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 Ime
Trang 1Volume 2009, Article ID 167403, 12 pages
doi:10.1155/2009/167403
Research Article
On Some Quasimetrics and Their Applications
Imed Bachar1 and Habib M ˆaagli2
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
Copyrightq 2009 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
Let D be a bounded smooth domain inRn , n ≥ 1, or D R n
: {x, x n ∈ Rn : x n > 0} the
half space We denote by G D the Green’s function of the operator u→ −Δα u with Dirichlet
or Navier boundary conditions, where α is a positive integer or 0 < α < 1.
The following inequality called 3G inequality has been proved by several authorssee
1,2 for α 1 or 3,4 for α ≥ 1 or 5 for 0 < α < 1.
There exists a constant C > 0 such that for each x, y, z ∈ D,
G D x, zG D
z, y
G D
x, y ≤ C
λ z
λ x G D x, z λ z
λ
y G D
y, z
where x → λx is a positive function which depends on the Euclidean distance between x and ∂D and the exponent α.
More precisely, to prove this inequality, the authors showed that the function
x, y → ρx, y λxλy/G D x, y is a quasi-metric on D seeDefinition 2.1
We emphasis that the generalized 3G inequality is crucial for various applicationssee e.g.,1, Theorem 1.2, 6, Lemma 7.1 It is also very interesting tools for analysts working on pde’s In7, Theorem 5.1, the authors used the standard 3G inequality see 7, Proposition 4.1 to prove that on the unit ball B in Rn , the inverse of polyharmonic operators that are
Trang 2perturbed 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 2 by using estimates on Green’s functions Recently, a refined version of the standard 3G inequality for polyharmonic operator is obtained in3, Theorem 2.8 and 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 one ρx, y λxλy/G D x, y.
In order to simplify our statements, we define some convenient notations
For s, t ∈ R, we denote by
s ∧ t mins, t, s ∨ t maxs, t. 1.2
The following properties will be used several times
For s, t ≥ 0, we have
st
s t ≤ mins, t ≤ 2 st
s t ,
1
2s t ≤ maxs, t ≤ s t,
min
2p−1, 1
s p t p ≤ s t p≤ max2p−1, 1
s p t p , p ∈ R.
1.3
For λ, μ > 0 and t > 0, we have
min
1, μ
λ
Log1 λt ≤ Log1 μt≤ max1, μ
λ
Log1 λt 1.4
Let f and g be two nonnegative functions on a set S We write f ∼ g, if there exists c > 0 such
that
1
c g x ≤ fx ≤ cgx, ∀x ∈ S. 1.5
Throughout this paper, we denote by c a positive generic constant whose value may vary
from line to line
2 Quasimetrics
Definition 2.1 Let E be a nonempty set A nonnegative function ρ x, y defined on E × E is called a quasi-metric on E if it satisfies the following properties.
i For all x, y ∈ E, ρx, y ρy, x.
ii There exists a constant c > 0 such that for all x, y, z ∈ E,
ρ
x, y
≤ cρ x, z ρz, y
Trang 3Example 2.2 1 Let d be a metric on a set E, then for each α ≥ 0, d α is a quasi-metric on E.
2 Let ρ be a quasi-metric on E and γ a non-negative symmetric function on E × E such that γ ∼ ρ, then γ is a quasi-metric on E.
3 Let ρ1and ρ2be two quasimetrics on E, then for all a ≥ 0, ρ1 aρ2and maxρ1, ρ2
are quasimetrics on E.
Next we denote byH the set of nonnegative nondecreasing functions f on 0, ∞
satisfying the following property
For each a > 0, there exists a constant c ca > 0 such that for each t, s ∈ 0, ∞,
f at s ≤ cf t fs. 2.2
Example 2.3 The following functions belong to the setH:
i ft αt β, with α ≥ 0 and β ≥ 0.
ii ft t p , with p ≥ 0.
iii ft Log1 αt, α > 0.
Remark 2.4 1 Let f be a function in H Then for each a > 0, there exists a constant c ca >
0 such that for each t ∈ 0, ∞,
1
c f t ≤ fat ≤ cft. 2.3
2 Let f be a nontrivial function in H Then for each t > 0, ft > 0.
Proposition 2.5 H is a convex cone which is stable by product and composition of functions.
Proof Let f, g ∈ H and λ ≥ 0 First, it is clear that f λg, fg and the composition of functions
g ◦ f are nonnegative nondecreasing functions on 0, ∞ So we need to prove that these
functions satisfy2.2 Let a > 0, then from 2.2, there exists a constant c > 0, such that for each t, s ∈ 0, ∞, we have
f at s ≤ cf t fs, g at s ≤ cg t gs. 2.4
So
f λgat s ≤ 1 λcf λgt f λgs . 2.5
HenceH is a convex cone
On the other hand, for each t, s ∈ 0, ∞, we have
fg
at s ≤ c2
f t fsg t gs
≤ c2
fg
t fg
s ftgs fsgt
≤ 2c2
fg
t fg
s .
2.6
Thus, we deduce that the coneH is stable by product
Trang 4Finally, for each t, s ∈ 0, ∞, we have by using again 2.2,
g
f at s≤ gc
f t fs≤ cg
f t gf s. 2.7 HenceH is stable by composition of functions
Remark 2.6 Let ρ be a quasi-metric on nonempty set E and f, g two functions in H Then it is clear that f ◦ ρ is a quasi-metric on E and so that for f ◦ ρ · g ◦ ρ f · g ◦ ρ In particular, for each α, β ∈ 0, ∞, f ◦ ρ α g ◦ ρ β is a quasi-metric on E.
Indeed, the last assertion follows from the fact that the functions t → ft α and t →
g t β belong to the cone H.
Before stating our main result, we need to introduce the following setF
Let F be the set of nonnegative nondecreasing functions f on 0, ∞ satisfying the
following two properties
There exists a constant c > 0, such that for all t ∈ 0, ∞,
1
c
t
and for all a > 0, there exists a constant c ca > 0, such that for each t ∈ 0, ∞,
Note that functions belonging to the setF satisfy also properties stated inRemark 2.4
Example 2.7 1 For 0 ≤ α ≤ 1, the function f α t : t/1 t αbelongs to the setF.
2 For λ > 0 and 0 ≤ α ≤ 1, the function f α,λ t : t1−αlog1 λtα belongs to the set F.
3 The function ft : arctant belongs to the set F.
Indeed, the function f satisfies2.8 with c 1 and for each a > 0 and t ≥ 0, we have
a
max1, a2f t ≤ fat ≤
a
min1, a2f t. 2.10
Throughout this paper,E, d denote a metric space Let Ω be a subset in E such that
∂ Ω / ∅, and let δx be the distance between x and ∂Ω Our main result is the following.
Theorem 2.8 Let f ∈ F, g ∈ H and h be a nontrivial function in H For x, y ∈ Ω × Ω, put
γ
x, y
gmax
d2
x, y
, δ xδy h
δ xδy
f
h
δ xδy
/h
d2
x, y . 2.11
Then γ is a quasi-metric on Ω.
For the proof, we need the following key lemmasee 9 For completeness of this paper, we reproduce the proof of this lemma here
Trang 5Lemma 2.9 see 9 Let x, y ∈ Ω Then one has the following properties.
1 If δxδy ≤ d2x, y, then δx ∨ δy ≤ √5 1/2dx, y.
2 If d2x, y ≤ δxδy, then 3−√5/2δx ≤ δy ≤ 3√5/2δx and dx, y ≤
√5 1/2δx ∧ δy.
Proof 1 We may assume that δx ∨ δy δy Then the inequalities
δ
y
≤ δx dx, y
, δ xδy
≤ d2
x, y
2.12 imply that
δ
y2
− δy
d
x, y
− d2
x, y
That is
⎛
⎜
⎝δy
√
5− 1
2 d
x, y
⎞
⎟
⎛
⎜
⎝δy
−
√
5 1
2 d
x, y
⎞
⎟
⎠ ≤ 0. 2.14
It follows that
δ x ∨ δy
≤
√
5 1
2 d
x, y
2 For each z ∈ ∂Ω, we have dy, z ≤ dx, y dx, z and since d2x, y ≤ δxδy,
we obtain
d
y, z
≤δ xδy
dx, z ≤d x, zdy, z
dx, z. 2.16
That is
⎛
⎜dy, z
√
5− 1 2
d x, z
⎞
⎟
⎛
⎜dy, z−
√
5 1 2
d x, z
⎞
⎟
⎠ ≤ 0. 2.17
It follows that
d
y, z
≤
3√5
Thus, interchanging the role of x and y, we have
3−√5 2
d x, z ≤ dy, z
≤
3√5 2
d x, z. 2.19
Trang 6Which gives that
3−√5 2
δ x ≤ δy
≤
3√5 2
Moreover, we have
d2
x, y
≤ δxδy
≤
√
5 1 2
2
δ x ∧ δy2
Proof of Theorem 2.8 It is clear that γ is a non-negative symmetric function onΩ × Ω
Put ρx, y gmaxd2x, y, δxδy, for x, y in Ω.
Since there exists a constant c > 0, such that for each t ≥ 0,
1
c
t
1 t ≤ ft ≤ ct, 2.22
we deduce that for each x, y in Ω,
1
c ρ
x, y
h
d2
x, y
≤ γx, y
≤ cρx, y
h
d2
x, y
hδ xδy
. 2.23
Let z in Ω We distinguish the following subcases.
i If δxδy ≤ d2x, y, then by 2.23 andRemark 2.6, we have
γ
x, y
≤ cgd2
x, y
h
d2
x, y
≤ cg
d2x, zh
d2x, z gd2
y, z
h
d2
y, z
≤ cγ x, z γy, z
.
2.24
ii If d2x, y ≤ δxδy, it follows byLemma 2.9that δx ∼ δy.
a If d2x, z ≤ δxδz or d2y, z ≤ δyδz, then fromLemma 2.9, we deduce that
δ x ∼ δy ∼ δz.
So
ρ
x, y
∼ ρx, z ∼ ρy, z
,
h
δ xδy
∼ hδxδz ∼ hδ
y
δ z. 2.25 Now, since
h
d2
x, y
≤ ch
d2x, z hd2
y, z
≤ ch
d2x, z∨ hd2
y, z
, 2.26
Trang 7we deduce from2.25 that
h δxδz
h d2x, z ∧
h
δ
y
δ z
h
d2
y, z ≤ c h
δ xδy
h
d2
x, y 2.27 Which implies from2.25 and 2.9 that
h
δ xδy
f
h
δ xδy
/h
d2
x, y
≤ c
h δxδz
f hδxδz/hd2x, z∨
h
δ
y
δ z
f
h
δ
y
δ z/h
d2
y, z
≤ c
h δxδz
f hδxδz/hd2x, z
h
δ
y
δ z
f
h
δ
y
δ z/h
d2
y, z
2.28
Hence
γ
x, y
≤ cγ x, z γy, z
b If d2x, z ≥ δxδz and d2y, z ≥ δyδz, then byLemma 2.9, it follows that
δx ∨ δz ≤ cdx, z, δ
y
∨ δz≤ cdy, z
Hence, by2.23 and 2.2, we have
γ
x, y
≤ cgδ xδy
h
δ xδy
≤ cg
δx2
h
δx2
g
δ
y2
h
δ
y2
≤ cg
d2x, zh
d2x, z gd2
y, z
h
d2
y, z
≤ cγ x, z γy, z
.
2.31
So there exists a constant c > 0, such that for each x, y, z ∈ Ω, we have
γ
x, y
≤ cγ x, z γy, z
By taking ft t inTheorem 2.8, we obtain the following corollary
Corollary 2.10 Let g and h be two functions in H Then the function
ρ
x, y : hd2
x, y
g max
d2
x, y
, δ xδy
2.33
is a quasi-metric on Ω.
Trang 8Example 2.11 Let Ω be a subset of E, d such that ∂Ω / ∅ We denote δΩx δx, the distance between x and ∂Ω.
1 For each α, β ≥ 0, the function
ρ
x, y :d
x, yβ
max
d2
x, y
, δ xδyα
2.34
is a quasi-metric onΩ.
2 For each λ, μ ≥ 0, the function
ρ
x, y
:max
d2
x, y
, δ xδyμ
δ xδyλ Log
1δ xδyλ
/
d
x, 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 operator u→ −Δα u with
Dirichlet or Navier boundary conditions, where α is 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 function GB
m,nof−Δmon
the unit ball B ofRn n ≥ 2, with Dirichlet boundary conditions ∂/∂ν j u 0, 0 ≤ j ≤ m − 1:
G B m,n
x, y
k m,nx − y2m−nx,y/|x −y|
1
v2− 1m−1
v n−1 dv, 3.1
where ∂/∂ν is the outward normal derivative, m is a positive integer, k m,n Γn/2/
22m−1 π n/2 m − 1!2 and x, y2 |x − y|2 1 − |x|21 − |y|2, for x, y in B.
Then we deduce that for each a∈ Rn and r > 0, we have
G B m,n a,r
x, y
r 2m−n G B m,n
x − a
r ,
y − a
r
, for x, y ∈ Ba, r, 3.2
where Ba, r denote the open ball in R n with radius r, centered at a.
Using a rescaling argument, one recovers from3.1 a similar Green function GRn
m,nof
−Δmon the half-spaceRn
: {x, x n ∈ Rn : x n > 0} see 11, page 165:
GRn
m,n
x, y
k m,nx − y2m−n|x −y|/|x −y|
1
v2− 1m−1
v n−1 dv, for x, y inRn
, 3.3
where y y1, , y n−1, −y n
Trang 9Indeed, let e 0, , 0, 1 ∈ R n and for p ∈ N, we denote by B p B2 p e, 2 p Then by
using3.2 and 3.1, we obtain for each x, y in R n
,
GRn
m,n
x, y
sup
p∈NG B p
m,n
x, y
k m,nx − y2m−n|x −y|/|x −y|
1
v2− 1m−1
v n−1 dv. 3.4
Now to prove the 3G inequality1.1 with λx δx m , for these Green’s functions G D
m,n ,
where D B or D R n
, we put ρ x, y : δxδy m /G D
m,n x, y and we need to show that
ρ is a quasi-metric on D To this end, we observe that from7 or 3 for D B and from 4
for D Rn
, we have the following estimates on G D
m,n:
G D
m,n
x, y
∼
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
δ xδym
x − yn −2mx − y2∨ δxδym , if n > 2m,
Log
1
δ xδym
x − y2m
, if n 2m,
δ xδym
x − y2∨δ xδyn/2 , if n < 2m.
3.5
From which we deduce that
ρ
x, y
∼
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
x − yn −2m
maxx − y2
, δ xδym
, if n > 2m,
δ xδym log
1
δ xδym
/x − y2m , if n 2m,
maxx − y2
, δ xδyn/2
, if n < 2m.
3.6
So by Example 2.11, we see that ρ is a quasi-metric on D, that is, G D
m,n satisfies the 3G inequality1.1 with λx δx m
3.2 Polyharmonic Laplacian Operator with Navier Boundary Conditions
Let D be a bounded smooth domain inRn or D Rn
the half space We denote by G D m,nthe
Green’s function of the polyharmonic operator u → −Δm u on D, with Navier boundary
conditions−Δk u| 0, for 0 ≤ k ≤ m − 1, where m is a positive integer.
Trang 10In12, for D a bounded smooth domain and in 13, for D R n
, the authors have
established the following estimates for the Green function G D
m,n:
G D m,n
x, y
∼
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
δ xδy
x − yn −2mx − y2
∨ δxδy , if n > 2m,
Log
1δ xδ
y
x − y2
, if n 2m,
δ xδy
x − y2∨ δxδy1/2 , if n 2m − 1.
3.7
So we deduce that the function ρx, y : δxδy/G D
m,n x, y satisfies
ρ
x, y
∼
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
x − yn −2mmaxx − y2
, δ xδy
, if n > 2m,
δ xδy Log
1 δxδy
/x − y2, if n 2m,
maxx − y2
, δ xδy1/2
, if n 2m − 1.
3.8
Hence, the function ρ is a quasi-metric on D, byExample 2.11 Therefore, the Green function
G D
m,nsatisfies the 3G inequality1.1 with λx δx.
3.3 Fractional Laplacian with Dirichlet Boundary Conditions
Let D be a bounded C 1,1domain inRn n ≥ 2 and G Dthe Green’s function of the fractional Laplacian−Δα/2 , with Dirichlet boundary conditions 0 < α ≤ 2 From 14, we have the
following estimates on G D:
G D
x, y
∼
δ xδyα/2
x − yn −α
maxx − y2
, δ xδyα/2 , 3.9
which implies that the function ρx, y : δxδy α/2 /G D x, y satisfies
ρ
x, y
∼x − yn −α
maxx − y2, δ xδyα/2
By Example 2.11, the function ρ is a quasi-metric on D, and so the Green’s function G D
satisfies the 3G inequality1.1 with λx δx α/2 Note that in this case, the 3G inequality
has been already proved by Chen and Song in5