Mathematical Inequalities - Chapter 4 ppsx

Poincaré- and Sobolev-Type Inequalities4.1 Introduction In the development of the theory of partial differential equations and in ing the foundations of the finite element analysis, the

Poincaré- and Sobolev-Type Inequalities

4.1 Introduction

In the development of the theory of partial differential equations and in ing the foundations of the finite element analysis, the fundamental role played bycertain inequalities and variational principles involving functions and their partialderivatives is well known In particular, the integral inequalities originally due

establish-to Poincaré and Sobolev and their various generalizations and variants have beenextensively used in the study of problems in the theory of partial differential equa-tions and finite element analysis Because of the dominance of such inequalities inthe qualitative analysis of partial differential equations and in finite element analy-sis, numerous studies have been made of various types of new inequalities related

to Poincaré- and Sobolev-type inequalities These investigations have achieved adiversity of desired goals Over the years a number of papers have appeared inthe literature which deals with the far-reaching generalizations, extensions andvariants of Poincaré and Sobolev inequalities and their various applications Thischapter deals with a number of new inequalities recently discovered in the litera-ture which claim their origin to the inequalities of Poincaré and Sobolev.Let R be the set of real numbers and B be a bounded domain in R n,

the n-dimensional Euclidean space, defined by B=#n

i=1[a i , b i ] For x i ∈ R,

x = (x1, , x n ) is a variable point in B and dx = dx1· · · dx n For any

con-tinuous real-valued function u(x) defined on B, we denote by B u(x) dx the n-fold integral b n

a n· · · b1

a1 u(x1, , x n ) dx1· · · dx n The notation b i

a i u(x1, , t i , , x n ) dt i for i = 1, , n we mean, for i = 1, it is b1

a1 u(t1, x2, , x n ) dt1and

so on, and for i = n, it is b n

a n u(x1, , x n−1, t n ) dt n For any continuous

real-valued function u(x) defined onRn, we denote by i u(x1, , t i , , x n ) dt i theintegral ∞

−∞u(x1, , t i , , x n ) dt i , i = 1, , n, taken along the whole line

381

through x = (x1, , x i , , x n ) parallel to the x i-axis, and denote by Rn u(x) dx

the n-fold integral ∞

−∞· · · −∞∞ u(x1, , x n ) dx1· · · dx n For any function u(x) defined on B orRn, we define| grad u(x)| = ( n

i=1|∂u(x)

∂x i |2) 1/2 We say that a

function is of compact support in S if it is nonzero only on a bounded main S
of the domain S, where S
lies at a positive distance ∂S, the boundary

subdo-of S We assume without further mention that all the integrals exist on the

respec-tive domains of their definitions

4.2 Inequalities of Poincaré, Sobolev and Others

There exists a vast literature on the various generalizations, extensions and ants of Poincaré’s inequality (10), see Introduction We start with the followinguseful version of Poincaré’s inequality given in Friedman [120, p 284]

2

+n

2σ2

Taking square on both sides of (4.2.2) and using the elementary inequality

In [247] Pachpatte has given the following variant of Theorem 4.2.1.

THEOREM4.2.2 Let Q be as defined in Theorem 4.2.1 and f, g be real-valued

functions belonging to C1(Q) Then



+n

4σ2

Writing (4.2.5) for the functions f and g, and then by multiplying the results and using the elementary inequalities ab1

2(a2+ b2), ( n

i=1a i )2 n n

i=1a2i (a, b, a i are reals) and Schwarz inequality, we obtain

f (x)g(x) + f (y)g(y) − f (x)g(y) − f (y)g(x)

The desired inequality (4.2.4) follows from inequality (4.2.7) 

REMARK4.2.1 We note that in the special case when g(x) = f (x), the

inequal-ity established in Theorem 4.2.2 reduces to the inequalinequal-ity given in Theorem 4.2.1

In [236] Pachpatte has established the following Poincaré-type inequality.

THEOREM4.2.3 Let Q be as defined in Theorem 4.2.1 and f, g be real-valued

dt i (4.2.11)Similarly, we obtain

dt i (4.2.12)

From (4.2.11), (4.2.12) and using the elementary inequalities ab1

2(a2+ b2), ( n

The proof is complete 

REMARK4.2.2 In the special case when g(x) = f (x), the inequality established

in Theorem 4.2.3 reduces to the following Poincaré-type integral inequality

Inequality (4.2.15) is known as Sobolev’s inequality, although the same name

is attached to the above inequality in n-dimensional Euclidean space

Inequal-ities of the form (4.2.15) or its variants have been applied with considerablesuccess to the study of many problems in the theory of partial differential equa-tions and in establishing the foundations of the finite element analysis There is avast literature which deals with various generalizations, extensions and variants ofinequality (4.2.15)

In 1964, Payne [362] has given the following version of inequality (4.2.15)

THEOREM4.2.4 Let u(x, y) be any smooth function of compact support in

∂t u(x, t ) dt. (4.2.18)

From (4.2.17) and (4.2.18), we obtain

THEOREM 4.2.5 Let E be a bounded domain inRn , n
2, and u be a

real-valued function such that u ∈ C1(E) and u = 0 on ∂E, the boundary of E, then

PROOF From the hypotheses, we have the following identities

dt1. (4.2.26)Similarly, we obtain

n/(n −1)

1

∂t ∂1u(t1, x2, , x n )

dt n

1/(n −1) (4.2.28)

We integrate both sides of (4.2.28) with respect to x1and use on the right-handside the general version of Hölder’s inequality (see [179, p 40])

dx1/(n −1)· · ·



E

∂x ∂ n u(x)

dx1/(n −1)

(4.2.30)

From (4.2.30) and using the elementary inequalities

dx1/n· · ·



E

∂x ∂ n u(x)

21/2 dx



i=1

∂x ∂ i u(x)

2

1/2 dx

REMARK 4.2.3 We note that on employing Schwarz inequality on the hand side of (4.2.23) we get the following inequality

where V (D) is the n-dimensional measure of E By taking n = 3 and u = φ2in(4.2.23) and using the Schwarz inequality, we obtain

In 1991, Pachpatte [290] has established the following inequality

THEOREM4.2.6 Let u be a real-valued sufficiently smooth function of compact

PROOF First we establish inequality (4.2.33) for p= 0, q = 1 and by taking u(x) = v(x) Since v(x) is a smooth function of compact support in E, we have

the following identities

dt1. (4.2.36)

for i = 2, , n Now, by following exactly the same steps as in the proof of

Theorem 4.2.5 below inequality (4.2.27), we obtain

If E |u(x)| (p +q)n/(n−q) dx= 0 then (4.2.33) is trivially true; otherwise, we divide

both sides of (4.2.40) by { |u(x)| (p +q)n/(n−q) dx}(q −1)/q and then raise both

sides to the power q and use the elementary inequality

the right-hand side to get (4.2.33) The proof is complete 

REMARK4.2.4 By taking p = 0, q = 2 and n
3 in (4.2.33) and then raising

the power 1/2 on both sides of the resulting inequality, we get

Further, by taking p = 1, q = 1 in (4.2.33) and raising the power 1/2 on both

sides of the resulting inequality, we get

inequal-4.3 Poincaré- and Sobolev-Type Inequalities I

The importance of the Poincaré and Sobolev inequalities in the theory of partialdifferential equations is well known, and over the years much effort has been de-voted to the study of these inequalities In this section we present some Poincaré-and Sobolev-type inequalities established by Pachpatte in [249,265,290]

In 1987, Pachpatte [265] established the following Poincaré-type inequality

THEOREM 4.3.1 Let p
2 be a constant and B =#n

i=1[0, a i ] be a bounded

domain inRn Let u be a real-valued function belonging to C1(B) which vanishes

on the boundary ∂B of B Then

PROOF From the hypotheses, we have the following identities

From (4.3.2) and (4.3.3), we observe that

dt i (4.3.4)From (4.3.4) and using the elementary inequality (see [79,211])

p

α p−1



α n

p dx



p/2 dx

This result is the desired inequality in (4.3.1) and the proof is complete 

The following theorem deals with the Poincaré-type inequality which is an tegral analogue of the discrete inequality given by Pachpatte in [242, Theorem 1]

in-THEOREM4.3.2 Let u, p, B, α be as in Theorem 4.3.1 Then

∂t i u(x1, , t i , , x n )

p/(p −1) dt i

(4.3.8)

Integrating both sides of (4.3.8) with respect to x1, , x n on B, using the tion of α and inequality (4.3.5) we observe that

From (4.3.9) and using Hölder’s inequality with indices 2(p − 1)/p, 2(p −

1)/(p − 2) and the definition of α we observe that

This result is the required inequality in (4.3.7) and the proof is complete 

The following theorem established in [265] deals with the Sobolev-type equality

in-THEOREM4.3.3 Let p
1 be a constant and u, B, α be as in Theorem 4.3.1.

2

dx

1/2

This result is the desired inequality in (4.3.10) and the proof is complete 

The following variant of Sobolev’s inequality is established in [290]

THEOREM 4.3.4 Let p
0, q
1 be constants and u, B, α be as in

q dx. (4.3.15)PROOF From the hypotheses, we have the following identities

dt i (4.3.18)

Integrating both sides of (4.3.18) over B and using the definition of α and

rewrit-ing the resultrewrit-ing inequality we have

If B |u(x)| p +q dx= 0 then (4.3.15) is trivially true; otherwise, we divide both

sides of (4.3.20) by { B |u(x)| p +q dx}(q −1)/q and then raise both sides to the

power q and use the elementary inequality ( n

i=1c i ) k  n k−1 n

i=1c k i (for c i
0

reals and k
1) to get (4.3.15) The proof is complete 

REMARK 4.3.1 We note that, in the special cases when (i) p = 2, q = 2 and

(ii) p= 0, inequality (4.3.15) reduces, respectively, to the following inequalities

THEOREM4.3.5 Let p, q
1 be constants and B =#n

i=1[a i , b i ] be a bounded

domain inRn Let u, v be sufficiently smooth functions defined on B which vanish

on the boundary ∂B of B Then

REMARK 4.3.2 In the special cases when p = q = 1 and a i = 0,

inequal-ities (4.3.23) and (4.3.24) reduce to the Poincaré-type inequality given inTheorem 4.2.3

PROOFS OF THEOREMS 4.3.5 AND 4.3.6 From the hypotheses of rem 4.3.5, we have the following identities

From (4.3.25) and (4.3.26), we observe that

dt i (4.3.27)

From (4.3.27) and using inequality (4.3.5), Hölder’s inequality with indices p,

p/(p − 1) (see [74, p 126]) and the definition of α, we obtain

2q dt i

(4.3.30)

Integrating both sides of (4.3.30) with respect to x1, , x n on B, using the nition of α and a suitable version of inequality (4.3.5) we get

2q

1/q q dx

The proof of Theorem 4.3.5 is complete

From the hypotheses of Theorem 4.3.6, for any x in B, we have the following

dt i (4.3.34)

From (4.3.33), (4.3.34) and using the elementary inequality cd 1

∂t i u(x1, , t i , , x n )

2dt i (4.3.35)

Integrating both sides of (4.3.35) with respect to x1, , x n on B, using the tion of α and Hölder’s inequality on the right-hand side with indices p, p/(p − 1)

defini-and q, q/(q − 1) (see [74, p 126]) we obtain

4.4 Poincaré- and Sobolev-Type Inequalities II

In the recent past, several authors have presented numerous integral inequalities

of Poincaré and Sobolev type In this section we present some Poincaré- andSobolev-type inequalities investigated by Pachpatte in [237,246]

In 1986, Pachpatte [237] has established the following inequalities of thePoincaré and Sobolev type, involving functions of several independent variables

THEOREM 4.4.1 Let B=#n

i=1[0, a i ] be a bounded domain in R n , n
3 Let

1 p < Q and u be a real-valued function belonging to C1(B) which vanishes

on the boundary ∂B of B Then

(Q −p)/(pQ)


B∇u(x)

Q

Q dx

REMARK4.4.1 In the special case when Q = 2 and p = n/(n − 1) (for n
3),

we see that 1 < p < Q holds and inequality (4.4.1) reduces to

THEOREM 4.4.2 Let p
1, P, Q > 1, P−1+ Q−1= 1, B be as in

Theo-rem 4.4.1 and u be a real-valued function belonging to C1(B) which vanishes on

the boundary ∂B of B Then

1/Q

where ( ∇u(x) Q ) Q is as defined in Theorem 4.4.1.

REMARK 4.4.2 By taking p = n/(n − 1) (for n
3), P = Q = 2 in (4.4.3)

and then squaring on both sides of the resulting inequality we have the followinginequality

PROOFS OFTHEOREMS4.4.1AND 4.4.2 If u ∈ C1(B), then we have the

From (4.4.6) and (4.4.7), we obtain

dt i (4.4.8)From (4.4.8) and using the elementary inequality (see [3, p 338])

p (4.4.10)

for any p
1 Applying Hölder’s inequality with indices P , Q > 1 (P−1+

Q−1= 1) to each integral in (4.4.10) we get

Q dt i

p/Q (4.4.11)

Integrating both sides of (4.4.11) over B we get

Q dt i

p/Q dx.

(4.4.12)

Now, applying Hölder’s inequality with indices p1= Q/(Q − p), q1= Q/p to

each integral on the right-hand side in (4.4.12), we get

Q dx

p/Q

Now, applying Hölder’s inequality to the sum on the right-hand side in (4.4.13)

with indices p1 and q1again, we obtain

Q dx

p/Q (4.4.14)

1/Q

.

The proof of Theorem 4.4.1 is complete

From the hypotheses of Theorem 4.4.2, if u ∈ C1(B) then we have the

Applying Hölder’s inequality with indices P , Q > 1 (P−1+ Q−1= 1) to each

integral on the right-hand side in (4.4.18) we get

Q dx

1/Q

.

Now, applying Hölder’s inequality to the sum on the right-hand side in the above

inequality with the same indices P , Q, we obtain

THEOREM4.4.3 Let u r , r = 1, , m, be sufficiently smooth functions of

REMARK4.4.3 We note that in the special case when m = 1, u1= u, inequality

(4.4.19) reduces to the following inequality

THEOREM4.4.4 Let u r , r = 1, , m, be sufficiently smooth functions of

com-pact support in E, the n-dimensional Euclidean space with n
2, and let p r
1

On taking n = 2 and p1= 2 in (4.4.21) and squaring both sides of the resulting

inequality, we obtain the sharpened version of Sobolev’s inequality established byPayne in [362]

The next two theorems established in [246] deal with the Poincaré- andSobolev-type inequalities in which the constants appearing depend on the size

of the domain of definitions of the function

THEOREM4.4.5 Let B=#n

i=1[a i , b i ] be a bounded domain in R n with n
2,

u r , r = 1, , m, be sufficiently smooth functions defined on B which vanish on

where α is as defined in Theorem 4.4.5.

REMARK 4.4.5 In the special case when m= 1, inequalities (4.4.22) and

(4.4.23) reduce respectively to the following Poincaré- and Sobolev-type ities

dx, p1
2, (4.4.24)

For similar inequalities, see [73,120,121,152–157,178,179,418].

PROOFS OF THEOREMS 4.4.3 AND 4.4.4 From the hypotheses of rem 4.4.3, we have the following identities

dt1. (4.4.28)Similarly, we obtain

n/(n −1)

1

∂t ∂1u r (t1, x2, , x n )

(for c i nonnegative reals and k
1), we obtain

dx1/(n −1)· · ·



E

∂x ∂ n u1(x)

dx1/(n −1)

dx1/(n −1)· · ·



E

∂x ∂ n u m (x)

dx1/(n −1)

(4.4.33)

From (4.4.33) and using inequalities (4.3.5), (4.4.31) and the inequality

dx1/n· · ·



E

∂x ∂ n u1(x)

dx1/n

dx1/n· · ·



E

∂x ∂ n u m (x)

From the assumptions of Theorem 4.4.4, we have the following identities

for r = 1, , m From (4.4.39) and inequality (4.4.31), we obtain

∂x n

u1(x) dx

PROOFS OF THEOREMS 4.4.5 AND 4.4.6 From the hypotheses of

Theo-rem 4.4.5, since u r (x) are smooth functions defined on B which vanish on the

boundary ∂B of B, we have the following identities

for r = 1, , m From (4.4.42) and (4.4.43), we observe that

dt i (4.4.44)From (4.4.44) and on using inequality (4.3.5), Hölder’s inequality with indices

p r , p r /(p r − 1) and the definition of α, we obtain

× α



B

 ∂x ∂1u1(x)

The proof of Theorem 4.4.5 is complete

From the assumptions on the functions u r (x) in Theorem 4.4.6, we have the

dt i



. (4.4.49)

From (4.4.49) and inequality (4.4.31), we obtain

dt i



Integrating both sides of (4.4.50) over B, using the definition of α, the Schwarz

inequality and inequality (4.4.34) we have

4.5 Inequalities of Dubinskii and Others

Integral inequalities of Poincaré and Sobolev type play a fundamental role inthe theory and applications of partial differential equations A large number ofinequalities related to these inequalities are established by several authors inthe literature In this section we deal with certain inequalities established byDubinskii [95], Alzer [10] and Pachpatte [345]

In what follows, we let x = (x1, , x n ) be a variable point in Rn, an

n-dimensional Euclidean space, G be a bounded region in Rn with

bound-ary ∂G satisfying the cone condition (see [95]), C m (G) is the space of functions u(x) with bounded derivatives in G (the closure of G) up to order m inclusive,

dx = dx1· · · dx n is the volume element, and ds is the surface element sponding to ∂G Constant quantities, not depending on u(x), will be denoted by the symbol K In different inequalities their meaning will be different.

corre-The inequalities in the following theorems are established by Dubinskii [95]

THEOREM 4.5.1 Let −∞ < α0< +∞, α1
1, u(x), |u(x)| α0+α1 ∈ C1(G).

Then the following inequality is valid



|u| α0+α1dx  K



|u| α0 ∂x ∂u α1dx+



|u| α0+α1ds (4.5.1)

for i = 1, , n The constant K depends on α0, α1and G.

PROOF From the divergence theorem we have

REMARK4.5.1 We note that, for the case when α0, α1are even and u|∂G= 0,

inequality (4.5.1) was obtained earlier by Visik [421]

THEOREM 4.5.2 Let −∞ < α0< +∞, α1
0, α2
0, α1+ α2
1, u(x),

|u(x)| α0+α1+α2∈ C1(G) Then the following inequality is valid



∂G |u| α0+α1+α2ds (4.5.5)

for i = 1, , n.

The proof follows by estimating the integral on the left-hand side of (4.5.5), by

using Young’s inequality with index p = (α1+ α2)α−1and Theorem 4.5.1.

From (4. 4 .49 ) and inequality (4. 4.31), we obtain

dt i



Integrating both sides of (4. 4.50) over B, using the... Theorem 4. 4.5 is complete

From the assumptions on the functions u r (x) in Theorem 4. 4.6, we have the

dt i



. (4. 4 .49 )...

inequality and inequality (4. 4. 34) we have