Volume 2009, Article ID 167403,12pages doi:10.1155/2009/167403
Research Article
On Some Quasimetrics and Their Applications
Imed Bachar
1and Habib M ˆaagli
21Mathematics Department, College of Sciences, King Saud University, P.O. Box 2455,
Riyadh 11451, Saudi Arabia
2D´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 Rn
: {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, dαis 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◦ραg◦ρβis a quasi-metric onE.
Indeed, the last assertion follows from the fact that the functionst → ftαandt →
gtβbelong 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/hd2x, y. 2.11
Thenγis a quasi-metric onΩ.
Lemma 2.9see9. Letx, y∈Ω.Then one has the following properties.
1Ifδxδy≤d2x, y,thenδx∨δy≤√51/2dx, y.
2Ifd2x, 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≤d2x, 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 sinced2x, 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 gmaxd2x, 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
iiIfd2x, y≤δxδy,it follows byLemma 2.9thatδx∼δy.
aIfd2x, z≤δxδzord2y, 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 hd2y, z ≤c
hδxδy
hd2x, y . 2.27
Which implies from2.25and2.9that
hδxδy
fhδxδy/hd2x, y
≤c
hδxδz
fhδxδz/hd2x, z∨
hδyδz
fhδyδz/hd2y, z
≤c
hδxδz
fhδxδz/hd2x, z
hδyδz
fhδyδz/hd2y, z
2.28
Hence
γx, y≤cγx, z γy, z. 2.29
bIfd2x, z≥δxδzandd2y, 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∂/∂νju0,0≤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−1πn/2m−1!2andx, y2|x−y|2 1− |x|21− |y|2,forx, yinB.
Then we deduce that for eacha∈Rnandr >0,we have
GBa,rm,n
x, yr2m−nGBm,n
x−a r ,
y−a r
, forx, y∈Ba, r, 3.2
whereBa, rdenote the open ball inRnwith radiusr,centered ata.
Using a rescaling argument, one recovers from3.1a similar Green functionGRn
m,nof
−Δmon the half-spaceRn
:{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∈Rnand 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:
GDm,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 inRnn≥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.
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