On Some Quasimetrics and Their Applications

Volume 2009, Article ID 167403,12pages doi:10.1155/2009/167403

Research Article

On Some Quasimetrics and Their Applications

Imed Bachar

_{and Habib M ˆaagli}

1_{Mathematics Department, College of Sciences, King Saud University, P.O. Box 2455,}

Riyadh 11451, Saudi Arabia

2_{D´epartement de Math´ematiques, Facult´e des Sciences de Tunis, Campus Universitaire,}

2092 Tunis, Tunisia

Correspondence should be addressed to Imed Bachar,abachar@ksu.edu.sa

Received 27 September 2009; Accepted 8 December 2009

Recommended by Shusen Ding

We aim at giving a rich class of quasi-metrics from which we obtain as an application an interesting inequality for the Greens function of the fractional Laplacian in a smooth domain inRn_.

Copyrightq2009 I. Bachar and H. Mˆaagli. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

LetDbe a bounded smooth domain inRn_{, n≥ 1,orD R}n

: {x, xn ∈Rn :xn > 0}the half space. We denote byGDthe Green’s function of the operatoru→−Δαuwith Dirichlet or Navier boundary conditions, whereαis a positive integer or 0< α <1.

The following inequality called 3G inequality has been proved by several authorssee

1,2forα1 or3,4forα≥1 or5for 0< α <1.

There exists a constantC >0 such that for eachx, y, z∈D,

GDx, zGD

z, y GD

x, y ≤C

λz

λxGDx, z λz

λyGD

y, z

, 1.1

wherex → λxis a positive function which depends on the Euclidean distance betweenx

and∂Dand the exponentα.

More precisely, to prove this inequality, the authors showed that the function

x, y → ρx, y λxλy/GDx, yis a quasi-metric onDseeDefinition 2.1.

perturbed by small lower order terms is positivity preserving. They also obtained similar results for systems of these operators. On the other hand, local maximum principles for solutions of higher-order differential inequalities in arbitrary bounded domains are obtained in8, Theorem 2by using estimates on Green’s functions. Recently, a refined version of the standard 3G inequality for polyharmonic operator is obtained in3, Theorem 2.8and 4, Theorem 2.9. This allowed the authors to introduce and study an interesting functional Kato class, which permits them to investigate the existence of positive solutions for some polyharmonic nonlinear problems.

In the present manuscript we aim at giving a generalization of these known 3G inequalities by proving a rich class of quasimetricssee Theorem 2.8 which in particular includes the oneρx, y λxλy/GDx, y.

In order to simplify our statements, we define some convenient notations. Fors, t∈R,we denote by

s∧tmins, t, s∨tmaxs, t. 1.2

The following properties will be used several times. Fors, t≥0,we have

st

st ≤mins, t≤2 st st,

1

2st≤maxs, t≤st,

min2p−1,1sptp≤stp≤max2p−1,1sptp, p∈R.

1.3

Forλ, μ >0 andt >0,we have

min1,μ λ

Log1λt≤Log1μt≤max1,μ λ

Log1λt. 1.4

Letfandgbe two nonnegative functions on a setS.We writef∼g,if there existsc >0 such that

1

cgx≤fx≤cgx, ∀x∈S. 1.5

Throughout this paper, we denote byca positive generic constant whose value may vary from line to line.

2. Quasimetrics

Definition 2.1. LetEbe a nonempty set. A nonnegative functionρx, ydefined onE×Eis called a quasi-metric onEif it satisfies the following properties.

iFor allx, y∈E, ρx, y ρy, x.

iiThere exists a constantc >0 such that for allx, y, z∈E,

Example 2.2. 1Letdbe a metric on a setE,then for eachα≥0, dα_{is a quasi-metric onE.}

2Let ρbe a quasi-metric onE andγ a non-negative symmetric function onE×E

such thatγ∼ρ,thenγis a quasi-metric onE.

3Letρ1andρ2be two quasimetrics onE,then for alla≥0, ρ1aρ2and maxρ1, ρ2

are quasimetrics onE.

Next we denote byH the set of nonnegative nondecreasing functions f on 0,∞

satisfying the following property

For eacha >0,there exists a constantcca>0 such that for eacht, s∈0,∞,

fats≤cft fs. 2.2

Example 2.3. The following functions belong to the setH:

ift αtβ,withα≥0 andβ≥0. iift tp_,withp≥0.

iii ft Log1αt, α >0.

Remark 2.4. 1Letfbe a function inH.Then for eacha >0,there exists a constantcca>

0 such that for eacht∈0,∞,

1

cft≤fat≤cft. 2.3

2Letfbe a nontrivial function inH.Then for eacht >0, ft>0.

Proposition 2.5. His a convex cone which is stable by product and composition of functions.

Proof. Letf, g∈ Handλ≥0.First, it is clear thatfλg, fgand the composition of functions

g ◦f are nonnegative nondecreasing functions on 0,∞.So we need to prove that these functions satisfy2.2. Leta >0,then from2.2, there exists a constantc > 0,such that for eacht, s∈0,∞,we have

fats≤cft fs, gats≤cgt gs. 2.4

So

fλgats≤1λcfλgt fλgs . 2.5

HenceHis a convex cone.

On the other hand, for eacht, s∈0,∞,we have

fgats≤c2ft fsgt gs

≤c2fgt fgs ftgs fsgt

≤2c2fgt fgs .

2.6

Finally, for eacht, s∈0,∞,we have by using again2.2,

gfats≤gcft fs≤cgftgfs. 2.7

HenceHis stable by composition of functions.

Remark 2.6. Letρbe a quasi-metric on nonempty setEandf, gtwo functions inH.Then it is clear thatf◦ρis a quasi-metric onEand so that forf◦ρ·g◦ρ f·g◦ρ.In particular,

for eachα, β∈0,∞,f◦ρα_g◦ρβ_{is a quasi-metric onE.}

Indeed, the last assertion follows from the fact that the functionst → ftα_{andt →}

gtβ_{belong to the coneH.}

Before stating our main result, we need to introduce the following setF.

Let F be the set of nonnegative nondecreasing functionsf on 0,∞satisfying the following two properties.

There exists a constantc >0,such that for allt∈0,∞,

1

c t

1t ≤ft≤ct, 2.8

and for alla >0,there exists a constantcca>0,such that for eacht∈0,∞,

fat≤cft. 2.9

Note that functions belonging to the setFsatisfy also properties stated inRemark 2.4.

Example 2.7. 1For 0≤α≤1,the functionfαt:t/1tαbelongs to the setF.

2Forλ >0 and 0≤α≤1,the functionfα,λt:t1−αlog1λtαbelongs to the setF.

3The function ft:arctantbelongs to the setF.

Indeed, the functionfsatisfies2.8withc1 and for eacha >0 andt≥0,we have

a

max1, a2ft≤fat≤

a

min1, a2ft. 2.10

Throughout this paper,E, ddenote a metric space. LetΩbe a subset inEsuch that

∂Ω/∅,and letδxbe the distance betweenxand∂Ω.Our main result is the following.

Theorem 2.8. Letf ∈ F, g∈ Handhbe a nontrivial function inH.Forx, y∈Ω×Ω,put

γx, ygmaxd2x, y, δxδy h

δxδy

fhδxδy/hd2_{x, y}. 2.11

Thenγis a quasi-metric onΩ.

Lemma 2.9see9. Letx, y∈Ω.Then one has the following properties.

1Ifδxδy≤d2_{x, y,thenδx∨δy≤}√_{51/2dx, y.}

2Ifd2_{x, y≤δxδy, then3−}√_{5/2δx≤δy≤3}√_{5/2δxanddx, y≤}

√51/2δx∧δy.

Proof. 1We may assume thatδx∨δy δy.Then the inequalities

δy≤δx dx, y, δxδy≤d2_{x, y 2.12}

imply that

δy2−δydx, y−d2x, y≤0. 2.13

That is

⎛ ⎜ ⎝δy

_√ 5−1

2 d

x, y

⎞ ⎟ ⎠

⎛ ⎜ ⎝δy−

_√ 51

2 d

x, y

⎞ ⎟

⎠≤0. 2.14

It follows that

δx∨δy≤

_√ 51

2 d

x, y. 2.15

2For eachz∈∂Ω,we havedy, z≤dx, y dx, zand sinced2_{x, y≤δxδy,}

we obtain

dy, z≤

δxδydx, z≤

dx, zdy, zdx, z. 2.16

That is ⎛ ⎜

⎝dy, z

_√ 5−1

2

dx, z

⎞ ⎟ ⎠

⎛ ⎜

⎝dy, z−

_√ 51

2

dx, z

⎞ ⎟

⎠≤0. 2.17

It follows that

dy, z≤

3√5

2 dx, z. 2.18

Thus, interchanging the role ofxandy,we have

3−√5 2

dx, z≤dy, z≤

3√5

2

Which gives that

3−√5

2

δx≤δy≤

3√5

2

δx. 2.20

Moreover, we have

d2x, y≤δxδy≤

√ 51

2 2

δx∧δy2. 2.21

Proof ofTheorem 2.8. It is clear thatγis a non-negative symmetric function onΩ×Ω Putρx, y gmaxd2_{x, y, δxδy,forx, yinΩ.}

Since there exists a constantc >0,such that for eacht≥0,

1

c t

1t ≤ft≤ct, 2.22

we deduce that for eachx, yinΩ,

1

cρ

x, yhd2x, y≤γx, y≤cρx, yhd2x, yhδxδy. 2.23

LetzinΩ.We distinguish the following subcases.

iIfδxδy≤d2x, y,then by2.23andRemark 2.6, we have

γx, y≤cgd2x, yhd2x, y

≤cgd2x, zhd2x, zgd2y, zhd2y, z

≤cγx, z γy, z.

2.24

iiIfd2_{x, y≤δxδy,it follows byLemma 2.9thatδx∼δy.}

aIfd2_{x, z≤δxδzord}2_{y, z≤δyδz,then fromLemma 2.9, we deduce that}

δx∼δy∼δz.

So

ρx, y∼ρx, z∼ρy, z,

hδxδy∼hδxδz∼hδyδz.

2.25

Now, since

we deduce from2.25that

hδxδz hd2x, z ∧

hδyδz hd2_{y, z} ≤c

hδxδy

hd2_{x, y} . 2.27

Which implies from2.25and2.9that

hδxδy

fhδxδy/hd2_{x, y}

≤c

hδxδz

fhδxδz/hd2_{x, z}∨

hδyδz

fhδyδz/hd2_{y, z}

≤c

hδxδz

fhδxδz/hd2_{x, z}

hδyδz

fhδyδz/hd2_{y, z}

2.28

Hence

γx, y≤cγx, z γy, z. 2.29

bIfd2_{x, z≥δxδzandd}2_{y, z≥δyδz,then byLemma 2.9, it follows that}

δx∨δz≤cdx, z, δy∨δz≤cdy, z. 2.30

Hence, by2.23and2.2, we have

γx, y≤cgδxδyhδxδy

≤cgδx2hδx2gδy2hδy2

≤cgd2x, zhd2x, zgd2y, zhd2y, z

≤cγx, z γy, z.

2.31

So there exists a constantc >0,such that for eachx, y, z∈Ω,we have

γx, y≤cγx, z γy, z. 2.32

By takingft tinTheorem 2.8, we obtain the following corollary.

Corollary 2.10. Letgandhbe two functions inH.Then the function

ρx, y:hd2x, ygmaxd2x, y, δxδy 2.33

Example 2.11. Let Ω be a subset of E, d such that ∂Ω/∅. We denote δΩx δx, the distance betweenxand∂Ω.

1For eachα, β≥0,the function

ρx, y:dx, yβmaxd2x, y, δxδyα 2.34

is a quasi-metric onΩ.

2For eachλ, μ≥0,the function

ρx, y:maxd2x, y, δxδyμ

δxδyλ

Log1δxδyλ/dx, y2λ

2.35

is a quasi-metric onΩ.

3. Applications

As applications ofTheorem 2.8, we will collect many forms of the 3G inequality1.1, which in fact depend on the shape of the domain and the choose of the operatoru→−Δαuwith Dirichlet or Navier boundary conditions, whereαis a positive integer or 0< α <1.

3.1. Polyharmonic Laplacian Operator with Dirichlet Boundary Conditions

In10, page 126, Boggio gave an explicit expression for the Green functionGB

m,nof−Δmon the unit ballBofRn _{n≥2,with Dirichlet boundary conditions∂/∂ν}j_{u0,0≤j≤m−1:}

GB m,n

x, ykm,nx−y2m−n

x,y/|x−y|

1

v2₋₁m−1

vn−1 dv, 3.1

where ∂/∂ν is the outward normal derivative, m is a positive integer, km,n Γn/2/

22m−1_πn/2_m−1!2_{andx, y}2_|x−y|2 _{1− |x|}2_{1− |y|}2_{,forx, yinB.}

Then we deduce that for eacha∈Rn_{andr >0,we have}

GBa,rm,n

x, yr2m−nGB_m,n

x−a r ,

y−a r

, forx, y∈Ba, r, 3.2

whereBa, rdenote the open ball inRn_{with radiusr,centered ata.}

Using a rescaling argument, one recovers from3.1a similar Green functionGRn

m,nof

−Δm_{on the half-spaceR}n

:{x, xn∈Rn:xn>0}see11, page 165:

GRn

m,n

x, ykm,nx−y2m−n

_{|x−y|/|x−y|}

1

v2₋₁m−1

vn−1 dv, forx, yinR

n

, 3.3

Indeed, lete 0, . . . ,0,1∈Rn_{and forp∈N,we denote byB}

p B2pe,2p.Then by using3.2and3.1, we obtain for eachx, yinRn

,

GRn

m,n

x, ysup p∈NG

Bp

m,n

x, ykm,nx−y2m−n

_|x−y|/|x−y|

1

v2₋₁m−1

vn−1 dv. 3.4

Now to prove the 3G inequality1.1withλx δxm,for these Green’s functionsGD m,n, whereDBorDRn

,we putρx, y: δxδym/GDm,nx, yand we need to show that

ρis a quasi-metric onD.To this end, we observe that from7or3forDBand from4

forDRn

,we have the following estimates onGDm,n:

GD m,n

x, y∼

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

δxδym

x−yn−2mx−y2∨δxδym

, ifn >2m,

Log

1

δxδym

_x−y2m

, ifn2m,

δxδym

x−y2∨δxδyn/2

, ifn <2m.

3.5

From which we deduce that

ρx, y∼

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x−yn−2mmaxx−y2, δxδym, ifn >2m,

δxδym

log1δxδym/x−y2m

, ifn2m,

maxx−y2, δxδyn/2, ifn <2m.

3.6

So by Example 2.11, we see that ρ is a quasi-metric on D, that is, GD

m,n satisfies the 3G inequality1.1withλx δxm.

3.2. Polyharmonic Laplacian Operator with Navier Boundary Conditions

In12, forD a bounded smooth domain and in13, forD Rn

,the authors have established the following estimates for the Green functionGD

m,n:

GD_m,nx, y∼

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

δxδy

x−yn−2mx−y2∨δxδy

, ifn >2m,

Log

1δxδ

y

_x−y2

, ifn2m,

δxδy

x−y2∨δxδy1/2

, ifn2m−1.

3.7

So we deduce that the functionρx, y:δxδy/GD

m,nx, ysatisfies

ρx, y∼

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x−yn−2mmaxx−y2, δxδy, ifn >2m,

δxδy

Log1δxδy/x−y2

, ifn2m,

maxx−y2, δxδy1/2, ifn2m−1.

3.8

Hence, the functionρis a quasi-metric onD,byExample 2.11. Therefore, the Green function

GD

m,nsatisfies the 3G inequality1.1withλx δx.

3.3. Fractional Laplacian with Dirichlet Boundary Conditions

LetDbe a boundedC1,1 _{domain inR}n_n≥2andG

Dthe Green’s function of the fractional Laplacian−Δα/2,with Dirichlet boundary conditions 0 < α ≤ 2.From14, we have the following estimates onGD:

GD

x, y∼

δxδyα/2

_x−yn−α

maxx−y2, δxδyα/2

, 3.9

which implies that the functionρx, y: δxδyα/2/GDx, ysatisfies

ρx, y∼x−yn−αmaxx−y2, δxδyα/2. 3.10

By Example 2.11, the function ρ is a quasi-metric on D, and so the Green’s function GD

3.4. On the Operator

−

1

/A

Au

on

0

,

1

with

u

0

u

1

0

LetA be a continuous function on0,1,which is positive and differentiable on 0,1.We assume that t → 1/At is integrable on0,1 and without loss of generality we may suppose that1₀1/Atdt1.

We denote byGx, ythe Green’s function of the operatoru → −1/AAuwith

u0 u1 0.Clearly we have for eachx, y∈0,1,

Gx, yAyρx∧y1−ρx∨y, 3.11

whereρx x₀1/Atdt.

Putλx ρx1−ρx,then we claim that the Green’s functionGx, ysatisfies the following 3G inequality:∀x, y, z∈0,1,

Gx, zGz, y Gx, y ≤

λz

λxGx, z λz λyG

y, z. 3.12

That is

Γx, zΓz, y Γx, y ≤

λz

λxΓx, z λz λyΓ

y, z, 3.13

whereΓx, y:ρx∧y1−ρx∨y.

To prove3.13, we need to show that the function

γx, y: λxλ

y Γx, y ρ

x∨y1−ρx∧y 3.14

is quasi-metric on0,1.

Indeed, since the functionγx, yis symmetric inx, y,we will discuss three cases:

iifz≤x≤y,then

γx, yρy1−ρx≤ρy1−ρzγz, y≤γx, z γz, y, 3.15

iiifx≤y≤z,then

γx, yρy1−ρx≤ρz1−ρxγx, z≤γx, z γz, y, 3.16

So we deduce that

γx, z γz, yαρx 1−αρy1−ρx

ρyα1−ρx 1−α1−ρy

≥1−αρy1−ρxαρy1−ρx ≥ρy1−ρxγx, y.

3.17

This completes the proof.

Acknowledgments

The authors thank the reviewers for their valuable suggestions. The research of I. Bachar is supported by King Saud University, College of Sciences Research Center, Project no

Math/2009/51.

References

1 N. J. Kalton and I. E. Verbitsky, “Nonlinear equations and weighted norm inequalities,”Transactions of the American Mathematical Society, vol. 351, no. 9, pp. 3441–3497, 1999.

2 M. Selmi, “Inequalities for Green functions in a Dini-Jordan domain inR2_{,”Potential Analysis, vol. 13,}

no. 1, pp. 81–102, 2000.

3 I. Bachar, H. Mˆaagli, S. Masmoudi, and M. Zribi, “Estimates for the Green function and singular solutions for polyharmonic nonlinear equation,”Abstract and Applied Analysis, vol. 2003, no. 12, pp. 715–741, 2003.

4 I. Bachar, H. Mˆaagli, and M. Zribi, “Estimates on the Green function and existence of positive solutions for some polyharmonic nonlinear equations in the half space,”Manuscripta Mathematica, vol. 113, no. 3, pp. 269–291, 2004.

5 Z. Q. Chen and R. Song, “General gauge and conditional gauge theorems,”The Annals of Probability, vol. 30, no. 3, pp. 1313–1339, 2002.

6 Y. Pinchover, “Maximum and anti-maximum principles and eigenfunctions estimates via perturba-tion theory of positive soluperturba-tions of elliptic equaperturba-tions,”Mathematische Annalen, vol. 314, no. 3, pp. 555–590, 1999.

7 H. C. Grunau and G. Sweers, “Positivity for equations involving polyharmonic operators with Dirichlet boundary conditions,”Mathematische Annalen, vol. 307, no. 4, pp. 589–626, 1997.

8 H. C. Grunau and G. Sweers, “Classical solutions for some higher order semilinear elliptic equations under weak growth conditions,”Nonlinear Analysis: Theory, Methods & Applications, vol. 28, no. 5, pp. 799–807, 1997.

9 H. Mˆaagli, “Inequalities for the Riesz potentials,”Archives of Inequalities and Applications, vol. 1, no. 3-4, pp. 285–294, 2003.

10 T. Boggio, “Sulle funzione di Green d’ordinem,”Rendiconti del Circolo Matem`atico di Palermo, vol. 20, pp. 97–135, 1905.

11 H. C. Grunau and G. Sweers, “The maximum principle and positive principal eigenfunctions for polyharmonic equations,” inReaction Diffusion Systems, G. Caristi and E. Mitidieri, Eds., vol. 194, pp. 163–182, Marcel Dekker, New York, NY, USA, 1998.

12 H. C. Grunau and G. Sweers, “Sharp estimates for iterated Green functions,”Proceedings of the Royal Society of Edinburgh A, vol. 132, no. 1, pp. 91–120, 2002.

13 I. Bachar, “Estimates for the Green function and characterization of a certain Kato class by the Gauss semigroup in the half space,”Journal of Inequalities in Pure and Applied Mathematics, vol. 6, no. 4, article 119, 14 pages, 2005.