Mathematical Inequalities - Chapter 4 ppsx

Chapter 4 Poincaré- and Sobolev-Type Inequalities 4.1 Introduction In the development of the theory of partial differential equations and in establish- ing the foundations of the finite element analysis, the fundamental role played by certain inequalities and variational principles involving functions and their partial derivatives is well known. In particular, the integral inequalities originally due to Poincaré and Sobolev and their various generalizations and variants have been extensively 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 in the 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 a diversity of desired goals. Over the years a number of papers have appeared in the literature which deals with the far-reaching generalizations, extensions and variants of Poincaré and Sobolev inequalities and their various applications. This chapter 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 ].Forx i ∈ R, x = (x 1 , ,x n ) is a variable point in B and dx = dx 1 ···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 ···  b 1 a 1 u(x 1 , ,x n ) dx 1 ···dx n . The notation  b i a i u(x 1 , ,t i , ,x n ) dt i for i =1, ,n we mean, for i =1, it is  b 1 a 1 u(t 1 ,x 2 , ,x n ) dt 1 and so on, and for i = n,itis  b n a n u(x 1 , ,x n−1 ,t n ) dt n . For any continuous real- valued function u(x) defined on R n , we denote by  i u(x 1 , ,t i , ,x n ) dt i the integral  ∞ −∞ u(x 1 , ,t i , ,x n ) dt i , i = 1, ,n, taken along the whole line 381 382 Chapter 4. Poincaré- and Sobolev-Type Inequalities through x =(x 1 , ,x i , ,x n ) parallel to the x i -axis, and denote by  R n u(x) dx the n-fold integral  ∞ −∞ ···  ∞ −∞ u(x 1 , ,x n ) dx 1 ···dx n . For any function u(x) defined on B or R n , we define |gradu(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 subdo- main S  of the domain S, where S  lies at a positive distance ∂S, the boundary 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 vari- ants of Poincaré’s inequality (10), see Introduction. We start with the following useful version of Poincaré’s inequality given in Friedman [120, p. 284]. T HEOREM 4.2.1. Let Q ={x = (x 1 , ,x n ) ∈ R n :0 x i  σ, i = 1, ,n} and let u be a real-valued function belonging to C 1 (Q). Then  Q u 2 (x) dx  1 σ n   Q u(x) dx  2 + n 2 σ 2  Q   gradu(x)   2 dx. (4.2.1) P ROOF. For any x = (x 1 , ,x n ), y = (y 1 , ,y n ) ∈ Q, the following identity holds u(x) −u(y) = n  i=1  x i y i ∂ ∂t i u(y 1 , ,y i−1 ,t i ,x i+1 , ,x n ) dt i . (4.2.2) Taking square on both sides of (4.2.2) and using the elementary inequality (  n i=1 a i ) 2  n  n i=1 a 2 i , where a i are reals and Schwarz inequality, we have u 2 (x) +u 2 (y) −2u(x)u(y)  nσ n  i=1  σ 0  ∂ ∂t i u(y 1 , ,y i−1 ,t i ,x i+1 , ,x n )  2 dt i . (4.2.3) Integrating (4.2.3) with respect to x 1 , ,x n , y 1 , ,y n , we get 2σ n  Q u 2 (x) dx −2   Q u(x) dx  2  nσ n+2 n  i=1  Q  ∂ ∂x i u(x)  2 dx, from which (4.2.1) follows.  4.2. Inequalities of Poincaré, Sobolev and Others 383 In [247] Pachpatte has given the following variant of Theorem 4.2.1. T HEOREM 4.2.2. Let Q be as defined in Theorem 4.2.1 and f, g be real-valued functions belonging to C 1 (Q). Then  Q f(x)g(x)dx  1 σ n   Q f(x)dx   Q g(x)dx  + n 4 σ 2  Q    gradf(x)   2 +   gradg(x)   2  dx. (4.2.4) P ROOF. For any x,y ∈Q and h ∈ C 1 (Q), the following identity holds h(x) −h(y) = n  i=1  x i y i ∂ ∂t i h(y 1 , ,y i−1 ,t i ,x i+1 , ,x n ) dt i . (4.2.5) Writing (4.2.5) for the functions f and g, and then by multiplying the results and using the elementary inequalities ab  1 2 (a 2 +b 2 ), (  n i=1 a i ) 2  n  n i=1 a 2 i (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)  n 2 σ n  i=1  σ 0  ∂ ∂t i f(y 1 , ,y i−1 ,t i ,x i+1 , ,x n )  2 +  ∂ ∂t i g(y 1 , ,y i−1 ,t i ,x i+1 , ,x n )  2  dt i . (4.2.6) Integrating both sides of (4.2.6) with respect to x 1 , ,x n , y 1 , ,y n , we get 2σ n  Q f(x)g(x)dx − 2   Q f(x)dx   Q g(x)dx   n 2 σ n+2  Q    gradf(x)   2 +   gradg(x)   2  dx. (4.2.7) The desired inequality (4.2.4) follows from inequality (4.2.7).  R EMARK 4.2.1. We note that in the special case when g(x) = f(x), the inequal- ity established in Theorem 4.2.2 reduces to the inequality given in Theorem 4.2.1. 384 Chapter 4. Poincaré- and Sobolev-Type Inequalities In [236] Pachpatte has established the following Poincaré-type inequality. T HEOREM 4.2.3. Let Q be as defined in Theorem 4.2.1 and f, g be real-valued functions belonging to C 1 (Q) which vanish on the boundary ∂Q of Q. Then  Q   f(x)     g(x)   dx  σ 2 8n  Q    gradf(x)   2 +   gradg(x)   2  dx. (4.2.8) P ROOF.Ifx ∈Q, then we have the following identities nf (x ) = n  i=1  x i 0 ∂ ∂t i f(x 1 , ,t i , ,x n ) dt i , (4.2.9) nf (x ) =− n  i=1  σ x i ∂ ∂t i f(x 1 , ,t i , ,x n ) dt i . (4.2.10) From (4.2.9) and (4.2.10), we obtain 2n   f(x)    n  i=1  σ 0     ∂ ∂t i f(x 1 , ,t i , ,x n )     dt i . (4.2.11) Similarly, we obtain 2n   g(x)    n  i=1  σ 0     ∂ ∂t i g(x 1 , ,t i , ,x n )     dt i . (4.2.12) From (4.2.11), (4.2.12) and using the elementary inequalities ab  1 2 (a 2 + b 2 ), (  n i=1 a i ) 2  n  n i=1 a 2 i (for a, b, a i reals) and Schwarz inequality, we obtain   f(x)     g(x)    1 8n σ  n  i=1  σ 0     ∂ ∂t i f(x 1 , ,t i , ,x n )     2 dt i + n  i=1  σ 0     ∂ ∂t i g(x 1 , ,t i , ,x n )     2 dt i  . (4.2.13) Integrating both sides of (4.2.13) with respect to x 1 , ,x n we get  Q   f(x)     g(x)   dx  σ 2 8n  Q    gradf(x)   2 +   gradg(x)   2  dx. 4.2. Inequalities of Poincaré, Sobolev and Others 385 The proof is complete.  R EMARK 4.2.2. 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  Q   f(x)   2 dx  σ 2 4n  Q   gradf(x)   2 dx. (4.2.14) One of the many mathematical discoveries of S.L. Sobolev is the following integral inequality (see [157, p. 101])  ∞ −∞  ∞ −∞ u 4 dx dy  α 2   ∞ −∞  ∞ −∞ u 2 dx dy   ∞ −∞  ∞ −∞ |gradu| 2 dx dy  , (4.2.15) where u(x,y) is any smooth function of compact support in two-dimensional Euclidean space and α is a dimensionless constant. 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 considerable success 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 a vast literature which deals with various generalizations, extensions and variants of inequality (4.2.15). In 1964, Payne [362] has given the following version of inequality (4.2.15). T HEOREM 4.2.4. Let u(x,y) be any smooth function of compact support in two- dimensional Euclidean space E 2 . Then  ∞ −∞  ∞ −∞ u 4 dx dy  1 2   ∞ −∞  ∞ −∞ u 2 dx dy   ∞ −∞  ∞ −∞ |gradu| 2 dx dy  . (4.2.16) P ROOF. From the hypotheses, we have the following identities u 2 (x, y) = 2  x −∞ u(s, y) ∂ ∂s u(s, y) ds =−2  ∞ x u(s, y) ∂ ∂s u(s, y) ds, (4.2.17) u 2 (x, y) = 2  y −∞ u(x, t) ∂ ∂t u(x, t) dt =−2  ∞ y u(x, t) ∂ ∂t u(x, t) dt. (4.2.18) 386 Chapter 4. Poincaré- and Sobolev-Type Inequalities From (4.2.17) and (4.2.18), we obtain u 2 (x, y)   ∞ −∞   u(s, y)       ∂ ∂s u(s, y)     ds (4.2.19) and u 2 (x, y)   ∞ −∞   u(x, t)       ∂ ∂t u(x, t)     dt. (4.2.20) From (4.2.19) and (4.2.20), we observe that  ∞ −∞  ∞ −∞ u 4 (x, y) dx dy   ∞ −∞  ∞ −∞   ∞ −∞   u(s, y)       ∂ ∂s u(s, y)     ds  ×   ∞ −∞   u(x, t)       ∂ ∂t u(x, t)     dt  dx dy. (4.2.21) By using the Schwarz inequality on the right-hand side of (4.2.21), we get  ∞ −∞  ∞ −∞ u 4 (x, y) dx dy    ∞ −∞  ∞ −∞ u 2 (x, y) dx dy  ×   ∞ −∞  ∞ −∞  ∂u ∂x  2 dx dy   ∞ −∞  ∞ −∞  ∂u ∂y  2 dx dy  1/2 . (4.2.22) Now an application of the arithmetic mean and geometric mean inequality on the last term on the right-hand side of (4.2.22) leads to the desired inequality in (4.2.16).  In 1963, Serrin [405] proved the following useful multidimensional integral inequality. T HEOREM 4.2.5. Let E be a bounded domain in R n ,n 2, and u be a real- valued function such that u ∈C 1 (E) and u =0 on ∂E, the boundary of E, then   E |u| n/(n−1) (x) dx  (n−1)/n   1 4n  1/2  E   gradu(x)   dx. (4.2.23) 4.2. Inequalities of Poincaré, Sobolev and Others 387 P ROOF. From the hypotheses, we have the following identities u(x) =  x 1 −∞ ∂ ∂t 1 u(t 1 ,x 2 , ,x n ) dt 1 , (4.2.24) u(x) =−  ∞ x 1 ∂ ∂t 1 u(t 1 ,x 2 , ,x n ) dt 1 . (4.2.25) From (4.2.24) and (4.2.25), we obtain   u(x)    1 2  1     ∂ ∂t 1 u(t 1 ,x 2 , ,x n )     dt 1 . (4.2.26) Similarly, we obtain   u(x)    1 2  i     ∂ ∂t i u(x 1 , ,t i , ,x n )     dt i (4.2.27) for i =2, 3, ,n. From (4.2.26) and (4.2.27), we observe that   u(x)   n/(n−1)   1 2  n/(n−1)   1     ∂ ∂t 1 u(t 1 ,x 2 , ,x n )     dt 1  1/(n−1) ×···×   n     ∂ ∂t n u(x 1 , ,x n−1 ,t n )     dt n  1/(n−1) . (4.2.28) We integrate both sides of (4.2.28) with respect to x 1 and use on the right-hand side the general version of Hölder’s inequality (see [179, p. 40])  i |f 1 ···f k |dk    i |f 1 | k dt  1/k ···   i |f k | k dt  1/k , (4.2.29) where k = n −1. We then integrate the resulting inequality with respect to x 2 and use inequality (4.2.29) on the right-hand side. We repeat this procedure, integrat- ing with respect to x 3 , ,x n , and obtain (see [121, Chapter 1, Theorem 9.3])  E   u(x)   n/(n−1) dx   1 2  n/(n−1)   E     ∂ ∂x 1 u(x)     dx  1/(n−1) ···   E     ∂ ∂x n u(x)     dx  1/(n−1) . (4.2.30) 388 Chapter 4. Poincaré- and Sobolev-Type Inequalities From (4.2.30) and using the elementary inequalities  n  i=1 c i  1/n  1 n n  i=1 c i for nonnegative reals c 1 , ,c n and n  1 and  n  i=1 c i  2  n n  i=1 c 2 i for c 1 , ,c n reals, we obtain   E   u(x)   n/(n−1) dx  (n−1)/n  1 2   E     ∂ ∂x 1 u(x)     dx  1/n ···   E     ∂ ∂x n u(x)     dx  1/n  1 2n n  i=1  E     ∂ ∂x i u(x)     dx = 1 2n  E  n  i=1     ∂ ∂x i u(x)      2  1/2 dx  1 2n  E  n n  i=1     ∂ ∂x i u(x)     2  1/2 dx =  1 4n  1/2  E   gradu(x)   dx. The proof is complete.  R EMARK 4.2.3. We note that on employing Schwarz inequality on the right- hand side of (4.2.23) we get the following inequality   E   u(x)   n/(n−1) dx  (n−1)/n   V(D) 4n  1/2   E   gradu(x)   2 dx  1/2 , (4.2.31) 4.2. Inequalities of Poincaré, Sobolev and Others 389 where V(D) is the n-dimensional measure of E. By taking n = 3 and u = φ 2 in (4.2.23) and using the Schwarz inequality, we obtain   E   φ(x)   3 dx  2/3  3 −1/2   E φ 2 (x) dx  1/2   E   gradφ(x)   2 dx  1/2 and so   E   φ(x)   3 dx   3 −3/4   E φ 2 (x) dx  3/4   E   gradφ(x)   2 dx  3/4 . (4.2.32) In 1991, Pachpatte [290] has established the following inequality. T HEOREM 4.2.6. Let u be a real-valued sufficiently smooth function of compact support in E, the n-dimensional Euclidean space with n  2, and p  0,q 1 and q<n. Then   E   u(x)   (p+q)n/(n−q) dx  (n−q)/n  M n  i=1  E   u(x)   p     ∂ ∂x i u(x)     q dx, (4.2.33) where M = 1 n  (p +q)(n −1) 2(n −q)  q . P ROOF. 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,wehave the following identities v(x) =  x 1 −∞ ∂ ∂t 1 v(t 1 ,x 2 , ,x n ) dt 1 , (4.2.34) v(x) =−  ∞ x 1 ∂ ∂t 1 v(t 1 ,x 2 , ,x n ) dt 1 . (4.2.35) From (4.2.34) and (4.2.35), we obtain   v(x)    1 2  1     ∂ ∂t 1 v(t 1 ,x 2 , ,x n )     dt 1 . (4.2.36) 390 Chapter 4. Poincaré- and Sobolev-Type Inequalities Similarly, we obtain   v(x)    1 2  i     ∂ ∂t i v(x 1 , ,t i , ,x n )     dt i (4.2.37) 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   E   v(x)   n/(n−1) dx  (n−1)/n  1 2n n  i=1  E     ∂ ∂x i v(x)     dx. (4.2.38) This result proves inequality (4.2.33) for p = 0, q = 1 and u(x) = v(x). To prove (4.2.33), we take v(x) =  u(x)  (p+q)(n−1)/(n−q) and hence ∂ ∂x i v(x) = (p +q)(n −1) n −q  u(x)  (p+q)(n−1)/(n−q)−1 ∂ ∂x i u(x) in inequality (4.2.38), and rewriting the resulting inequality we have   E   u(x)   (p+q)n/(n−q) dx  (n−1)/n  (p +q)(n −1) 2n(n −q) n  i=1  E   u(x)   p/q     ∂ ∂x i u(x)       u(x)   (p+q)(n−1)/(n−q)−1−p/q dx. (4.2.39) Using Hölder’s inequality with indices q, q/(q − 1) on the right-hand side of (4.2.39) we obtain   E   u(x)   (p+q)n/(n−q) dx  (n−1)/n  (p +q)(n −1) 2n(n −q) n  i=1   E   u(x)   p     ∂ ∂x i u(x)     q dx  1/q ×   E   u(x)   (p+q)n/(n−q) dx  (q−1)/q . (4.2.40) 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 {  E |u(x)| (p+q)n/(n−q) dx} (q−1)/q and then raise both [...]... inequalities of the forms (4. 4 .4) and (4. 4.5), see [152,155] (4. 4.5) 40 4 Chapter 4 Poincar - and Sobolev-Type Inequalities P ROOFS OF T HEOREMS 4. 4.1 AND 4. 4.2 If u ∈ C 1 (B), then we have the following identities n xi nu(x) = ∂ u(x1 , , ti , , xn ) dti , ∂ti i=1 0 n ai nu(x) = − i=1 xi (4. 4.6) ∂ u(x1 , , ti , , xn ) dti ∂ti (4. 4.7) ∂ u(x1 , , ti , , xn ) dti ∂ti (4. 4.8) From (4. 4.6)... following Poincar - and Sobolev-type inequalities u1 (x) B p1 dx 1 α n 2 p1 grad u1 (x) B p1 dx, p1 2, (4. 4. 24) 40 9 4. 4 Poincar - and Sobolev-Type Inequalities II and u1 (x) (pr +2)/2 dx B p1 + 2 α √ 2 2 n p1 u1 (x) 1/2 1/2 2 grad u1 (x) dx dx B , p1 1 B (4. 4.25) For similar inequalities, see [73,120,121,152–157,178,179 ,41 8] P ROOFS OF T HEOREMS 4. 4.3 AND 4. 4 .4 From the hypotheses of Theorem 4. 4.3, we have... u1 (x) E p1 /2 1/(n−1) ∂ u1 (x) dx ∂xn 1/(n−1) 41 4 Chapter 4 Poincar - and Sobolev-Type Inequalities + ··· + um (x) pm /2 ∂ um (x) dx ∂x1 1/(n−1) pm /2 ∂ um (x) dx ∂xn 1/(n−1) E × ··· × um (x) E (4. 4 .41 ) From (4. 4 .41 ), (4. 3.5), (4. 4.31), the Schwarz inequality and (4. 4. 34) , we obtain ur (x) E (n−1)/n 1/m m ((pr +2)/2)(n/(n−1)) dx r=1 m (n−1)/n 1 m pr + 2 4 r=1 × u1 (x) p1 /2 E × ··· × 1/m ∂ u1 (x) dx... inequality (4. 4.20) and the proof of Theorem 4. 4 .4 is complete P ROOFS OF T HEOREMS 4. 4.5 AND 4. 4.6 From the hypotheses of Theorem 4. 4.5, since ur (x) are smooth functions defined on B which vanish on the boundary ∂B of B, we have the following identities n nur (x) = xi i=1 ai ∂ ur (x1 , , ti , , xn ) dti ∂ti (4. 4 .42 ) ∂ ur (x1 , , ti , , xn ) dti , ∂ti (4. 4 .43 ) and n nur (x) = − bi i=1 xi 41 6 Chapter. .. (4. 4 .42 ) ∂ ur (x1 , , ti , , xn ) dti , ∂ti (4. 4 .43 ) and n nur (x) = − bi i=1 xi 41 6 Chapter 4 Poincar - and Sobolev-Type Inequalities for r = 1, , m From (4. 4 .42 ) and (4. 4 .43 ), we observe that n bi ∂ ur (x1 , , ti , , xn ) dti ∂ti 2n ur (x) i=1 ai (4. 4 .44 ) From (4. 4 .44 ) and on using inequality (4. 3.5), Hölder’s inequality with indices pr , pr /(pr − 1) and the definition of α, we obtain ur... ∂t1 (4. 4.36) for r = 1, , m From (4. 4.35) and (4. 4.36), we obtain ur (x) (pr +2)/2 pr + 2 4 ur (t1 , x2 , , xn ) pr /2 1 ∂ ur (t1 , x2 , , xn ) dt1 ∂t1 (4. 4.37) Similarly, we obtain ur (x) (pr +2)/2 pr + 2 4 ur (x1 , , ti , , xn ) i pr /2 ∂ ur (x1 , , ti , , xn ) dti ∂ti (4. 4.38) for i = 2, , n From (4. 4.37) and (4. 4.38), we observe that ur (x) ((pr +2)/2)(n/(n−1)) pr + 2 4 ×... pr dx, r=1 B (4. 4.22) where α = max{b1 − a1 , , bn − an } T HEOREM 4. 4.6 Let B, ur , r = 1, , m, be as in Theorem 4. 4.5 and pr constants Then 1/m m ur (x) B 1 be (pr +2)/2 dx r=1 α √ 2m n m r=1 1/m pr + 2 2 m × ur (x) pr 1/2 B r=1 1/2 2 grad ur (x) dx dx , (4. 4.23) B where α is as defined in Theorem 4. 4.5 R EMARK 4. 4.5 In the special case when m = 1, inequalities (4. 4.22) and (4. 4.23) reduce respectively... established in [ 246 ] 40 7 4. 4 Poincar - and Sobolev-Type Inequalities II T HEOREM 4. 4.3 Let ur , r = 1, , m, be sufficiently smooth functions of compact support in E, the n-dimensional Euclidean space with n 2 Then ur (x) E (n−1)/n 1/m m n/(n−1) dx i=1 (n−1)/n m 1 1 √ m 4n grad ur (x) dx (4. 4.19) r=1 E R EMARK 4. 4.3 We note that in the special case when m = 1, u1 = u, inequality (4. 4.19) reduces to... × E ∂ u1 (x) dx ∂x1 1/(n−1) ··· E ∂ u1 (x) dx ∂xn 1/(n−1) + E ∂ um (x) dx ∂x1 1/(n−1) ··· E ∂ um (x) dx ∂xn 1/(n−1) (4. 4.33) 41 1 4. 4 Poincar - and Sobolev-Type Inequalities II From (4. 4.33) and using inequalities (4. 3.5), (4. 4.31) and the inequality 2 n ci n 2 ci , n i=1 (4. 4. 34) i=1 where c1 , , cn are reals, we obtain ur (x) E (n−1)/n 1/m m n/(n−1) dx r=1 1 1 2 m (n−1)/n E ∂ u1 (x) dx ∂x1 1/n... x1 ∂ ur (t1 , x2 , , xn ) dt1 , ∂t1 −∞ ∞ ur (x) = − (4. 4.26) ∂ ur (t1 , x2 , , xn ) dt1 , ∂t1 x1 (4. 4.27) for r = 1, , m From (4. 4.26) and (4. 4.27), we obtain 1 2 ur (x) ∂ ur (t1 , x2 , , xn ) dt1 ∂t1 (4. 4.28) ∂ ur (x1 , , ti , , xn ) dti ∂ti (4. 4.29) 1 Similarly, we obtain ur (x) 1 2 i for i = 2, , n From (4. 4.28) and (4. 4.29), we observe that ur (x) n/(n−1) 1 2 n/(n−1) 1 × ··· . get   E   u(x)   2n/(n−1) dx  (n−1)/(2n)  1 √ n   E   u(x)   n  i=1     ∂ ∂x i u(x)     dx  1/2 . (4. 2 .42 ) We note that inequality (4. 2 .41 ) is established by Nirenberg in [229] and inequal- ity (4. 2 .42 ) provides a new estimate on the Nirenberg-type inequality. 4. 3 Poincar - and Sobolev-Type. Theorem 4. 2.1. 3 84 Chapter 4. Poincar - and Sobolev-Type Inequalities In [236] Pachpatte has established the following Poincaré-type inequality. T HEOREM 4. 2.3. Let Q be as defined in Theorem 4. 2.1. ,x n )     p/(p−1) dt i . (4. 3.8) 3 94 Chapter 4. Poincar - and Sobolev-Type Inequalities Integrating both sides of (4. 3.8) with respect to x 1 , ,x n on B, using the defini- tion of α and inequality (4. 3.5) we