Volume 2008, Article ID 475957, 15 pagesdoi:10.1155/2008/475957 Research Article Representation of Multivariate Functions via the Potential Theory and Applications to Inequalities Floric
Trang 1Volume 2008, Article ID 475957, 15 pages
doi:10.1155/2008/475957
Research Article
Representation of Multivariate Functions
via the Potential Theory and Applications to
Inequalities
Florica C Cˆırstea 1 and Sever S Dragomir 2
1 Department of Mathematics, Australian National University, Canberra, ACT 0200, Australia
2 School of Computer Science and Mathematics, Victoria University, P.O Box 14428, Melbourne City, Victoria 8001, Australia
Correspondence should be addressed to Sever S Dragomir, sever.dragomir@vu.edu.au
Received 12 February 2007; Revised 2 August 2007; Accepted 9 November 2007
Recommended by Siegfried Carl
We use the potential theory to give integral representations of functions in the Sobolev spaces
W 1,p Ω, where p ≥ 1 and Ω is a smooth bounded domain inRN N ≥ 2 As a byproduct, we
obtain sharp inequalities of Ostrowski type.
Copyright q 2008 F C Cˆırstea and S S Dragomir 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 and main results
Let N ≥ 2 and let ·, · denote the canonical inner product on R N× RN If ω N stands for the area of the surface of theN − 1-dimensional unit sphere, then ω N 2π N/2 / ΓN/2, where Γ
is the gamma function defined byΓs 0∞e −t t s−1dt for s > 0see 1, Proposition 0.7
Let E denote the normalized fundamental solution of Laplace equation:
E x
⎧
⎪
⎨
⎪
⎪
1
2π ln|x|, x / 0 if N 2,
1
2 − Nω N |x| N−2, x / 0 if N ≥ 3.
1.1
Unless otherwise stated, we assume throughout that Ω ⊂ RN is a bounded domain
with C2 boundary ∂Ω Let ν denote the unit outward normal to ∂Ω and let dσ indicate
theN−1-dimensional area element in ∂Ω The Green-Riemann formula says that any function
Trang 2f ∈ C2Ω ∩ C1Ω satisfying Δf ∈ CΩ can be represented in Ω as follows see 2, Section 2.4:
f y
∂Ω
∂ν x − y − ∂f
∂ν xEx − y
dσ
ΩE x − yΔfxdx, ∀y ∈ Ω,
1.2 where∂f/∂νx is the normal derivative of f at x ∈ ∂Ω In particular, if f ∈ C∞
0Ω the set of
functions in C∞Ω with compact support in Ω, then 1.2 leads to the representation formula
f y
For a continuous function h on ∂Ω, the double-layer potential with moment h is defined by
u h y
∂Ωh x ∂E
Expression1.4 may be interpreted as the potential produced by dipoles located on ∂Ω; the direction of which at any point x ∈ ∂Ω coincides with that of the exterior normal ν, while its intensity is equal to hx The double-layer potential is well defined in R N and it satisfies the Laplace equation Δu 0 in R N \ ∂Ω see Proposition 2.8 For other properties of the double-layer potential, seeLemma 2.9andProposition 2.10
The double-layer potential plays an important role in solving boundary value prob-lems of elliptic equations The representation of the solution of the interior/exterior Dirichlet problem for Laplace’s equation is sought as a double-layer potential with unknown
density h An application of property2.14 leads to a Fredholm equation of the second kind
on ∂Ω in order to determine the function h see, e.g., 3
In many problems of mathematical physics and variational calculus, it is not sufficient
to deal with classical solutions of differential equations One needs to introduce the notion of weak derivatives and to work in Sobolev spaces, which have become an indispensable tool in the study of partial differential equations
For 1≤ p ≤ ∞, we denote by W 1,pΩ the Sobolev space defined by
W 1,pΩ
⎧
⎪
⎨
⎪u ∈ L pΩ
Ωu ∂φ
∂x i
0 Ω, ∀i ∈ {1, 2, , N}
⎫
⎪
⎬
⎪. 1.5
For u ∈ W 1,p Ω, we define g i ∂u/∂x iand write∇u ∂u/∂x1, ∂u/∂x2, , ∂u/∂x N The
Sobolev space W 1,pΩ is endowed with the norm
W 1,pΩ L pΩ
N
i1
∂x ∂u i
where L pΩ stands for the usual norm on L p Ω The closure of C∞
0 Ω in the norm of
W 1,p Ω is denoted by W 1,p
0 Ω For details on Sobolev spaces, we refer to 2,4, or 5
Trang 3SinceΩ is bounded, we have C1Ω ⊂ W 1,∞ Ω ⊆ W 1,p Ω for every p ∈ 1, ∞.
The following representation holds for functions f in W01,p Ω with p ≥ 1 see Remark
2.3:
f y −
Ω
∇Ex − y, ∇fxdx a.e y ∈ Ω. 1.7
We first give an integral representation of functions in W 1,p Ω for any p ≥ 1.
Theorem 1.1 For any g ∈ W 1,p Ω with p ≥ 1, there is a sequence g n in C∞Ω such that
g y lim
n→∞
∂Ωg n x ∂E
∂ν x − ydσx −
Ω
∇Ex − y, ∇gxdx a.e y ∈ Ω, 1.8
0 lim
n→∞
∂Ωg n x ∂E
∂ν x − ydσx −
Ω
∇Ex − y, ∇gxdx, ∀y ∈ R N \ Ω. 1.9
0Ω for which 1.8 holds Thus, we regain1.7 for any function f in W 1,p
0 Ω
Under a suitable smoothness condition, the representation ofTheorem 1.1can be refined
for functions in W 1,p Ω with p > N seeTheorem 1.3 Using Morrey’s inequality, one can
prove that functions in the Sobolev space W 1,p Ω with p > N are classically differentiable
almost everywhere inΩ cf 2, page 176 or 4 ByProposition 2.13, any function in W 1,pΩ
with N < p < ∞ is uniformly H¨older continuous in Ω with exponent 1 − N/p after possibly
being redefined on a set of measures 0 In particular, any function in W1,p Ω with p > N is
continuous onΩ, and thus it has a well-defined trace which is bounded
The proof ofTheorem 1.1relies on the density of C∞Ω in W 1,pΩ as well as the fol-lowing result
Theorem 1.3 Assume that f ∈ W 1,p Ω ∩ C1Ω \ A, where p ≥ 1 and A a ii ∈I is a finite family
a If p > N, then f can be represented as follows:
f y
⎧
⎪
⎪
⎪
⎪
u f y −
Ω
∇Ex − y, ∇fxdx, ∀y ∈ Ω,
2
u f y −
Ω
∇Ex − y, ∇fxdx
, ∀y ∈ ∂Ω.
1.10
b If p ≥ 1 and f ∈ CΩ, then
0 u f y −
Ω
∇Ex − y, ∇fxdx, ∀y ∈ R N \ Ω. 1.11
Trang 4Remark 1.4 i If f 1 on Ω, thenTheorem 1.3recovers Gauss formulaseeLemma 2.9.
ii Theorem 1.3 leads to the mean value theorems for harmonic functions see
Remark 5.4
iii If f ∈ C2Ω ∩ C1Ω such that Δf ∈ CΩ, then by combiningTheorem 1.3 and
Proposition 2.7, we regain the Green-Riemann representation formula1.2
This paper is organized as follows In Section 2, we include some known results that are necessary later in the paper Section 3 is dedicated to the proof of Theorem 1.3 Based
on it, we prove Theorem 1.1 in Section 4 We conclude the paper with a representation of
smooth functions in W 1,p Ω with p > N in terms of the integral mean value over the domain
see Theorem 5.1in Section 5 As a byproduct of our main results, we obtain a sharp esti-mate of the difference between the value of a function f and the double-layer potential with
moment f.
2 Preliminaries
Lemma 2.1 see 4, Theorem IV.9 Let ω ⊂ RN be an open set Let h n be a sequence in L p ω,
1≤ p ≤ ∞, and let h ∈ L p
n L p ω → 0.
a h n k x → hx a.e in ω,
b |h n k x| ≤ ϕx for all k, a.e in ω.
For fixed y∈ RN, we define the operatorKj by
Kj u
y
Ω
x j − y j
|x − y| N u xdx, j ∈ {1, 2, , N}. 2.1
Lemma 2.2 i If 1 ≤ p ≤ N, then the operator Kj : L p Ω → L p Ω is compact.
ii If p > N, then the operator K j : L p Ω → CΩ is compact.
0 Ω with p ≥ 1, then 1.7 holds
Indeed, Ex given by 1.1 has weak derivatives and ∂/∂x j Ex − y 1/ω n x j − y j /|x −
y|N for every j ∈ {1, 2, , N} If f ∈ C∞
0 Ω, then by the definition of weak derivatives, we have
ΩE x − yΔfxdx −N
j1
Ω
∂E x − y
∂x j
∂f
Ω
∇Ex − y, ∇fxdx. 2.2
Thus, using 1.3, we find 1.7 for every y ∈ Ω Now, if f ∈ W01,pΩ, we take a sequence
f nn≥1in C∞0 Ω such that f n → f in W 1,p Ω as n → ∞ Thus, for each f n with n≥ 1, we have
f n y − 1
N
j1
Kj
∂f
n
∂x j
Trang 5
By Lemma 2.2, each operatorKj is compact from L p Ω to L p Ω Thus, ∂f n /∂x j → ∂f/∂x j
in L p Ω as n → ∞ implies that K j ∂f n /∂x j → Kj ∂f/∂x j in L p Ω as n → ∞ By Lemma
2.1, we have up to a subsequence of f n limn→∞Kj ∂f n /∂x j y K j ∂f/∂x j y and
limn→∞f n y fy a.e y ∈ Ω since f n → f in L p Ω as n → ∞ By passing to the limit in
2.3, we conclude 1.7
Lemma 2.4 see 5, Lemma 5.47 Let y ∈ RN and let ω be a domain of finite volume inRN
If 0 ≤ γ < N, then
ω
where the constant K depends on γ and N but not on y or ω.
By a vector field, we understand an RN-valued function on a subset of RN If Z
ω is defined by
div ZN
i1
∂z i
Proposition 2.5 the divergence theorem If ω ⊂ R N is a bounded domain with C1boundary and
ω div Zydy
∂ω
If ω is a domain to which the divergence theorem applies, then we have the following.
Proposition 2.6 Green’s first identity If u, v ∈ C2ω ∩ C1ω, then the following holds:
ω
v
ω
∇ux, ∇vxdx
∂ω
∂ν xdσx. 2.7
Proposition 2.7 Let Ω be a bounded domain with C1 boundary If f ∈ C2Ω ∩ C1Ω such that
Δf ∈ CΩ, then for every y ∈ R N \ ∂Ω, one has
Ω
∇Ex − y, ∇fxdx
∂Ω
∂f
∂ν xEx − ydσx −
ΩE x − yΔfxdx. 2.8
Proposition 2.6applied on Ω \ B y, we find
Ω\B y E x − yΔfxdx
∂Ω
∂f
∂ν xEx − ydσx −
∂B y
∂f
∂ν xEx − ydσx
−
Ω\B y
∇fx, ∇Ex − ydx.
2.9
Trang 6SinceΔf ∈ CΩ and f ∈ C1Ω, we have that x → Ex − yΔfx and x →Ω∇Ex −
y , ∇fxdx are integrable on Ω We see that
∂B y
∂f
Indeed, for some constant C > 0, we have
⎧
⎪
⎪
⎨
⎪
⎪
⎩
1
2π
∂B y
∂f
∂ν x ln |x − y|
1
ω N N − 2
∂B y
∂f
∂ν x |x − y| dσ x N−2 ≤ Cω N ≥ 3.
2.11
→ 0 in 2.9 and using 2.10, we obtain 2.8
We next give some properties of the double-layer potential u h y defined by 1.4 see
1
Proposition 2.8 If h is a continuous function on ∂Ω, then
i u h y given by 1.4 is well defined for all y ∈ R N ,
ii Δu h y 0 for all y ∈ R N \ ∂Ω.
Lemma 2.9 Let v be the double-layer potential with moment h ≡ 1, that is,
v y
∂Ω
∂E
Then, one has
v y
⎧
⎪
⎨
⎪
⎪
1 if y ∈ Ω,
1
2 if y ∈ ∂Ω,
0 if y∈ RN \ Ω.
2.13
Proposition 2.10 If h is continuous on ∂Ω and y0∈ ∂Ω, then
lim
Ωy→y0
u h y 1
2h
y0
h
y0
RN \Ωy→y0
u h y −1
2h
y0
h
y0
Indeed, by Propositions2.8and2.10, the function ϕ : Ω → R defined by ϕy u h y for y ∈ Ω and ϕy0 1/2hy0 h y0 for y0 ∈ ∂Ω is continuous on Ω It follows that
u h ∈ C∂Ω and ϕ ∈ L∞Ω But ϕ ≡ u honΩ so that u h ∈ L∞Ω Thus, for each 1 ≤ m < ∞, we
have
Ω u h m dx≤u hm
which shows that u h ∈ L mΩ
Trang 7Definition 2.12 A Lipschitz domainor domain with Lipschitz boundary is a domain in RN
whose boundary can be locally represented as the graph of a Lipschitz continuous function Many of the Sobolev embedding theorems require that the domain of study be a Lips-chitz domain All smooth and many piecewise smooth boundaries are LipsLips-chitz boundaries
Proposition 2.13 see 2, Theorem 7.26 Let ω be a Lipschitz domain in RN If N < p < ∞, then
W 1,p ω is continuously embedded in C 0,α ω with α 1 − N/p.
Proposition 2.14 see 2, page 155 If ω is a Lipschitz domain, then C∞ω is dense in W 1,p ω
3 Proof of Theorem 1.3
SinceΩ is bounded, we can assume without loss of generality that p < ∞.
F x f x − fy∇Ex − y f x − fy
ω N |x − y| N x − y. 3.1
Note that F / ∈ C1
B y, respectively, B a i a i ∈ A\{y}, is contained within Ω and every two such closed balls are disjoint Therefore, F ∈ C1D ∩ CD , where D Ω \ i ∈I B a i ∪ B y.
UsingProposition 2.5, we arrive at
D
div Fdx
∂Ω
f x − fy ∂E
∂ν x − ydσx − N1−1−α
∂B y
f x − fy
ω N |x − y| α dσ x
− 1
i ∈I,a i / y
∂B a i
f x − fy
|x − y| N
x − y, x − a i
We see that
lim
→0
1
N −1−α
∂B y
f x − fy
Indeed, byProposition 2.13, there exists a constant L > 0 such that
0≤ N1−1−α
∂B y
f x − fy
|x − y| α dσ x
≤ N L −1−α
∂B y dσ x Lω N α
3.4
Notice that, for each i ∈ I with a i / y, there exists a constant C i > 0 such that
f x − fy i |x − y| N−1, ∀x ∈ B a i
3.5
Trang 8since y/∈B a i Hence, if i ∈ I such that a i / y, then
∂B a i
f x − fy
|x − y| N
x − y, x − a i
∂B a i
f x − fy
|x − y| N−1 dσ x ≤ C i ω N N−1. 3.6
By3.2–3.6 and Gauss lemma, it follows that
lim
→0
D div Fxdx
∂Ω
f x − fy ∂E
∂ν x − ydσx
∂Ωf x ∂E
∂ν x − ydσx − fy.
3.7
Recall that x → Ex − y is harmonic on R N \ {y} Thus, from 3.1, we derive that
div Fx ∇fx, ∇Ex − y, ∀x ∈ D 3.8 FromLemma 2.2ii, we know that
Ω
∇Ex − y, ∇fxdx is continuous on Ω. 3.9
From3.7 and 3.8, we find
Ω
∇fx, ∇Ex − ydx lim
→0
D div Fxdx
∂Ωf x ∂E
∂ν x − ydσx − fy, 3.10 which concludes the proof of1.10 for y ∈ Ω.
3.9 and the continuity of f on Ω, we obtain
f y lim
Ωt→y f t lim
Ωt→y u f t −
Ω
∇Ex − y, ∇fxdx. 3.11 FromProposition 2.10, we know that
lim
Ωt→y u f t f y
By combining3.11 and 3.12, we attain 1.10
We define the vector field Z :Ω → RNby
Z x fx∇Ex − y f x
ω N |x − y| N x − y, ∀x ∈ Ω. 3.13
Trang 9Clearly, Z ∈ C1 a i ⊂ Ω for every i ∈ I and B a i∩
B a j ∅ for all i, j ∈ I with i / j Set Ω : Ω \ i ∈I B a i By applyingProposition 2.5to
Z :Ω → RN, we obtain
Ω div Zxdx
∂Ωf x ∂E
∂ν x − ydσx − 1
i ∈I
∂B a i
f xx − y, x − a i
|x − y| N dσ x.
3.14
If M i dist y, B a i , then M i > 0 for every i ∈ I since y/∈Ω Hence, for each i ∈ I,
∂B a i
f xx − y, x − a i
|x − y| N dσ x
∂B a i
|x − y| N−1dσ x ≤ L∞Ω
N−1. 3.15
By3.14 and 3.15, it follows that
lim
→0
Ω div Zxdx
∂Ωf x ∂E
Note that x → |x − y|1−N is continuous onΩ By H¨older’s inequality, x → ∇fx, ∇Ex − y
is integrable onΩ Since x → Ex − y is harmonic on R N \ {y}, we find
div Zx ∇fx, ∇Ex − y, ∀x ∈ Ω 3.17 Therefore, using3.16, we obtain
Ω
∇fx, ∇Ex − ydx lim
→0
Ω div Zxdx
∂Ωf x ∂E
∂ν x − ydσx. 3.18 This completes the proof ofTheorem 1.3
4 Proof of Theorem 1.1
As before, we can assume that g ∈ W 1,p Ω with p < ∞ ByProposition 2.14, there exists a
sequence g n ∈ C∞Ω such that g n → g in W 1,pΩ, that is,
lim
n→∞g n − g
L pΩ 0, lim
n→∞
∂g n
∂x i
L pΩ 0, ∀i ∈ {1, 2, , N}. 4.1 FromLemma 2.1, we know that, up to a subsequencerelabeled g n,
Since C1Ω ⊆ W 1,q Ω for every q ≥ 1, we can applyTheorem 1.3to each g nand obtain
∂Ωg n x ∂E
∂ν x − ydσx −
Ω
∇Ex − y, ∇g n xdx
g n y, ∀y ∈ Ω,
0, ∀y ∈ R N \ Ω. 4.3
Trang 10Using the definition ofKjin2.1, we write
Ω
∇Ex − y, ∇g n xdx 1
N
j1
Ω
x j − y j
|x − y| N
∂x j xdx 1
N
j1
Kj
∂g
n
∂x j
y. 4.4 From4.1 andLemma 2.2, it follows that for every j ∈ {1, 2, , N},
lim
n→∞
Kj
∂g
n
∂x j
− Kj
∂g
∂x j
L pΩ
0 if 1 ≤ p ≤ N,
Kj
∂g
n
∂x j
−→ Kj
∂g
∂x j
in CΩ as n −→ ∞ if p > N. 4.5
Hence, passing eventually to a subsequencedenoted again by g n, we have
lim
n→∞Kj
∂g
n
∂x j
y K j
∂g
∂x j
y a.e y ∈ Ω, ∀j ∈ {1, 2, , N}. 4.6 This, jointly with4.4, implies that
lim
n→∞
Ω
∇Ex − y, ∇g n xdx
Ω
∇Ex − y, ∇gxdx a.e y ∈ Ω. 4.7
Hence, passing to the limit n→ ∞ in 4.3 and using 4.2, we reach 1.8
denote the conjugate exponent to p 1 By H¨older’s inequality,
≤ 1
Ω
dx
|x − y| N−1p
1/p
Ω n − gx p
dx
1/p
.
4.8
Thus, using4.1 andLemma 2.4, we infer that
lim
n→∞
Ω
∇Ex − y, ∇g n xdx
Ω
∇Ex − y, ∇gxdx, ∀y ∈ R N \ Ω. 4.9
Letting n→ ∞ in 4.3, we conclude 1.9 This finishes the proof ofTheorem 1.1
5 Other results and applications to inequalities
If f : a, b → R is absolutely continuous on a, b, then the Montgomery identity holds:
f x 1
b
a
b
a
p t, xftdt for x ∈ a, b, 5.1