1
.
(k − 1) (k + 1)p−2
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First we note that
g3 (k) ≡ 1
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when p = 3, implying that c1 = 1 with equality for all ξ1 = kξ2 , 1 ≤ k < ∞.
Moreover, we have that
g30
(3 − p) (1 − k p−1 ) + (p − 1) (k − k p−2 )
(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
p−1
k
−1
= (p − 1) 22−p , for 2 < p < 3.
k→1 (k − 1) (k + 1)p−2
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
p−1
k
−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.
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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)
Sharp Constants for Some
Inequalities Connected to the
p-Laplace Operator
Johan Byström
Then
f4 (k, γ) > 0.
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Moreover,
2k ((p − 2) A (k) + B (k)) A (k)
∂f4
=
> 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
g4 (k) =
p
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p−1
f4 (k, −1) =
k
+1
.
(k + 1)p−1
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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 .
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4.
Proof of the Main Theorems
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 ,
Sharp Constants for Some
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p-Laplace Operator
Johan Byström
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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 ,
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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.
Sharp Constants for Some
Inequalities Connected to the
p-Laplace Operator
Johan Byström
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References
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[3] J. BYSTRÖM, J. ENGSTRÖM AND P. WALL, Reiterated homogenization
of degenerate nonlinear elliptic equations, Chin. Ann. Math. Ser. B, 23(3)
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[4] V. CHIADÒ PIAT AND A. DEFRANCESCHI, Homogenization of quasilinear equations with natural growth terms, Manuscripta Math., 68(3)
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[5] G. DAL MASO AND A. DEFRANCESCHI, Correctors for the homogenization of monotone operators, Differential Integral Equations, 3(6)
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[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.
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[9] S. FUČIK AND A. KUFNER, Nonlinear Differential Equations, Elsevier
Scientific, New York, 1980.
[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.
Sharp Constants for Some
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p-Laplace Operator
Johan Byström
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