POINCARÉ TYPE INEQUALITIES FOR VARIABLE
EXPONENTS
Poincaré Type Inequalities
FUMI-YUKI MAEDA
4-24 Furue-higashi-machi, Nishiku
Hiroshima, 733-0872 Japan
EMail: fymaeda@h6.dion.ne.jp
Fumi-Yuki Maeda
vol. 9, iss. 3, art. 68, 2008
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Contents
Received:
04 March, 2008
Accepted:
04 August, 2008
Communicated by:
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B. Opić
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2000 AMS Sub. Class.:
26D10, 26D15.
Page 1 of 13
Key words:
Poincaré inequality, variable exponent.
Abstract:
We consider Poincaré type inequalities of integral form for variable exponents.
We give conditions under which these inequalities do not hold as well as conditions under which they hold.
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Contents
1
Introduction and preliminaries
3
2
Invalidity of Poincaré type inequalities
5
3
Validity of Poincaré Type Inequalities in One-Dimensional Case
7
Poincaré Type Inequalities
4
Validity of Poincaré Type Inequalities in Higher-Dimensional Case
11
Fumi-Yuki Maeda
vol. 9, iss. 3, art. 68, 2008
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1.
Introduction and preliminaries
One of the classical Poincaré inequalities states
Z
Z
p
|ϕ(x)| dx ≤ C(N, p, |G|) |∇ϕ(x)|p dx,
G
∀ϕ ∈ C01 (G),
G
where G is a bounded open set in RN (N ≥ 1) and p ≥ 1.
In Fu [2], this inequality with p replaced by a bounded variable exponent p(x) is
given as a lemma. Namely, let p(x) be a bounded measurable function on G such
that p(x) ≥ 1 for all x ∈ G. We shall say that the Poincaré inequality (PI, for short)
holds on G for p(·) if there exists a constant C > 0 such that
Z
Z
p(x)
(PI)
|ϕ(x)|
dx ≤ C
|∇ϕ(x)|p(x) dx
G
Poincaré Type Inequalities
Fumi-Yuki Maeda
vol. 9, iss. 3, art. 68, 2008
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G
for all ϕ ∈ C01 (G). Fu’s lemma asserts that (PI) always holds. However, as was
already remarked in [1, pp. 444-445, Example] in the one dimensional case, this is
false. We shall give some types of p(·) for which (PI) does not hold.
We remark here that the following norm-form of the Poincaré inequality holds
for variable exponents (cf. [3, Theorem 3.10]):
kϕkLp(·) (G) ≤ Ck|∇ϕ|kLp(·) (G)
for all ϕ ∈ C01 (G) provided that p(x) is continuous on G, where k · kLp(·) (G) denotes
the (Luxemburg) norm in the variable exponent Lebesgue space Lp(·) (G) (see [3] for
definition). Thus, our results show that we must distiguish between norm-form and
integral-form when we consider the Poincaré inequalities for variable exponents.
We also consider a slightly weaker form: we shall say that the weak Poincaré
inequality (wPI, for short) holds on G for p(·) if there exists a constant C > 0 such
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that
Z
p(x)
|ϕ(x)|
(wPI)
G
Z
dx ≤ C 1 +
p(x)
|∇ϕ(x)|
dx
G
for all ϕ ∈ C01 (G). We shall see that this weak Poincaré inequality does not always
hold either.
The main purpose of this paper is to give some sufficient conditions on p(·) under
which (PI) or (wPI) holds, and our results show that (PI) holds for a fairly large class
of non-constant p(x) and (wPI) holds for p(x) in a larger class.
Poincaré Type Inequalities
Fumi-Yuki Maeda
vol. 9, iss. 3, art. 68, 2008
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2.
Invalidity of Poincaré type inequalities
For a measurable function p(x) on G and E ⊂ G, let
−
p+
E = ess sup p(x) and pE = ess inf p(x).
x∈E
x∈E
Lemma 2.1. Let p(x) and q(x) be measurable functions on G such that 0 < p−
G ≤
−
+
p+
<
∞
and
0
<
q
≤
q
<
∞.
G
G
G
1. If there exist a compact set K and open sets G1 , G2 such that K ⊂ G1 b G2 ⊂
−
G, |K| > 0 and qK
> p+
, then there exists a sequence {ϕn } in C01 (G)
G2 \G1
such that
R
Z
|∇ϕn (x)|p(x) dx
q(x)
RG
|ϕn (x)| dx → ∞
and
→0
q(x) dx
|ϕ
(x)|
n
G
G
as n → ∞.
2. If there exist a compact set K and open sets G1 , G2 such that K ⊂ G1 b G2 ⊂
+
G, |K| > 0 and qK
< p−
, then there exists a sequence {ψn } in C01 (G)\{0}
G2 \G1
such that
R
Z
|∇ψn (x)|p(x) dx
RG
|∇ψn (x)|p(x) dx → 0
and
→0
|ψn (x)|q(x) dx
G
G
as n → ∞.
Proof. Choose ϕ1 ∈ C01 (G) such that ϕ1 = 1 on G1 and Spt ϕ1 ⊂ G2 .
−
−
(1) Suppose qK
> p+
. For simplicity, write q1 = qK
and p2 = p+
. Let
G2 \G1
G2 \G1
ϕn = nϕ1 , n = 1, 2, . . .. Then
Z
Z
Z
p(x)
p(x)
p(x)
p2
|∇ϕn |
dx =
n |∇ϕ1 |
dx ≤ n
|∇ϕ1 |p(x) dx
G
G2 \G1
G
Poincaré Type Inequalities
Fumi-Yuki Maeda
vol. 9, iss. 3, art. 68, 2008
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and
Z
q(x)
|ϕn |
Z
nq(x) dx ≥ nq1 |K|.
dx ≥
G
K
These inequalities show that the sequence {ϕn } has the required properties.
+
+
(2) Suppose qK
< p−
. Write q2 = qK
and p1 = p−
. Let ψn = (1/n)ϕ1 , n =
G2 \G1
G2 \G1
1, 2, . . .. Then
Z
Z
Z
−p(x)
p(x)
−p1
p(x)
n
|∇ϕ1 |
dx ≤ n
|∇ϕ1 |p(x) dx
|∇ψn |
dx =
G2 \G1
G
and
Z
q(x)
|ψn |
G
Z
dx ≥
G
n−q(x) dx ≥ n−q2 |K|.
Poincaré Type Inequalities
Fumi-Yuki Maeda
vol. 9, iss. 3, art. 68, 2008
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Thus the sequence {ψn } has the required properties.
By taking p(x) = q(x) in this lemma, we readily obtain
Proposition 2.2.
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1. If there exist a compact set K and open sets G1 , G2 such that K ⊂ G1 b G2 ⊂
+
G, |K| > 0 and p−
K > pG \G , then (wPI) does not hold for p(·) on G.
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2. If there exist a compact set K and open sets G1 , G2 such that K ⊂ G1 b G2 ⊂
−
G, |K| > 0 and p+
K < pG \G , then (PI) does not hold for p(·) on G.
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3.
Validity of Poincaré Type Inequalities in One-Dimensional Case
We shall say that f (t) on (t0 , t1 ) is of type (L) if there is τ ∈ (t0 , t1 ) such that f (t)
is non-increasing on (t0 , τ ) and non-decreasing on (τ, t1 ).
Proposition 3.1. Let N = 1 and G = (a, b).
1. If p(t) is monotone (i.e., non-decreasing or non-increasing) or of type (L) on
G, then
Z b
Z b
|G|
p+
p(t)
+ max(|G|, |G| )
|f 0 (t) |p(t) dt
|f (t)| dx ≤
2
a
a
2. If p(t) is monotone on G, then
Z b
Z b
p(t)
|f (t)| dx ≤ C
|f 0 (t) |p(t) dt
a
for f ∈ C01 (G), where the constant C depends only on p+ and |G|.
Proof. (I) First, we consider the case G = (0, 1). Let f ∈ C01 (G).
(I-1) Suppose p(t) is non-increasing on (0, τ ), 0 < τ ≤ 1. Then, for 0 < t < τ ,
p(t)
|f (t)|
Z
≤
t
0
|f (s)| ds
0
Z
≤
0
t
p(t)
Z
≤
Fumi-Yuki Maeda
vol. 9, iss. 3, art. 68, 2008
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for f ∈ C01 (G), where |G| = b − a and p+ = p+
G.
a
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t
0
p(t)
|f (t) | ds
Z 1
0
p(s)
ds ≤ t +
1 + |f (s) |
|f 0 (s) |p(s) ds.
0
0
Hence
τ
Z 1
τ2
|f (t)| dt ≤
+τ
|f 0 (s) |p(s) ds.
2
0
0
Similarly, if p(t) is non-decreasing on (τ, 1), 0 ≤ τ < 1, then
Z 1
Z 1
(1 − τ )2
p(t)
|f (t)| dt ≤
+ (1 − τ )
|f 0 (s) |p(s) ds.
2
0
τ
Z
p(t)
Hence, if p(t) is monotone or of type (L) on G, then
Z 1
Z 1
1
p(t)
(3.1)
|f 0 (t) |p(t) dt.
|f (t)| dt ≤ +
2
0
0
R
1
(I-2) The case kf 0 k1 := 0 |f 0 (t)| dt ≥ 1.
In this case,
Z
Z 1
1 1
0
1≤
|f (t)| dt =
|2f 0 (t)| dt
2 0
0
Z
Z 1
1 1 1
1
0
p(t)
p+ −1
|2f (t) | dt ≤ + 2
|f 0 (t) |p(t) dt,
≤ +
2 2 0
2
0
so that
1
+
≤ 2p −1
2
Z
1
|f 0 (t) |p(t) dt.
0
0
in case kf k1 ≥ 1.
Fumi-Yuki Maeda
vol. 9, iss. 3, art. 68, 2008
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Hence, by (3.1), we have
Z 1
Z
p(t)
p+ −1
(3.2)
|f (t)| dt ≤ (1 + 2
)
0
Poincaré Type Inequalities
0
1
|f 0 (t) |p(t) dt
(I-3) The case p(t) is monotone and kf 0 k1 < 1.
We may assume that p(t) is non-decreasing. Set
E1 = {t ∈ (0, 1); |f 0 (t)| ≤ 1}, E2 = {t ∈ (0, 1); |f 0 (t)| > 1},
Z
Z
0
g1 (t) =
|f (s)| ds and g2 (t) =
|f 0 (s)| ds.
(0,t)∩E1
(0,t)∩E2
Poincaré Type Inequalities
Then for 0 < t < 1
Fumi-Yuki Maeda
|f (t)|p(t)
Z
p(t)
t
p(t)
≤
|f 0 (s)| ds
= g1 (t) + g2 (t)
0
p+ −1
≤2
g1 (t)p(t) + g2 (t)p(t) .
Since p(s) ≤ p(t) for 0 < s < t and |f (s)| ≤ 1 for s ∈ E1 ,
Z
Z
Z
p(t)
0
p(t)
0
p(s)
g1 (t) ≤
|f (s) | ds ≤
|f (s) | ds ≤
(0,t)∩E1
(0,t)∩E1
|f 0 (s) |p(s) ds.
On the other hand, since g2 (t) ≤ kf 0 k1 < 1 and |f 0 (s)| > 1 for s ∈ E2 ,
Z
Z
p(t)
0
g2 (t) ≤ g2 (t) =
|f (s)| ds ≤
|f 0 (s) |p(s) ds.
E2
Hence
p(t)
|f (t)|
p+ −1
Z
≤2
1
|f 0 (s) |p(s) ds
0
for all 0 < t < 1, and hence
Z
Z 1
p(t)
p+ −1
|f (t)| dt ≤ 2
0
0
1
|f 0 (s) |p(s) ds
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E1
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in case kf 0 k1 < 1.
(I-4) Combining (I-2) and (I-3), we have (3.2) for all f ∈ C01 (G) if p(t) is monotone.
(II) The general case: Let G = (a, b) and f ∈ C01 (G). Let
g(t) = f (a + t(b − a)) and q(t) = p(a + t(b − a))
Poincaré Type Inequalities
for 0 < t < 1. Then
Fumi-Yuki Maeda
Z
b
|f (s)|p(s) ds = (b − a)
a
Z
1
|g(t)|q(t) dt
vol. 9, iss. 3, art. 68, 2008
0
and
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Z
0
1
1
|g (t)| dt =
b−a
0
Z
b
Contents
p(s)
|(b − a)f 0 (s)|
ds
a
Z b
p+ −1
≤ max(1, (b − a)
)
|f 0 (s) |p(s) ds.
a
Hence, applying (3.1) and (3.2) to g(t) and q(t), we obtain the required inequalities
+
+
of the proposition. (In fact, we can take C = (1 + 2p −1 ) max(|G|, |G|p ).)
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4.
Validity of Poincaré Type Inequalities in Higher-Dimensional
Case
Theorem 4.1. Let N ≥ 2 and G ⊂ G0 × (a, b) with a bounded open set G0 ⊂ RN −1
and set Gx0 = {t ∈ (a, b) : (x0 , t) ∈ G} for x0 ∈ G0 .
1. If t 7→ p(x0 , t) is monotone or of type (L) on each component of Gx0 for a.e.
x0 ∈ G0 (with respect to the (N − 1)-dimensional Lebesgue measure), then
(wPI) holds for p(·) on G.
2. If t 7→ p(x0 , t) is monotone on each component of Gx0 for a.e. x0 ∈ G0 (with
respect to the (N − 1)-dimensional Lebesgue measure), then (PI) holds for p(·)
on G.
Proof. Fix x0 ∈ G0 for a moment and let Ij be the components of Gx0 . If ϕ ∈
C01 (G), then t 7→ ϕ(x0 , t) belongs to C01 (Ij ) for each j. Thus, by Proposition 3.1, if
t 7→ p(x0 , t) is monotone or of type (L) on each Ij , then
Z
Z
p(x0 ,t)
p(x0 ,t)
0
p+
|ϕ(x , t)|
dt ≤ |Ij | + max(1, |Ij | )
|∇ϕ(x0 , t)|
dt,
Ij
Poincaré Type Inequalities
Fumi-Yuki Maeda
vol. 9, iss. 3, art. 68, 2008
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so that
Z
p(x0 ,t)
0
|ϕ(x , t)|
p+
Z
dt ≤ |Gx0 | + max(1, (b − a) )
Gx0
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0
|∇ϕ(x , t)|
Gx0
and if t 7→ p(x0 , t) is monotone on each Ij then
Z
Z
p(x0 ,t)
p(x0 ,t)
0
+
|ϕ(x , t)|
dt ≤ C(p , Ij )
|∇ϕ(x0 , t)|
dt,
Ij
Ij
p(x0 ,t)
dt;
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so that
Z
0
p(x0 ,t)
|ϕ(x , t)|
Gx0
+
Z
dt ≤ C(p , b − a)
p(x0 ,t)
|∇ϕ(x0 , t)|
dt.
Gx0
Hence, integrating over G0 with respect to x0 , we obtain the assertion of the theorem.
The following proposition is easily seen by a change of variables:
Poincaré Type Inequalities
Fumi-Yuki Maeda
Proposition 4.2. (PI) and (wPI) are diffeomorphically invariant. More precisely,
let G1 and G2 be bounded open sets and Φ(x) = (φ1 (x), . . . , φN (x)) be a (C 1 )diffeomorphism of G1 onto G2 . Suppose |∇φj |, j = 1, . . . , N and |∇ψj |, j =
1, . . . , N are all bounded, where Φ−1 (y) = (ψ1 (y), . . . , ψN (y)), and suppose 0 <
α ≤ JΦ (x) ≤ β for all x ∈ G1 . Let p1 (x) = p2 (Φ(x)) for x ∈ G1 . Then, (PI) (resp.
(wPI)) holds for p1 (·) on G1 if and only if it holds for p2 (·) on G2 .
Combining Theorem 4.1 with this Proposition, we can find a fairly large class of
p(x) for which (PI) (as well as (wPI)) holds.
vol. 9, iss. 3, art. 68, 2008
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References
[1] X. FAN AND D. ZHAO, On the spaces Lp(x) (Ω) and W m,p(x) (Ω), J. Math. Anal.
Appl., 263 (2001), 424–446.
[2] Y. FU, The existence of solutions for elliptic systems with nonuniform growth,
Studia Math., 151 (2002), 227–246.
[3] O. KOVÁČIK AND J. RÁKOSNÍK, On spaces Lp(x) and W k,p(x) , Czechoslovak
Math. J., 41 (1991), 592–618.
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Fumi-Yuki Maeda
vol. 9, iss. 3, art. 68, 2008
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