Volume 9 (2008), Issue 3, Article 68, 5 pp.
POINCARÉ TYPE INEQUALITIES FOR VARIABLE EXPONENTS
FUMI-YUKI MAEDA
4-24 F URUE - HIGASHI - MACHI , N ISHIKU
H IROSHIMA , 733-0872 JAPAN
fymaeda@h6.dion.ne.jp
Received 04 March, 2008; accepted 04 August, 2008
Communicated by B. Opić
A BSTRACT. 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.
Key words and phrases: Poincaré inequality, variable exponent.
2000 Mathematics Subject Classification. 26D10, 26D15.
1. I NTRODUCTION 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
N
where G is a bounded open set in R (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
G
C01 (G).
for all ϕ ∈
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
067-08
2
F UMI -Y UKI M AEDA
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 that
Z
Z
p(x)
p(x)
(wPI)
|ϕ(x)|
dx ≤ C 1 +
|∇ϕ(x)|
dx
G
G
C01 (G).
for all ϕ ∈
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.
2. I NVALIDITY OF P OINCARÉ
TYPE INEQUALITIES
For a measurable function p(x) on G and E ⊂ G, let
p+
E = ess sup p(x) and
x∈E
p−
E = ess inf p(x).
x∈E
+
Lemma 2.1. Let p(x) and q(x) be measurable functions on G such that 0 < p−
G ≤ pG < ∞
−
+
and 0 < qG
≤ qG
< ∞.
(1) If there exist a compact set K and open sets G1 , G2 such that K ⊂ G1 b G2 ⊂ G,
−
, then there exists a sequence {ϕn } in C01 (G) such that
|K| > 0 and qK
> p+
G2 \G1
R
Z
|∇ϕn (x)|p(x) dx
q(x)
G
R
|ϕn (x)| dx → ∞
and
→0
|ϕn (x)|q(x) dx
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} such that
G2 \G1
R
Z
|∇ψn (x)|p(x) dx
p(x)
G
R
|∇ψn (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 ϕn = nϕ1 , n =
G2 \G1
G2 \G1
1, 2, . . .. Then
Z
Z
Z
p(x)
p(x)
p(x)
p2
|∇ϕn |
dx =
n |∇ϕ1 |
dx ≤ n
|∇ϕ1 |p(x) dx
G2 \G1
G
and
Z
q(x)
|ϕn |
G
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 = 1, 2, . . ..
G2 \G1
G2 \G1
Then
Z
Z
Z
p(x)
−p(x)
p(x)
−p1
|∇ψn |
dx =
n
|∇ϕ1 |
dx ≤ n
|∇ϕ1 |p(x) dx
G2 \G1
G
and
Z
q(x)
|ψn |
G
Z
dx ≥
G
n−q(x) dx ≥ n−q2 |K|.
K
Thus the sequence {ψn } has the required properties.
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 68, 5 pp.
http://jipam.vu.edu.au/
P OINCARÉ T YPE I NEQUALITIES
3
By taking p(x) = q(x) in this lemma, we readily obtain
Proposition 2.2.
(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 > pG2 \G1 , then (wPI) does not hold for p(·) on G.
(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.
2
1
3. VALIDITY OF P OINCARÉ T YPE I NEQUALITIES IN O NE -D IMENSIONAL C ASE
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(t)
p+
|f (t)| dx ≤
+ max(|G|, |G| )
|f 0 (t) |p(t) dt
2
a
a
for f ∈ C01 (G), where |G| = b − a and p+ = p+
G.
(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),
a
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 ,
Z t
p(t) Z t
p(t)
0
|f (t)| ≤
|f (s)| ds
≤
|f 0 (t) |p(t) ds
0
0
Z t
Z 1
0
p(s)
≤
1 + |f (s) |
ds ≤ t +
|f 0 (s) |p(s) ds.
0
Hence
0
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
τ
0
Z
τ
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 (t)| dt ≤ +
|f 0 (t) |p(t) dt.
2
0
0
R1 0
0
(I-2) The case kf k1 := 0 |f (t)| dt ≥ 1.
In this case,
Z 1
Z
1 1
0
1≤
|f (t)| dt =
|2f 0 (t)| dt
2 0
0
Z 1
Z 1
1 1
1
0
p(t)
p+ −1
≤ +
|2f (t) | dt ≤ + 2
|f 0 (t) |p(t) dt,
2 2 0
2
0
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 68, 5 pp.
http://jipam.vu.edu.au/
4
F UMI -Y UKI M AEDA
so that
1
+
≤ 2p −1
2
Hence, by (3.1), we have
Z
(3.2)
1
Z
|f 0 (t) |p(t) dt.
0
1
p(t)
|f (t)|
dt ≤ (1 + 2
p+ −1
Z
)
0
1
|f 0 (t) |p(t) dt
0
0
in case kf k1 ≥ 1.
(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
Then for 0 < t < 1
p(t)
p(t)
≤
|f (s)| ds
= g1 (t) + g2 (t)
0
p+ −1
≤2
g1 (t)p(t) + g2 (t)p(t) .
t
Z
p(t)
|f (t)|
0
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.
E1
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.
(0,t)∩E2
E2
Hence
p(t)
|f (t)|
p+ −1
Z
1
≤2
|f 0 (s) |p(s) ds
0
for all 0 < t < 1, and hence
Z
1
p(t)
|f (t)|
p+ −1
1
Z
|f 0 (s) |p(s) ds
dt ≤ 2
0
0
0
in case kf 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))
for 0 < t < 1. Then
Z
b
p(s)
|f (s)|
Z
ds = (b − a)
a
1
|g(t)|q(t) dt
0
and
Z
0
1
1
|g (t)| dt =
b−a
0
Z
b
p(s)
|(b − a)f 0 (s)|
ds
a
Z b
p+ −1
≤ max(1, (b − a)
)
|f 0 (s) |p(s) ds.
a
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 68, 5 pp.
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P OINCARÉ T YPE I NEQUALITIES
5
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 ).)
4. VALIDITY OF P OINCARÉ T YPE I NEQUALITIES IN H IGHER -D IMENSIONAL C ASE
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
Ij
so that
Z
p(x0 ,t)
0
|ϕ(x , t)|
p+
Z
dt ≤ |Gx0 | + max(1, (b − a) )
Gx0
p(x0 ,t)
|∇ϕ(x0 , t)|
dt;
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
so that
Z
Ij
0
p(x0 ,t)
|ϕ(x , t)|
+
Z
dt ≤ C(p , b − a)
Gx0
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:
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.
R EFERENCES
[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.
J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 68, 5 pp.
http://jipam.vu.edu.au/