POINCARÉ TYPE INEQUALITIES FOR VARIABLE EXPONENTS

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Poincaré Type Inequalities Fumi-Yuki Maeda vol. 9, iss. 3, art. 68, 2008

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POINCARÉ TYPE INEQUALITIES FOR VARIABLE EXPONENTS

FUMI-YUKI MAEDA

4-24 Furue-higashi-machi, Nishiku Hiroshima, 733-0872 Japan EMail:fymaeda@h6.dion.ne.jp

Received: 04 March, 2008

Accepted: 04 August, 2008

Communicated by: B. Opi´c 2000 AMS Sub. Class.: 26D10, 26D15.

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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1. Introduction and preliminaries

One of the classical Poincaré inequalities states Z

G

|ϕ(x)|^pdx≤C(N, p,|G|) Z

G

|∇ϕ(x)|^pdx, ∀ϕ∈C₀¹(G), whereGis a bounded open set inR^N (N ≥1) andp≥1.

In Fu [2], this inequality withpreplaced by a bounded variable exponent p(x)is given as a lemma. Namely, let p(x) be a bounded measurable function onG such thatp(x)≥1for allx∈ G. We shall say that the Poincaré inequality (PI, for short) holds onGforp(·)if there exists a constantC >0such that

(PI)

Z

G

|ϕ(x)|^p(x)dx≤C Z

G

|∇ϕ(x)|^p(x)dx

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 ofp(·)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]):

kϕk_L^p(·)_(G) ≤Ck|∇ϕ|k_L^p(·)_(G)

for allϕ∈C₀¹(G)provided thatp(x)is continuous onG, wherek · k_L^p(·)_(G)denotes the (Luxemburg) norm in the variable exponent Lebesgue spaceL^p(·)(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 onGforp(·)if there exists a constantC > 0such

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that (wPI)

Z

G

|ϕ(x)|^p(x)dx≤C

1 + Z

G

|∇ϕ(x)|^p(x)dx

for allϕ ∈ 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 onp(·)under which (PI) or (wPI) holds, and our results show that (PI) holds for a fairly large class of non-constantp(x)and (wPI) holds forp(x)in a larger class.

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2. Invalidity of Poincaré type inequalities

For a measurable functionp(x)onGandE ⊂G, let p⁺_E = ess sup

x∈E

p(x) and p⁻_E = ess inf

x∈E p(x).

Lemma 2.1. Letp(x)andq(x)be measurable functions onGsuch that0 < p⁻_G ≤ p⁺_G<∞and0< q_G⁻≤q⁺_G <∞.

1. If there exist a compact setKand open setsG₁,G₂such thatK ⊂G₁ bG₂ ⊂ G, |K| > 0and q_K⁻ > p⁺_G

2\G1 , then there exists a sequence {ϕn}in C₀¹(G) such that

Z

G

|ϕ_n(x)|^q(x)dx→ ∞ and R

G|∇ϕn(x)|^p(x)dx R

G|ϕ_n(x)|^q(x)dx →0 asn→ ∞.

2. If there exist a compact setKand open setsG₁,G₂such thatK ⊂G₁ bG₂ ⊂ G,|K|>0andq_K⁺ < p⁻

G2\G₁, then there exists a sequence{ψ_n}inC₀¹(G)\ {0}

such that Z

G

|∇ψ_n(x)|^p(x)dx→0 and R

G|∇ψ_n(x)|^p(x)dx R

G|ψ_n(x)|^q(x)dx →0 asn→ ∞.

Proof. Chooseϕ₁ ∈C₀¹(G)such thatϕ₁ = 1onG₁and Sptϕ₁ ⊂G₂. (1) Suppose q⁻_K > p⁺_G

2\G1. For simplicity, write q₁ = q_K⁻ and p₂ = p⁺_G

2\G1. Let ϕ_n =nϕ₁, n= 1,2, . . .. Then

Z

G

|∇ϕ_n|^p(x)dx= Z

G2\G1

n^p(x)|∇ϕ₁|^p(x)dx≤n^p² Z

G

|∇ϕ₁|^p(x)dx

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and Z

G

|ϕ_n|^q(x)dx≥ Z

K

n^q(x)dx≥n^q¹|K|.

These inequalities show that the sequence{ϕn}has the required properties.

(2) Supposeq⁺_K < p⁻

G2\G1. Writeq₂ =q_K⁺ andp₁ =p⁻

G2\G1. Letψ_n = (1/n)ϕ₁, n= 1,2, . . .. Then

Z

G

|∇ψ_n|^p(x)dx= Z

G2\G1

n^−p(x)|∇ϕ₁|^p(x)dx≤n^−p¹ Z

G

|∇ϕ₁|^p(x)dx

and Z

G

|ψ_n|^q(x)dx≥ Z

K

n^−q(x)dx≥n^−q²|K|.

Thus the sequence{ψ_n}has the required properties.

By takingp(x) =q(x)in this lemma, we readily obtain Proposition 2.2.

1. If there exist a compact setKand open setsG1,G2such thatK ⊂G1 bG2 ⊂ G,|K|>0andp⁻_K > p⁺_G

2\G1, then (wPI) does not hold forp(·)onG.

2. If there exist a compact setKand open setsG₁,G₂such thatK ⊂G₁ bG₂ ⊂ G,|K|>0andp⁺_K < p⁻_G

2\G1, then (PI) does not hold forp(·)onG.

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3. Validity of Poincaré Type Inequalities in One-Dimensional Case

We shall say thatf(t)on(t0, t1)is of type (L) if there isτ ∈(t0, t1)such thatf(t) is non-increasing on(t₀, τ)and non-decreasing on(τ, t₁).

Proposition 3.1. LetN = 1andG= (a, b).

1. If p(t)is monotone (i.e., non-decreasing or non-increasing) or of type (L) on G, then

Z b

a

|f(t)|^p(t)dx≤ |G|

2 + max(|G|, |G|^p⁺) Z b

a

|f⁰(t)|^p(t)dt forf ∈C₀¹(G), where|G|=b−aandp⁺=p⁺_G.

2. Ifp(t)is monotone onG, then Z b

a

|f(t)|^p(t)dx≤C Z b

a

|f⁰(t)|^p(t)dt

forf ∈C₀¹(G), where the constantCdepends only onp⁺and|G|.

Proof. (I) First, we consider the caseG= (0, 1). Letf ∈C₀¹(G).

(I-1) Supposep(t)is non-increasing on(0, τ),0< τ ≤1. Then, for0< t < τ,

|f(t)|^p(t) ≤ Z t

0

|f⁰(s)|ds p(t)

≤ Z t

0

|f⁰(t)|^p(t)ds

≤ Z t

0

1 +|f⁰(s)|^p(s)

ds≤t+ Z 1

0

|f⁰(s)|^p(s)ds.

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Hence

Z τ

0

|f(t)|^p(t)dt≤ τ² 2 +τ

Z 1

0

|f⁰(s)|^p(s)ds.

Similarly, ifp(t)is non-decreasing on(τ,1),0≤τ <1, then Z 1

τ

|f(t)|^p(t)dt ≤ (1−τ)²

2 + (1−τ) Z 1

0

|f⁰(s)|^p(s)ds.

Hence, ifp(t)is monotone or of type (L) onG, then (3.1)

Z 1

0

|f(t)|^p(t)dt ≤ 1 2 +

Z 1

0

|f⁰(t)|^p(t)dt.

(I-2) The casekf⁰k₁ :=R1

0 |f⁰(t)|dt ≥1.

In this case, 1≤

Z 1

0

|f⁰(t)|dt = 1 2

Z 1

0

|2f⁰(t)|dt

≤ 1 2+1

2 Z 1

0

|2f⁰(t)|^p(t)dt ≤ 1

2 + 2^p⁺⁻¹ Z 1

0

|f⁰(t)|^p(t)dt, so that

1

2 ≤2^p⁺⁻¹ Z 1

0

|f⁰(t)|^p(t)dt.

Hence, by (3.1), we have (3.2)

Z 1

0

|f(t)|^p(t)dt ≤(1 + 2^p⁺⁻¹) Z 1

0

|f⁰(t)|^p(t)dt in casekf⁰k1 ≥1.

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(I-3) The casep(t)is monotone andkf⁰k₁ <1.

We may assume thatp(t)is non-decreasing. Set

E₁ ={t ∈(0,1); |f⁰(t)| ≤1}, E₂ ={t∈(0,1); |f⁰(t)|>1}, g1(t) =

Z

(0,t)∩E₁

|f⁰(s)|ds and g2(t) = Z

(0,t)∩E₂

|f⁰(s)|ds.

Then for0< t <1

|f(t)|^p(t) ≤ Z t

0

|f⁰(s)|ds p(t)

= g₁(t) +g₂(t)p(t)

≤2^p⁺⁻¹ g₁(t)^p(t)+g₂(t)^p(t) . Sincep(s)≤p(t)for0< s < tand|f(s)| ≤1fors∈E₁,

g1(t)^p(t) ≤ Z

(0,t)∩E₁

|f⁰(s)|^p(t)ds≤ Z

(0,t)∩E₁

|f⁰(s)|^p(s)ds≤ Z

E1

|f⁰(s)|^p(s)ds.

On the other hand, sinceg2(t)≤ kf⁰k1 <1and|f⁰(s)|>1fors ∈E2, g₂(t)^p(t) ≤g₂(t) =

Z

(0,t)∩E2

|f⁰(s)|ds ≤ Z

E2

|f⁰(s)|^p(s)ds.

Hence

|f(t)|^p(t)≤2^p⁺⁻¹ Z 1

0

|f⁰(s)|^p(s)ds for all0< t <1, and hence

Z 1

0

|f(t)|^p(t)dt≤2^p⁺⁻¹ Z 1

0

|f⁰(s)|^p(s)ds

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in casekf⁰k₁ <1.

(I-4) Combining (I-2) and (I-3), we have (3.2) for allf ∈C₀¹(G)ifp(t)is monotone.

(II) The general case: LetG= (a, b)andf ∈C₀¹(G). Let

g(t) = f(a+t(b−a)) and q(t) =p(a+t(b−a)) for0< t <1. Then

Z b

a

|f(s)|^p(s)ds = (b−a) Z 1

0

|g(t)|^q(t)dt and

Z 1

0

|g⁰(t)|dt = 1 b−a

Z b

a

|(b−a)f⁰(s)|^p(s) ds

≤max(1,(b−a)^p⁺⁻¹) Z b

a

|f⁰(s)|^p(s)ds.

Hence, applying (3.1) and (3.2) tog(t)andq(t), we obtain the required inequalities of the proposition. (In fact, we can takeC = (1 + 2^p⁺⁻¹) max(|G|,|G|^p⁺).)

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4. Validity of Poincaré Type Inequalities in Higher-Dimensional Case

Theorem 4.1. LetN ≥2andG⊂G⁰×(a, b)with a bounded open setG⁰ ⊂R^N⁻¹ and setG_x⁰ ={t∈(a, b) : (x⁰, t)∈G}forx⁰ ∈G⁰.

1. If t 7→ p(x⁰, t) is monotone or of type (L) on each component of G_x⁰ for a.e.

x⁰ ∈ G⁰ (with respect to the (N −1)-dimensional Lebesgue measure), then (wPI) holds forp(·)onG.

2. If t 7→ p(x⁰, t)is monotone on each component of G_x⁰ for a.e. x⁰ ∈ G⁰ (with respect to the(N−1)-dimensional Lebesgue measure), then (PI) holds forp(·) onG.

Proof. Fix x⁰ ∈ G⁰ for a moment and let I_j be the components of G_x⁰. If ϕ ∈ C₀¹(G), thent 7→ ϕ(x⁰, t)belongs toC₀¹(Ij)for eachj. Thus, by Proposition3.1, if t7→p(x⁰, t)is monotone or of type (L) on eachIj, then

Z

Ij

|ϕ(x⁰, t)|^p(x⁰^,t) dt≤ |I_j|+ max(1,|I_j|^p⁺) Z

Ij

|∇ϕ(x⁰, t)|^p(x⁰^,t) dt, so that

Z

G_x0

|ϕ(x⁰, t)|^p(x⁰^,t) dt≤ |G_x⁰|+ max(1,(b−a)^p⁺) Z

G_x0

|∇ϕ(x⁰, t)|^p(x⁰^,t) dt;

and ift7→p(x⁰, t)is monotone on eachI_j then Z

Ij

|ϕ(x⁰, t)|^p(x⁰^,t) dt≤C(p⁺, I_j) Z

Ij

|∇ϕ(x⁰, t)|^p(x⁰^,t) dt,

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so that Z

G_x0

|ϕ(x⁰, t)|^p(x⁰^,t) dt≤C(p⁺, b−a) Z

G_x0

|∇ϕ(x⁰, t)|^p(x⁰^,t) dt.

Hence, integrating overG⁰with respect tox⁰, 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), . . . , φ_N(x)) be a (C¹- )diffeomorphism of G₁ onto G₂. Suppose |∇φ_j|, j = 1, . . . , N and |∇ψ_j|, j = 1, . . . , N are all bounded, where Φ⁻¹(y) = (ψ₁(y), . . . , ψ_N(y)), and suppose0 <

α≤J_Φ(x)≤βfor allx ∈G₁. Letp₁(x) =p₂(Φ(x))forx∈G₁. Then, (PI) (resp.

(wPI)) holds forp₁(·)onG₁ if and only if it holds forp₂(·)onG₂.

Combining Theorem4.1with this Proposition, we can find a fairly large class of p(x)for which (PI) (as well as (wPI)) holds.

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References

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[2] Y. FU, The existence of solutions for elliptic systems with nonuniform growth, Studia Math., 151 (2002), 227–246.

[3] O. KOVÁ ˇCIKANDJ. RÁKOSNÍK, On spacesL^p(x)andW^k,p(x), Czechoslovak Math. J., 41 (1991), 592–618.