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The opial inequality in RN
2001,21B(4):572-576
THE OPIAL INEQUALITY IN R N Deng Yin bin ( ~ ~I'"
)
1
College oj Mathematical and Statistics Science, Wuhan University, Wuhan 490072, China
Abstract
This paper discusses the minimal eigenvalue Al(p, q, 0) and its best estimate
of the following nonlinear eigenvalue problem
Where 0 is a bounded smooth domain in R N , ti.p is the p-laplace operator, p [p, :/!p)' By this result, the famous Opial inequality in R N are deduced.
Key words
Opial inequality, eigenvalue, p-Iaplace
1991 MR Subject Classification
1
2: 2, q E
35JI0, 35J20, 35J60, 35J65
Introduction
In this paper, we discuss the minimal eigenvalue A1(P, q, 0) and its best estimate of the following nonlinear eigenvalue problem uE 0,
Where 0 is a bounded smooth domain in RN,Llpu =
N
L: L(IV'ulp-2Jt!!..) ;=1 ax, ax,
(1)
is the p-laplace
operator, P 2:: 2, q E [p, :!p) are some given real number. By this result, we deduce the famous Opial inequality in R N • As we know, the solvabiliy of problem (1) results from whether the minimum (2)
can be achieved. If (2) is achieved, the attained function r,o(x) is the nontrivial solution of problem (1) corresponding to A A1 (p,q, 0) and A A1 (p, q, 0) is the corresponding minimum eigenvalue. Since q is smaller than the critical Sobolev embedding exponent, the embedding W~,P(o) '---+ Lq(O) is compact. It is easy to verify that the minimum problem (2) can be achieved by using variational methods. So the minimum eigenvalue of problem (1) exists. By
=
=
1 Received September 18,2000. The research supported by Natural Science Fundation of China and the Excellent Teachers Fundation of Ministry of Education of China
573
Deng: THE OPIAL INEQUALITY IN RN
No.4
virtue of isopermetric inequality and co-area formulalll, one can show that there exists a ball
BR(O) = {x E RNllxl < R} with Vol(BR(O)) =Vol(O) such that (3) and the equality holds as 0 = BR(O). In (3), if let p = q = 2, then by using some properties of Bessel function and Sturm comparision theory for ordinary differential equations, we can obtain the best estimate of the first eigenvalue Al(O) for the following eigenvalue problem
-~u = AU
U
E 0,
{ ulan = O. That is there exists R
~
0 with Vol(BR(O)) =Vol(O) such that
A (0) 1
> -
'2(N-2) -2R2
J1
(4)
and the equality holds as 0 = BR(O). Where h( N2"2) is the first zero point of the Bessel function ordered
N
2" 2 [2]•
Remark 1 When 0 is N dimension compact manifold without boundary, the result of Zong and Yang[4] gives us that eigenvalue Al(O) is larger than or equal to ~;. Where d is the diameter of O. Remark 2 If N = 3, then il(N2"2) = il(!) = 11" (ref.[2)[3]). From (4), we have, Al(O) ~ ~ ~ ~ as N = 3. By virtue of Sturm comparsion theory we can deduce that
where d denotes the diameter of O. Our results are quit different from those in [4] if 0 C R N is a bounded domain. Opial inequality was first presented by Z.Opial in 1960[5]. He proved that
l
0
for any
f
h
hlh
If(x)f'(x)ldx::; 1f'(x)1 2dx 40
E CJ([O, h]). Later C.Olech[6], P.R.Bessack[7J and C.L.Mallow[5] simplified the proof
and extended the results. Some different proofs were also given by Z.J .Liang[ll] and X.G.He[lO]. But the Opial inequality in R N , as we know, has not been discussed yet. In this paper, we are going to discuss the Opial inequality in some Sobolev space W~'P(O) with 0 E R N • As a special case, we will show that for any u E W~,2(0), we have:
where il( N 22) is the first zero pint of the Bessel function ordered ball BR(O), whose volume is eaqul to the volume of O.
N
22, R
is the radius of the
574
2
ACTA MATHEMATICA SCIENTIA
Vo1.21 Ser.B
The Best Embedding Constant
In this section, we will discuss the best embedding constant from W~'P(O) to Lq(O). Lemma 1 The minimum problem (2) can be achieved Proof Since q E [P, ~) is smaller than the critical Sobolev exponent. It is easy to verify this lemma by using the standard variational methods and Sobolev compact embedding theorem. Lemma 2 Let >'l(p, q, 0) be the minimum eigenvalue of problem (1), then there exsits a ball BR(O) with Vol(O) =Vol(B.n(O)) such that
The equality holds when 0 = BR(O). Proof The case where P = q = 2 was proved by Fabe-Krahnl'l. Now we are going to prove the case when q 2: P > 2. From Lemma 1 the minimum of >'l(p, q, 0) can be attained by some function tp > O. Since if tp is the attained function of >'l(p,q, 0), so is Itpl, we may assume that tp 2: 0 in O. Let "p(x) be the symmetrization fucntion of tp(x). i.e. 1/, : B R -> R+ Satisfies Vol(tp 2: c) = Vol("p 2: c)
(7)
for any e = const. 2: 0, and "p(x) = "p(r) is a radial function, where r = Ixl and r E [0, R]. It is obvious that for any e 2: 0, the set {x E R N : "p(r) 2: e} is a ball and "p( R) = O. By some calculations, we can obtain
As "p(r) = e =const., we have IV"p1 = So Area{"p = e} =
1.p=c
I~I
= const.
[1.p=c
1dx =
.!.
IV"pIP-1dX]
p
[1.p=c
l.=..!.
1 I] V"p
_I
P
(9)
By using the Isoperimetric inequality[l) we get ~
(i=c IVtpIP-ldX) (i=c l;tp!dX) P
= Area{tp = e}
2: Area{"p = e} =
1.p=c
l.=..!. P
2:
i=c 1dx
1dx
(10)
-- [1.p=c IV"p1P-l];; [1.p=c IV"pldx 7 1
dz
]
On the other hand, form co-area formula[l) d(Vol{tp > e}) = de and
1
rp=c
_l_
IVtpl
dx ,
d(Vol{tp 2: e}) de
--'---7-=-"":"':'"
d(Vol{"p de
> e}) =
2: e} = dVol{'ljJ de
1
rp=c
_l_
IV"p1
dx
575
Deng: THE OPIAL INEQUALITY IN R N
No.4
)P-I - (1 (1
Thus
1
1 --dx
1/J=c IY''f/I I
)P-I
'
from (10), we have
i=c lY'cplp-1dx ~ i=c 1Y''f/Ilp-1dx.
(11)
By using co-area formula again, we deduce that
(12)
From (8) and (12), we obtain that
AI(P,q,BR)
• IBR lY'ulPdx In lY'cplPdx = W~,P(BR) mf f BR Iu Iq d x ::;.hn Icp Iq d x = AI(p,q,f!).
It is easy to show that Vol(BR(O)) =Vol(f!) from (7).
Let W/,P(BR(O)) is radial Sobolev space in W~,P(BR(O)). Then
AI(P,q, BR) ::;
. mf
W;,P(BR(O»
lIY'ull~ ~ * IliPI - A1(p, q, BR). u q
(13)
Let cp(x) ~ 0 be the eigenfunction corresponding to AI(P,q,BR),'f/I(X) be the symmtrization function of cp(x) given by (7). From (8) and (12), we get (14)
Therefore by (13) and (14), we obtain Ai(p, q, BR~ = AI(p, q, B R). Appiying Lemma 1, Lemma 2 and the above fact, we may have the following theorem Theorem 1 There exists a ball BR(O) C R N with Vol(BR(O)) =Vol(f!) such that AI(p, ,q, f!) ~ AI(p, q, BR(O)) fo: any q E [p, ,p ~ 2 and the equality holds if n = BR(O). In the following, we are going to discuss the special case when p = q = 2. For convenience, we denote that AI(f!) = AI(2, 2, f!), Ai(R) = Ai(2, 2, BR(O)). By Lemma 2, there exists a R > 0 satisfying Vol(f!) =Vol(BR(O)) such that
:!p)
From [3], we know Ai(R) = i2il (N;-2). Thus AI(f!) ~ i2ir (N;-2) , where il(N;-2) is the .first zero of Bessel fuction with ordered N ;-2. Therefore we have: Theorem 2 There exists a ball BR(O) C R N with Vol(BR(O)) =Vol(f!) such that Al ~ bir( N22) and the equality holds if f! = BR(O).
3
Opial Inequality in R N
Opial inequality in R 1 has been studied by many authors recently. In this section, we are going to extend this inequality to the case of higher dimension. Our main result is as follows:
576
ACTA MATHEMATICA SCIENTIA
Theorem 3 Let a > 0, {3 > 0, a for anyu E W~,u+13(n), we have
+ (3 > 1, n be a
Vo1.21 Ser.B
bounded smooth domain in R N . Then
Where R is the radius of the ball BR(O) whose volume is equal to the volume of
n, Ai(B R) is
given by (13). Proof By virtue of Holder inequlity, Pioncare inequality, and Lemma 2 we have
L
lululVul 13dx :s;
(L
(L L L
_"_
lu1u+13dX) ,,+/1
IVu.1u+13dX)
<
[A1(p~q,n)] ,,+/1
<
(.~i(p, q,IBR(O)) ) <>+/1
-L. "+/I
u+13dx IVul u+13dx. IVul
Where AHp,q,BR(O)) is given by (13). Remark 3 By using poincare inequality[9], we have
L
So we have
luluIVul13dx:s; R U
Remark 4
If a
L
!Vulu+13dx.
= (3 = 1, by Theorem 2 and Theorem 3, we deduce:
Where j1( N22) is the first zero point of Bessel fuction ordered N22. References 1 Yau S T. Differential geometry. Beijing:Science Press 2 Watson G N. A treatise on the theorey of bessel function. The syndics of the cambridge university press, 1952 3 Deng Y B. The estimates of eigenvalues for laplace operator. Lecture notes in pure and applied mathematics, 176.15-20 4 Zong J Q, Yang H C. On the estimate of the first eigenvalue of compact Riemennian manifold. Scientia, 1983,A9:812-820 5 Opial Z. Sur nne inegalte. Ann Polon Math, 1960,8:29-32 6 Olech C. A simple proof of a certain result of Z Opial, Ann Polon Math, 1960,8:61-63 7 Bessaek P R. On an integral inequlity of Z Opial. Trans Amer Math Soc, 1960,104:470-475 8 Mallow. An even simpler proof of Opial inequality. Proc Amer Math Soc, 1960,15:565-566 9 Gilbarg D, Trudinger N S. Elliptic partial differential equatations of second order. Second Edition. Berlin, Heidelberg, New York, Tokyo: Springer-verlag, 1983 10 He X G. A short proof of a generalization of Opial inequality. J Math Anal Appl, 1984,182:299-300 11 Liang Z J. The Opial-Hua inequality. J C C N U, 1980,2:33-37
THE OPIAL INEQUALITY IN R N Deng Yin bin ( ~ ~I'"
)
1
College oj Mathematical and Statistics Science, Wuhan University, Wuhan 490072, China
Abstract
This paper discusses the minimal eigenvalue Al(p, q, 0) and its best estimate
of the following nonlinear eigenvalue problem
Where 0 is a bounded smooth domain in R N , ti.p is the p-laplace operator, p [p, :/!p)' By this result, the famous Opial inequality in R N are deduced.
Key words
Opial inequality, eigenvalue, p-Iaplace
1991 MR Subject Classification
1
2: 2, q E
35JI0, 35J20, 35J60, 35J65
Introduction
In this paper, we discuss the minimal eigenvalue A1(P, q, 0) and its best estimate of the following nonlinear eigenvalue problem uE 0,
Where 0 is a bounded smooth domain in RN,Llpu =
N
L: L(IV'ulp-2Jt!!..) ;=1 ax, ax,
(1)
is the p-laplace
operator, P 2:: 2, q E [p, :!p) are some given real number. By this result, we deduce the famous Opial inequality in R N • As we know, the solvabiliy of problem (1) results from whether the minimum (2)
can be achieved. If (2) is achieved, the attained function r,o(x) is the nontrivial solution of problem (1) corresponding to A A1 (p,q, 0) and A A1 (p, q, 0) is the corresponding minimum eigenvalue. Since q is smaller than the critical Sobolev embedding exponent, the embedding W~,P(o) '---+ Lq(O) is compact. It is easy to verify that the minimum problem (2) can be achieved by using variational methods. So the minimum eigenvalue of problem (1) exists. By
=
=
1 Received September 18,2000. The research supported by Natural Science Fundation of China and the Excellent Teachers Fundation of Ministry of Education of China
573
Deng: THE OPIAL INEQUALITY IN RN
No.4
virtue of isopermetric inequality and co-area formulalll, one can show that there exists a ball
BR(O) = {x E RNllxl < R} with Vol(BR(O)) =Vol(O) such that (3) and the equality holds as 0 = BR(O). In (3), if let p = q = 2, then by using some properties of Bessel function and Sturm comparision theory for ordinary differential equations, we can obtain the best estimate of the first eigenvalue Al(O) for the following eigenvalue problem
-~u = AU
U
E 0,
{ ulan = O. That is there exists R
~
0 with Vol(BR(O)) =Vol(O) such that
A (0) 1
> -
'2(N-2) -2R2
J1
(4)
and the equality holds as 0 = BR(O). Where h( N2"2) is the first zero point of the Bessel function ordered
N
2" 2 [2]•
Remark 1 When 0 is N dimension compact manifold without boundary, the result of Zong and Yang[4] gives us that eigenvalue Al(O) is larger than or equal to ~;. Where d is the diameter of O. Remark 2 If N = 3, then il(N2"2) = il(!) = 11" (ref.[2)[3]). From (4), we have, Al(O) ~ ~ ~ ~ as N = 3. By virtue of Sturm comparsion theory we can deduce that
where d denotes the diameter of O. Our results are quit different from those in [4] if 0 C R N is a bounded domain. Opial inequality was first presented by Z.Opial in 1960[5]. He proved that
l
0
for any
f
h
hlh
If(x)f'(x)ldx::; 1f'(x)1 2dx 40
E CJ([O, h]). Later C.Olech[6], P.R.Bessack[7J and C.L.Mallow[5] simplified the proof
and extended the results. Some different proofs were also given by Z.J .Liang[ll] and X.G.He[lO]. But the Opial inequality in R N , as we know, has not been discussed yet. In this paper, we are going to discuss the Opial inequality in some Sobolev space W~'P(O) with 0 E R N • As a special case, we will show that for any u E W~,2(0), we have:
where il( N 22) is the first zero pint of the Bessel function ordered ball BR(O), whose volume is eaqul to the volume of O.
N
22, R
is the radius of the
574
2
ACTA MATHEMATICA SCIENTIA
Vo1.21 Ser.B
The Best Embedding Constant
In this section, we will discuss the best embedding constant from W~'P(O) to Lq(O). Lemma 1 The minimum problem (2) can be achieved Proof Since q E [P, ~) is smaller than the critical Sobolev exponent. It is easy to verify this lemma by using the standard variational methods and Sobolev compact embedding theorem. Lemma 2 Let >'l(p, q, 0) be the minimum eigenvalue of problem (1), then there exsits a ball BR(O) with Vol(O) =Vol(B.n(O)) such that
The equality holds when 0 = BR(O). Proof The case where P = q = 2 was proved by Fabe-Krahnl'l. Now we are going to prove the case when q 2: P > 2. From Lemma 1 the minimum of >'l(p, q, 0) can be attained by some function tp > O. Since if tp is the attained function of >'l(p,q, 0), so is Itpl, we may assume that tp 2: 0 in O. Let "p(x) be the symmetrization fucntion of tp(x). i.e. 1/, : B R -> R+ Satisfies Vol(tp 2: c) = Vol("p 2: c)
(7)
for any e = const. 2: 0, and "p(x) = "p(r) is a radial function, where r = Ixl and r E [0, R]. It is obvious that for any e 2: 0, the set {x E R N : "p(r) 2: e} is a ball and "p( R) = O. By some calculations, we can obtain
As "p(r) = e =const., we have IV"p1 = So Area{"p = e} =
1.p=c
I~I
= const.
[1.p=c
1dx =
.!.
IV"pIP-1dX]
p
[1.p=c
l.=..!.
1 I] V"p
_I
P
(9)
By using the Isoperimetric inequality[l) we get ~
(i=c IVtpIP-ldX) (i=c l;tp!dX) P
= Area{tp = e}
2: Area{"p = e} =
1.p=c
l.=..!. P
2:
i=c 1dx
1dx
(10)
-- [1.p=c IV"p1P-l];; [1.p=c IV"pldx 7 1
dz
]
On the other hand, form co-area formula[l) d(Vol{tp > e}) = de and
1
rp=c
_l_
IVtpl
dx ,
d(Vol{tp 2: e}) de
--'---7-=-"":"':'"
d(Vol{"p de
> e}) =
2: e} = dVol{'ljJ de
1
rp=c
_l_
IV"p1
dx
575
Deng: THE OPIAL INEQUALITY IN R N
No.4
)P-I - (1 (1
Thus
1
1 --dx
1/J=c IY''f/I I
)P-I
'
from (10), we have
i=c lY'cplp-1dx ~ i=c 1Y''f/Ilp-1dx.
(11)
By using co-area formula again, we deduce that
(12)
From (8) and (12), we obtain that
AI(P,q,BR)
• IBR lY'ulPdx In lY'cplPdx = W~,P(BR) mf f BR Iu Iq d x ::;.hn Icp Iq d x = AI(p,q,f!).
It is easy to show that Vol(BR(O)) =Vol(f!) from (7).
Let W/,P(BR(O)) is radial Sobolev space in W~,P(BR(O)). Then
AI(P,q, BR) ::;
. mf
W;,P(BR(O»
lIY'ull~ ~ * IliPI - A1(p, q, BR). u q
(13)
Let cp(x) ~ 0 be the eigenfunction corresponding to AI(P,q,BR),'f/I(X) be the symmtrization function of cp(x) given by (7). From (8) and (12), we get (14)
Therefore by (13) and (14), we obtain Ai(p, q, BR~ = AI(p, q, B R). Appiying Lemma 1, Lemma 2 and the above fact, we may have the following theorem Theorem 1 There exists a ball BR(O) C R N with Vol(BR(O)) =Vol(f!) such that AI(p, ,q, f!) ~ AI(p, q, BR(O)) fo: any q E [p, ,p ~ 2 and the equality holds if n = BR(O). In the following, we are going to discuss the special case when p = q = 2. For convenience, we denote that AI(f!) = AI(2, 2, f!), Ai(R) = Ai(2, 2, BR(O)). By Lemma 2, there exists a R > 0 satisfying Vol(f!) =Vol(BR(O)) such that
:!p)
From [3], we know Ai(R) = i2il (N;-2). Thus AI(f!) ~ i2ir (N;-2) , where il(N;-2) is the .first zero of Bessel fuction with ordered N ;-2. Therefore we have: Theorem 2 There exists a ball BR(O) C R N with Vol(BR(O)) =Vol(f!) such that Al ~ bir( N22) and the equality holds if f! = BR(O).
3
Opial Inequality in R N
Opial inequality in R 1 has been studied by many authors recently. In this section, we are going to extend this inequality to the case of higher dimension. Our main result is as follows:
576
ACTA MATHEMATICA SCIENTIA
Theorem 3 Let a > 0, {3 > 0, a for anyu E W~,u+13(n), we have
+ (3 > 1, n be a
Vo1.21 Ser.B
bounded smooth domain in R N . Then
Where R is the radius of the ball BR(O) whose volume is equal to the volume of
n, Ai(B R) is
given by (13). Proof By virtue of Holder inequlity, Pioncare inequality, and Lemma 2 we have
L
lululVul 13dx :s;
(L
(L L L
_"_
lu1u+13dX) ,,+/1
IVu.1u+13dX)
<
[A1(p~q,n)] ,,+/1
<
(.~i(p, q,IBR(O)) ) <>+/1
-L. "+/I
u+13dx IVul u+13dx. IVul
Where AHp,q,BR(O)) is given by (13). Remark 3 By using poincare inequality[9], we have
L
So we have
luluIVul13dx:s; R U
Remark 4
If a
L
!Vulu+13dx.
= (3 = 1, by Theorem 2 and Theorem 3, we deduce:
Where j1( N22) is the first zero point of Bessel fuction ordered N22. References 1 Yau S T. Differential geometry. Beijing:Science Press 2 Watson G N. A treatise on the theorey of bessel function. The syndics of the cambridge university press, 1952 3 Deng Y B. The estimates of eigenvalues for laplace operator. Lecture notes in pure and applied mathematics, 176.15-20 4 Zong J Q, Yang H C. On the estimate of the first eigenvalue of compact Riemennian manifold. Scientia, 1983,A9:812-820 5 Opial Z. Sur nne inegalte. Ann Polon Math, 1960,8:29-32 6 Olech C. A simple proof of a certain result of Z Opial, Ann Polon Math, 1960,8:61-63 7 Bessaek P R. On an integral inequlity of Z Opial. Trans Amer Math Soc, 1960,104:470-475 8 Mallow. An even simpler proof of Opial inequality. Proc Amer Math Soc, 1960,15:565-566 9 Gilbarg D, Trudinger N S. Elliptic partial differential equatations of second order. Second Edition. Berlin, Heidelberg, New York, Tokyo: Springer-verlag, 1983 10 He X G. A short proof of a generalization of Opial inequality. J Math Anal Appl, 1984,182:299-300 11 Liang Z J. The Opial-Hua inequality. J C C N U, 1980,2:33-37