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On best constants in Hardy inequalities with a remainder term
Nonlinear Analysis 74 (2011) 5784–5792
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On best constants in Hardy inequalities with a remainder term Salvatore Cuomo 1 , Adamaria Perrotta ∗ Dipartimento di Matematica ed applicazioni ‘‘R.Caccioppoli’’, Complesso Universitario M. S. Angelo, Via Cinthia, 80126 Napoli, Italy
article
abstract
info
Let Ω be a bounded open set of RN containing the origin. We compute the best value of the constant C (α, |Ω |) in
Article history: Received 19 March 2010 Accepted 21 May 2011 Communicated by Ravi Agarwal
∫
|∇ u|2 dx −
(N − 2 ) 2 4
Ω
MSC: 35J20 26D10 46E35
u2
∫
| x| 2
Ω
dx ≥ C (α, |Ω |) ‖u‖2
L N2N −α ,2
,
with α < 2 and u ∈ H01 (Ω ). Then we get the optimal value of C (|Ω |) in
∫
Keywords: Hardy inequality One-dimensional Calculus of Variations Best constants
|∇ u|3 dx −
N −3
Ω 1,3
where u ∈ W0
3
3 ∫
u3 Ω
| x| 3
dx ≥ C (|Ω |) ‖u‖3L3 ,
(Ω ).
© 2011 Elsevier Ltd. All rights reserved.
1. Introduction Let N > 2 and let Ω ⊆ RN be a bounded domain containing the origin. The well known Hardy–Sobolev inequality (see [1,2]) reads
∫
p
|∇ u| dx ≥
N −p
p ∫
p
Ω
up Ω
|x|p
dx,
∀u ∈ W01,p (Ω ) .
(1.1)
This inequality and its various improvements are used in many contexts, as in the study of stability of solutions of semilinear elliptic and parabolic equations (see [3–5]), or in the analysis of the asymptotic behavior of the heat equation with singular potentials (see [6]). N −p p
The constant CN ,p =
p
1 ,p
is the best one and there is no function u ∈ W0
(Ω ) for which it is achieved. For this
reason several authors have improved inequality (1.1) by adding a nonnegative correction term. In case p = 2 the first result is due to Brezis and Vazquez; in [3] they prove the so-called Hardy–Poincaré inequality
∫
|∇ u|2 dx − Ω
(N − 2) 2 4
u2
∫ Ω
|x|
dx ≥ Λ2 2
ωN |Ω |
N2
‖u‖2L2 ,
∀u ∈ H01 (Ω ) .
(1.2)
Here Λ2 denotes the first eigenvalue of the Laplace operator in the two dimensional unit disk, and ωN and |Ω | are respectively the N-dimensional Lebesgue measure of the unit ball B1 (0) ⊆ RN and of the set Ω . The value Λ2 optimal in the ball but it is not achieved in
∗
H01
(Ω ) .
Corresponding author. Tel.: +39 081675703; fax: +39 081675636. E-mail addresses: [email protected] (S. Cuomo), [email protected] (A. Perrotta).
1 Tel.: +39 081675624; fax: +39 081675636. 0362-546X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2011.05.069
ωN |Ω |
N2
is
S. Cuomo, A. Perrotta / Nonlinear Analysis 74 (2011) 5784–5792
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Inequality (1.2) can be also generalized by replacing the L2 -norm of u either with an Lq -norm of u with 1 ≤ q < 2∗ = N2N −2 (see [3]), or with a norm of u in a suitable Lorentz space (see [7,8]). The problem of finding the best constant in the remaining term in some of inequalities is still open. A share in this direction has been given in [9], where the authors get the optimality of the constant in the right hand sides of
∫
(N − 2) 2
2
|∇ u| dx −
4
Ω
u2
∫
|x|2
Ω
dx ≥
ωN |Ω |
N1
V0 ‖u‖2
L N2N −1 ,2
,
(1.3)
here Ω is a bounded domain of RN containing the origin, 1 ≤ p < 2∗ if N > 2, 1 ≤ p < +∞ if N = 2, u ∈ H01 (Ω ), and V0 is the first zero of the function
√
V (r ) = J0 2 r =
+∞ −
(−1)k
k=0
rk
(k!)2
.
We remark that inequality (1.3) can be seen as an Hardy–Sobolev inequality whose remainder term is the norm of u in the
Lorentz space L
2N N −1
N1 , 2 . The constant |ωΩN| V0 is not achieved.
A related Lp version of (1.1) and its improvements are also well known. In [10] Gazzola et al. show that an extra Lp -norm of u can be added to (1.1) for every 1 < p < N ; in other words they prove that there exists a constant C (p, |Ω |) > 0 such that
∫
|∇ u|p dx −
N −p
p ∫
p
Ω
up Ω
p
dx ≥ C (p, |Ω |) ‖u‖Lp ,
|x|p
∀u ∈ W01,p (Ω ) .
(1.4)
In particular, they study the asymptotic behavior of C (p, |Ω |), not its sharp value. Closely related questions are studied simultaneously and independently by Adimurthi et al. in [11]. Inequality (1.1) can be also improved by adding an Lq norm of the gradient of u; see [6,9] for questions related to optimality. Finally, for the sake of completeness, we refer to the following papers, which deal with Hardy inequalities with different type of remainder terms. See [12–16]. In this paper we are interested in finding the expression of the best constant in inequalities of the type
∫
|∇ u|p dx −
N −p
p ∫
p
Ω
up
|x|p
Ω
dx ≥ C [Q (u)]p ,
∀u ∈ W01,p (Ω ) ,
that involve a Lorentz or a Lebesgue norm of u as remainder term. Here, the definition of the best constant must be under1,p stood as follows. For any u ∈ W0 (Ω ), u ̸≡ 0, let
F (u) =
Ω
|∇ u|p dx −
N −p p
p
up
Ω | x| p
[Q (u)]p
,
and set
λ(Ω ) =
inf
1,p u∈W0 (Ω ) u̸≡0
F (u).
The best constant C is then defined as C = inf λ(Ω ′ ), where Ω ′ varies in the class of all bounded, open subset of RN containing the origin, such that Ω ′ = |Ω | . The first part of this paper is devoted to get the best value of the constantC (α, |Ω |) in a generalized version of (1.3), obtained by adding to the right-hand side the norm of u in the Lorentz space L N2N , 2 with α < 2 −α
∫
|∇ u|2 dx −
(N − 2) 2 4
Ω
u2
∫ Ω
|x|2
dx ≥ C (α, |Ω |) ‖u‖2
L N2N −α ,2
,
∀u ∈ H01 (Ω ) .
(1.5)
Then we compute the optimal constant C (p, |Ω |) in (1.4) when p = 3, that is
∫
|∇ u|3 dx − Ω
N −3 3
3 ∫
u3 Ω
|x|3
dx ≥ C (|Ω |) ‖u‖3L3 ,
∀u ∈ W01,3 (Ω ) .
(1.6)
In order to reach both results, as usual we firstly reduce the study to spherically symmetric functions defined on Ω # , simply by replacing u with u# . The Pólya–Szego principle (see [17]) and Hardy–Littlewood inequality (see [2]) ensure that the previous assumption is not restrictive. Then, the optimal values of C (α, |Ω |) and C (|Ω |) are obtained as minima of timely
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S. Cuomo, A. Perrotta / Nonlinear Analysis 74 (2011) 5784–5792
functionals associated respectively to inequalities (1.5) and (1.6). To get such minima, we have to apply classical arguments of Calculus of Variations. So, we simplify the inequalities with a suitable change of variables (see [3]) and we try to solve the Euler equations linked to such inequalities. This is possible for what it concerns the Euler equation of (1.5) but cannot be obtained for the second one. To overcome this difficulty, we study only the qualitative behaviour of such a solution. We point out that the properties of the solution to the Euler equation we found with the qualitative analysis are sufficient to apply classical arguments of Calculus of Variations to (1.6), as already done for inequality (1.5). 2. Best constant in Hardy–Sobolev inequalities with a remainder term In this section we are interested in determining the optimal value of the constants C (α, |Ω |) in (1.5) with α < 2 and C (|Ω |) in (1.6). To this aim, we firstly recall the definition of the spherically decreasing rearrangement of a function u and some related properties. Let Ω be a measurable subset of RN and u : Ω → R a measurable function in Ω . The distribution function of u is the decreasing map µ from [0, +∞[ into [0, +∞[ defined at any point t ≥ 0 as the measure of a level set of u, {x ∈ Ω : |u(x)| ≥ t }. The decreasing rearrangement u∗ of u is the distribution function of µ u∗ (s) = sup {t ≥ 0 : µ(t ) > s} ,
s ∈ (0, |Ω |) .
The main property of rearrangements is the fact that the distribution of u∗ is µ, in other words u and u∗ are equidistributed. Let us by ωN the measure of the unit ball of RN and by Ω # the ball of RN centered at the origin such that denote |Ω | = Ω # . For every x ∈ Ω # , the spherically decreasing rearrangement of u is defined as the decreasing rearrangement u∗ valued in ωN |x|N
u# (x) = u∗ ωN |x|N ,
x ∈ Ω #.
# Obviously, and spherically symmetric; moreover u and u# are equidistributed, and the level set u is decreasing # # x ∈ Ω : u (x) ≥ t is the ball centered at the origin and whose measure is µ(t ). For an exhaustive treatment of rearrangements see, for example, [18–21]. Here we just recall the Hardy–Littlewood inequality
∫ Ω#
∫
u# (x) v# (x) dx ≤
Ω
|u (x) v (x)| dx ≤
∫ Ω#
u# (x) v # (x) dx,
(2.1)
with u, v measurable functions (see [2]), and the Pólya–Szego principle (see [17])
∫ RN
# p ∇ u (x) dx ≤ 1,p
∫
|∇ u (x)|p dx
RN
(2.2)
where u ∈ W0 RN , 1 < p < N . Now let Ω be a bounded open set of RN containing the origin, N ≥ 2, 1 ≤ p < 2∗ if N > 2, 1 ≤ p < +∞ if N = 2 and u ∈ H01 (Ω ). We prove the following
Theorem 2.1. Let Ω be a bounded open set of RN , with N ≥ 2. There exists a constant C (α, |Ω |) such that inequality (1.5) holds for every α < 2 and u ∈ H01 (Ω ). The optimal value of C (α, |Ω |) is given by 2
ωNN
C (α, |Ω |) = w
2−α 0
|Ω |
2−α N
,
where w0 is the first zero of the function
W (r ) = J0
α
r 1− 2
=
1 − α2
+∞ −
(−1)k
k=0
r (2−α)k
(2 − α)2k (k!)2
.
Here J0 denotes, as usual, the Bessel function of zeroth order. Proof. The first step consists into the reduction of the problem to a spherically symmetric one. We replace u with u# and Ω with Ω # . By Hardy–Littlewood inequality (2.1) and Pólya–Szego principle (2.2), the left hand side of (1.5) decreases, while the right side increases. This implies that it is enough to prove the theorem only in the radial case. By using a suitable homothety, which transforms Ω # in the ball BR centered at the origin and with radius C (α, |Ω |)α−2 R, inequality (1.5) can be written as
∫
|∇ u|2 dx − BR
(N − 2) 2 4
u2
∫ BR
dx ≥ 2
|x|
u2
∫ BR
|x|α
dx,
∀u ∈ H01 (BR ) ,
with R to be determined. We first prove our result in dimension two.
(2.3)
S. Cuomo, A. Perrotta / Nonlinear Analysis 74 (2011) 5784–5792
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Fig. 1. Closed path.
If N = 2 inequality (2.3) reads as R
∫
′ 2 u
R
∫
u2 r 1−α dr ,
rdr ≥
0
∀u ∈ H01 (0, R) .
(2.4)
0
Let us consider the functional associated to (2.4) J (u) =
R
∫
′ 2 u
r − u2 r 1−α dr ,
(2.5)
0
and its Euler equation u′′ r + u′ + ur 1−α = 0.
(2.6)
As well known, the functions for which equality in (2.4) holds satisfy (2.6). Through a power expansion, we obtain the following solution of (2.6) α
W ( r ) = J0
r 1− 2
1 − α2
=
+∞ − k=0
(−1)k
r (2−α)k
(2 − α)2k (k!)2
.
The properties of the Bessel function of zeroth order, J0 , imply that W (r ) is decreasing, satisfies conditions W (0) = 1, W ′ (0) = 0 and has a zero, denoted with w0 . Our target is to show that the one parameter family of extremals u(r ) = cW (r ), with c ∈ R, minimizes the functional (2.5). We take R = w0 and for instance u ∈ C 1 (0, R); we embed W (r ) in the extremal field defined by the vector (1, Φ (r , u)), where Φ (r , u) =
u(r )W ′ (r ) W (r )
(see [22] Lemma 3.1). Denoted with f (r , u, u′ ) the integrand in (2.5), the differential form
ζ = P (r , u)dr + Q (r , u)du, whose coefficients are P (r , u) = f (r , u, Φ (r , u)) − Φ (r , u)fu′ (r , u, Φ (r , u)), Q (r , u) = fu′ (r , u, Φ (r , u)), is exact (see [22]), so the integral of ζ along the closed path represented in Fig. 1 is equal to zero. This ensures that, for every ε > 0 and for every w ∈ C 1 (0, R) with w(R) = 0 J (w) =
R
∫
f (r , w, w ′ ) − f (r , w, Φ (r , w)) + (Φ (r , w) − w ′ )fu′ (r , w, Φ (r , w)) dr
0
ε
∫ +
[f (r , W (ε), Φ (r , W (ε))) − Φ (r , W (ε))fu′ (r , W (ε), Φ (r , W (ε)))] dr +
0
R
∫ ε
f (r , W , W ′ )dr .
It can be shown that
∫ lim
ε→0 0
ε
[f (r , W (ε), Φ (r , W (ε))) − Φ (r , W (ε))fu′ (r , W (ε), Φ (r , W (ε)))] dr +
R
∫ ε
f (r , W , W ′ )dr = 0.
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S. Cuomo, A. Perrotta / Nonlinear Analysis 74 (2011) 5784–5792
Moreover we want to prove that f (r , w, ξ ) − f (r , w, ξ1 ) + (ξ1 − ξ ) fu′ (r , w, ξ ) ≥ 0, for any ξ1 , ξ in R (Weierstrass condition). We have f (r , w, ξ ) − f (r , w, ξ1 ) + (ξ1 − ξ ) fu′ (r , w, ξ )
[ =
′ 2 (N − 2)2 w2 w2 2 (N − 2)2 w12 w2 − 2 − ξ1′ + + 21 − 2 (ξ − ξ1 ) ξ1 ξ − 2 2 4
r
r
4
r
]
r
2 2 2 = ξ ′ − ξ1′ . The fact that W (r ) can be embedded in (1, Φ (r , u)) and the Weierstrass condition imply that W (r ) yields a minimum for J (u) (see [22], Theorem 3.3.1). Now let us consider the case N ≥ 3. If we perform the following change of variable (see [3]),
v(r ) = r −
N −2 2
u(r ),
r = |x| ,
with u(0) = 0 (this condition ensures that v ∈ H01 (Ω )), the Hardy–Sobolev inequality (2.3) can be read as the Sobolev inequality in the plane (2.4) with u replaced by v . Taking for instance u ∈ C01 (BR ) and using again the previous change of variable, we go back to u and obtain (2.3) with R = w0 . The restriction u ∈ C01 (BR ) can be removed by a density argument. A dimensional analysis on the constant shows that the optimal value of C (α, |Ω |) in (1.5) is 2
ωNN
C (α, |Ω |) = w
2−α 0
|Ω |
2−α N
.
Finally, we observe that such a constant is not attained; this is due to the fact that it corresponds to equality in (2.4), which is realized when v(r ) = cW (r ) and so u(|x|) = c |x|−
N −2 2
W (|x|),
but u(|x|) is not in H because W (0) = 1. 1
The result obtained in Theorem 2.1 suggested us the idea of using the same procedure to get the optimal value of the constant C (|Ω |) in (1.6) too. The strong difference between the two results consists in the fact that we cannot solve the Euler equation associated to (1.6). To overcome this difficulty we analyze such equation in a different way; through a numerical approach we don’t get the explicit expression but the qualitative behavior of its solution. In Theorem 2.2 we shall show how to use these information to get the best value of C (|Ω |). Theorem 2.2. Let Ω be a bounded open set of RN , with N > 3. There exists a constant C (|Ω |) such that inequality (1.6) holds 1 ,3 for every u ∈ W0 (Ω ). The optimal value of the constant C (|Ω |) is given by C (|Ω |) = v
3 0
ωN |Ω |
N3
,
where v0 is the first zero of the solution of the Euler equation of the functional associated to (1.6). Proof. As in Theorem 2.1, the first step consists in the reduction of the problem to a spherically symmetric one. To this aim, we replace u with u# and Ω with Ω # . By Hardy–Littlewood inequality (2.1) and Pólya–Szego principle (2.2), the left hand side of (1.6) decreases, while the right side does not change. This implies that it is enough to prove the theorem only in the radial case. Using a suitable homothety, which transforms Ω # in the ball BR centered at the origin and with radius C (|Ω |)3 R, inequality (1.6) can be written as
∫
|∇ u|3 dx −
N −3
BR
3
3 ∫
u3 BR
∫
dx ≥ 3
| x|
u3 dx,
∀u ∈ W01,3 (Ω ) ,
(2.7)
BR
with radius R to be determined. If we perform the following change of variable N −3 u(r ) = r − 3 v(r ),
r = |x| ,
with v(0) = 0 and take for instance u ∈
(2.8) C01
(BR ), the Hardy–Sobolev inequality (2.7) becomes
S. Cuomo, A. Perrotta / Nonlinear Analysis 74 (2011) 5784–5792 R
∫
2 3 dr ≥ r 2 v ′ + (N − 3)r v v ′
5789
R
∫
r 2 v 3 dr .
(2.9)
0
0
Let us consider the functional associated to (2.9) J (v) =
∫
R
2 3 r 2 v ′ + (N − 3)r v v ′ − r 2 v 3 dr ,
(2.10)
0
and its Euler equation 2r v v − 2 ′ ′′
2 3
(N − 3)r vv + ′′
9−N
3
2
r v′
2
− (N − 3)vv ′ − r 2 v 2 = 0. 3
(2.11)
We have to solve such equation; unfortunately, a power expansion does not work in this case because it gives only a recursive relation between its coefficients. For this reason, we reformulate (2.11) in terms of the initial value problem
′ y1 = y2 y21 ′ − 9−3 N y22 + 23 (N − 3) y1ry2 r y2 = 2ry2 − 32 (N − 3)y1 y1 (0) = 1 y2 (0) = 0
in BR in BR ,
(2.12)
where v(r ) := y1 (r ), v ′ (r ) := y2 (r ) and the initial data represent the first two coefficients of the series. In Section 3 we show that (2.12) has a unique solution; obviously this ensures that there exists a solution V (r ) to (2.11). Moreover, thanks to a numerical analysis, we can deduce that V (r ) has a zero, v0 , and that V (r ) is decreasing in the interval [0, v0 ] (see Section 3.1). Even without the explicit expression of V (r ), the previous properties are enough to apply classical arguments of the Calculus of Variations to get a minimum of (2.10). Firstly we choose R = v0 ; proceeding as in Theorem 2.1 is possible to prove that V (r ) can be embedded in the extremal field defined by the vector (1, Φ (r , v)), where Φ (r , v) = in (2.10), we get
v(r )V ′ (r ) V (r )
(see [22] Lemma 3.1). Denoting by f (r , v, v ′ ) the integrand
f (r , w, ξ ) − f (r , w, ξ1 ) + (ξ1 − ξ ) fv ′ (r , w, ξ ) ≥ 0, for any ξ1 , ξ in R (Weierstrass condition). In fact f (r , w, ξ ) − f (r , w, ξ1 ) + (ξ1 − ξ ) fv ′ (r , w, ξ ) = ξ13 − ξ 3 − 3ξ12 (ξ1 − ξ ). The Weierstrass condition follows from the fact that V (r ) is decreasing in [0, R]. We conclude that V (r ) yields a minimum for J (v) (see [22] Theorem 3.3.1), and so we obtain (2.9) with R = v0 . Performing again the change of variables (2.8) and coming back to u we obtain (2.7) with R = v0 . The restriction u ∈ C01 (BR ) can be removed by density. A dimensional analysis of the constant shows that the optimal value of C (|Ω |) in (1.6) is C (|Ω |) = v03
ωN |Ω |
N3
.
Finally, we observe that such a constant is not attained; the reason of this is the fact that it corresponds to equality in (1.4), which happens for v(r ) = cV (r ) and so u(|x|) = c |x|−
N −3 3
1,3
but u(|x|) is not in W0
V (|x|), because V (0) = 1.
3. Numerical issues on the Euler equation The main problem in the proof of Theorem 2.2 is to solve the Euler equation (2.11). As pointed out in the previous section, it is not possible to get the explicit expression of such a solution V (r ). To overcome this difficulty, we rewrite the Eq. (2.11) in terms of the initial value problem (2.12) and analyze it from a numerical point of view. This kind of approach ensures not only that there exists a solution V (r ) to (2.11), but also gives us its qualitative behavior. This section is organized as follows: in Section 3.1 we get the solution of the problem (2.12) through a Numerical Method; in Section 3.2 we show the qualitative behavior of V (r ).
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S. Cuomo, A. Perrotta / Nonlinear Analysis 74 (2011) 5784–5792
Fig. 2. Lax–Richtmyer scheme.
3.1. Numerical solution of the Euler equation In this subsection we show how to apply the scheme just described to analyze the initial value problem (2.12). To this aim we first prove that (2.12) has a unique solution; then, fixed any consistentnumerical method for the discrete problem, if this satisfies the Lipschitz condition, then the method is also zero-stable. From the Lax Scheme we deduce that the method is convergent and hence the solution to discrete problem associated to (2.12) converges to the continuous one (see [23], Fig. 2). Let us denote by g (r , y1 , y2 ) the right hand side of (2.12),
g (r , y1 , y2 ) = y2 ,
y21 r
−
9−N 3
y22 + 32 (N − 3)
y1 y2 r
2ry2 − 23 (N − 3)y1
,
the Jacobian matrix of g is
0
Jg = 18t y1 y2 − 3t (N − 3) 2
1
+ (N − 3) 2 [3ty2 − (N − 3)y1 ]2 y21
2 2 y2
3t (N − 9) 2
y22
− 2t (N − 9)(N − 3)y1 y2 − 2(N − 3) + 9t
2
3
y21
.
2t [3ty2 − (N − 3)y1 ]
2
Extending by continuity at the origin both g and Jg , we deduce that the problem (2.12) satisfies the Cauchy Theorem, so there exists a unique solution to (2.12). To describe the behavior of V (r ) we choose, for example, the explicit Euler Method and we get the following discrete version of (2.12)
zn+1 = zn + hqn zn2 − 9−3 N q2n + 32 (N − 3) znrnqn rn qn+1 = qn + h , 2rn wn − 32 (N − 3)zn z0 = 1 q0 = 0 where ri = r0 + ih, i = 1, . . . , n, h = Rn , zi = y1 (ri ) and qi = y2 (ri ). The sequences zn and qn represent an approximation of V (r ) := y1 (r ) and V ′ (r ) := y2 (r ), respectively. From a direct computation we deduce (i) the sequence qn < 0, i.e. V (r ) is decreasing; (ii) z0 = V (0) = 1 and moreover there exists an r ∈ [0, R] such that zn (r ) = V (r ) < 0. Because of the convergence of the Euler Method and of the continuity of the solution V (r ), we trivially conclude that there exists a v0 ∈ [0, R] such that zn (v0 ) = V (v0 ) = 0. This procedure works for every N > 3, so the solution V (r ) to the Euler equation (2.11) has a zero v0 and is decreasing in [0, v0 ] in any dimension N > 3. Fig. 3 and Table 1 refer to the use of the Euler method to analyze problem (2.12) in the model case N = 11. 3.2. More experiments In this subsection we want experimentally confirm that the qualitative analysis described in the previous is independent of the election of the numerical method used to solve the problem (2.12) and from dimension N. The numerical simulations are carried out on a Intel(R) Pentium(R) M with 1.7 GHz processor by means the ODE suite numerical solvers of initial value problems for ordinary differential equations of Matlab R14 [24]. We remark that the expression ‘‘V (r ) has a zero in r’’ means that the function V (r ) assumes in r a value that is close to zero, with an |v −r | approximation which depends on the accurancy required; moreover 0v < 10−k . 0
S. Cuomo, A. Perrotta / Nonlinear Analysis 74 (2011) 5784–5792
5791
Fig. 3. Numerical solution and derivative behavior.
Table 1 Sequences of Euler method. rn
Numerical solution zn
Derivative qn
0.0010 0.3010 0.9010 1.2010 2.4010 2.7010 3.9010 4.2010 4.5010 5.1010
1.0000 0.9994 0.9849 0.9644 0.7415 0.6472 0.1655 0.0319 −0.1048 −0.4078
−0.0057 −0.0501 −0.0876 −0.2899 −0.3374 −0.4410 −0.4496 −0.4633 −0.5751
0
Table 2 ODE numerical integration. Numerical solver
Zero approximation
Number of steps
ode23 ode45 ode113
(3.671, 1.574e−02) (3.677, 1.279e−02) (3.706, 2.854e−05)
29 45 34
Table 3 v0 localization. N
v0
Zero approximation
7 9 13 15
3.706 3.795 4.147 4.810
2.854e−05 9.634e−02 1.566e−01 3.403e−02
In a first test we fix the value of N = 7, the Absolute error on the computed solution as AbsTol = 1e − 6 and ODE interval of integration is [0, 5]. Table 2 refer to the use of different numerical methods as explicit Runge–Kutta (4,5), AdamsBashforth-Moulton, Gear’s method (see [23]). In the second column, the first term represents an approximation of r and the second one corresponds to the value V (r ). We observe that a general purpose solver as ode23 is more efficient in terms of number of points used to reach the accuracy. On the contrary, ode113 is to prefer in terms of accuracy. In the following test (see Table 3 and Fig. 4) we fix ode113 as numerical solver, the Absolute error on the computed solution as AbsTol = 1e − 6. We numerically show that the location of the zero v0 of the function V (r ) depends on the dimension N.
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S. Cuomo, A. Perrotta / Nonlinear Analysis 74 (2011) 5784–5792
Fig. 4. V (r ) behavior for different values of N.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
G.H. Hardy, Notes on some points in the integral calculus, Messenger Math. 48 (1919) 107–112. G.H. Hardy, J.E. Littlewood, G. Pólya, Inequalities, Cambridge University Press, 1934. H. Brezis, J.L. Vazquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid 2 (10) (1997) 443–469. X. Cabré, Y. Martel, Weak eigenfunctions for the linearization of extremal elliptic problems, J. Funct. Anal. 156 (1998) 30–56. J.L. Vazquez, Domain of existence and blow-up for the exponential reaction diffusion equation, Indiana Univ. Math. J. 48 (1999) 677–709. J.L. Vazquez, E. Zuazua, The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse square potential, J. Funct. Anal. 173 (2000) 103–153. N. Chaudhuri, M. Ramaswamy, Existence of positive solutions of some semilinear elliptic equations with singular coefficients, Proc. Roy. Soc. Edinburgh Sect. A 6 (131) (2001) 1275–1295. N. Ghoussoub, A. Moradifam, On the best possible remaining term in the Hardy inequality, Proc. Natl. Acad. Sci. USA 105 (2008) 13746–13751. A. Alvino, R. Volpicelli, B. Volzone, On Hardy inequalities with a remainder term, Ric. Mat. 59 (2010) 265–280. F. Gazzola, H.C. Granau, E. Mitidieri, Hardy inequalities with optimal constants and remainder terms, Trans. Amer. Math. Soc. 356 (6) (2004) 2149–2168. Adimurthi, N. Chaudhuri, M. Ramaswamy, An improved Hardy–Sobolev inequality and its application, Proc. Amer. Math. Soc. 130 (2002) 485–505. Adimurthi, S. Filippas, A. Tertikas, On the best constant of Hardy–Sobolev inequalities, Nonlinear Anal. 70 (2009) 2826–2833. G. Barbatis, S. Filippas, A. Tertikas, A unified approach to improved Lp Hardy inequalities with best constants, Trans. Amer. Math. Soc. 356 (2004) 2169–2196. H. Brezis, M. Marcus, Hardy’s inequalities revisited, Ann. Sc. Norm. Super Pisa Cl. Sci. 25 (4) (1997) 217–237. S. Filippas, V.G. Maz’ja, A. Tertikas, Sharp Hardy–Sobolev inequalities, C. R. Math. Acad. Sci. Paris 339 (2004) 483–486. S. Filippas, A. Tertikas, Optimizing improved Hardy inequalities, J. Funct. Anal. 192 (2002) 186–233. G. Talenti, Best constant in Sobolev inequalities, Ann. Mat. Pura Appl. 110 (1976) 353–372. K.M. Chong, N.M. Rice, Equimeasurable rearrangements of functions, in: Queen’s Papers in Pure and Applied Mathematics, vol. 28, Queen’s University, Kingston, Ontario, 1971. B. Kawhol, Rearrangements and convexity of level sets in P.D.E., in: Lecture Notes in Mathematics, vol. 1150, Springer, Berlin, 1985. V.G. Maz’ya, On weak solutions of the Dirichlet and Neumann problems, Trans. Moscow Math. Soc. 20 (1969) 135–172. G. Talenti, Linear elliptic P.D.E.’s: level sets, rearrangements and a priori estimates of solutions, Boll. Unione Mat. Ital. 4-B (1985) 917–949. H. Sagan, Introduction to the Calculus of Variations, Dover Publications, Inc, New York, 1969. L.F. Shampine, Numerical Solution of Ordinary Differential Equations, Chapman and Hall, New York, 1994. L.F. Shampine, M.W. Reichelt, The MATLAB ODE Suite, SIAM J. Sci. Comput. 18 (1997) 1–22.
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On best constants in Hardy inequalities with a remainder term Salvatore Cuomo 1 , Adamaria Perrotta ∗ Dipartimento di Matematica ed applicazioni ‘‘R.Caccioppoli’’, Complesso Universitario M. S. Angelo, Via Cinthia, 80126 Napoli, Italy
article
abstract
info
Let Ω be a bounded open set of RN containing the origin. We compute the best value of the constant C (α, |Ω |) in
Article history: Received 19 March 2010 Accepted 21 May 2011 Communicated by Ravi Agarwal
∫
|∇ u|2 dx −
(N − 2 ) 2 4
Ω
MSC: 35J20 26D10 46E35
u2
∫
| x| 2
Ω
dx ≥ C (α, |Ω |) ‖u‖2
L N2N −α ,2
,
with α < 2 and u ∈ H01 (Ω ). Then we get the optimal value of C (|Ω |) in
∫
Keywords: Hardy inequality One-dimensional Calculus of Variations Best constants
|∇ u|3 dx −
N −3
Ω 1,3
where u ∈ W0
3
3 ∫
u3 Ω
| x| 3
dx ≥ C (|Ω |) ‖u‖3L3 ,
(Ω ).
© 2011 Elsevier Ltd. All rights reserved.
1. Introduction Let N > 2 and let Ω ⊆ RN be a bounded domain containing the origin. The well known Hardy–Sobolev inequality (see [1,2]) reads
∫
p
|∇ u| dx ≥
N −p
p ∫
p
Ω
up Ω
|x|p
dx,
∀u ∈ W01,p (Ω ) .
(1.1)
This inequality and its various improvements are used in many contexts, as in the study of stability of solutions of semilinear elliptic and parabolic equations (see [3–5]), or in the analysis of the asymptotic behavior of the heat equation with singular potentials (see [6]). N −p p
The constant CN ,p =
p
1 ,p
is the best one and there is no function u ∈ W0
(Ω ) for which it is achieved. For this
reason several authors have improved inequality (1.1) by adding a nonnegative correction term. In case p = 2 the first result is due to Brezis and Vazquez; in [3] they prove the so-called Hardy–Poincaré inequality
∫
|∇ u|2 dx − Ω
(N − 2) 2 4
u2
∫ Ω
|x|
dx ≥ Λ2 2
ωN |Ω |
N2
‖u‖2L2 ,
∀u ∈ H01 (Ω ) .
(1.2)
Here Λ2 denotes the first eigenvalue of the Laplace operator in the two dimensional unit disk, and ωN and |Ω | are respectively the N-dimensional Lebesgue measure of the unit ball B1 (0) ⊆ RN and of the set Ω . The value Λ2 optimal in the ball but it is not achieved in
∗
H01
(Ω ) .
Corresponding author. Tel.: +39 081675703; fax: +39 081675636. E-mail addresses: [email protected] (S. Cuomo), [email protected] (A. Perrotta).
1 Tel.: +39 081675624; fax: +39 081675636. 0362-546X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2011.05.069
ωN |Ω |
N2
is
S. Cuomo, A. Perrotta / Nonlinear Analysis 74 (2011) 5784–5792
5785
Inequality (1.2) can be also generalized by replacing the L2 -norm of u either with an Lq -norm of u with 1 ≤ q < 2∗ = N2N −2 (see [3]), or with a norm of u in a suitable Lorentz space (see [7,8]). The problem of finding the best constant in the remaining term in some of inequalities is still open. A share in this direction has been given in [9], where the authors get the optimality of the constant in the right hand sides of
∫
(N − 2) 2
2
|∇ u| dx −
4
Ω
u2
∫
|x|2
Ω
dx ≥
ωN |Ω |
N1
V0 ‖u‖2
L N2N −1 ,2
,
(1.3)
here Ω is a bounded domain of RN containing the origin, 1 ≤ p < 2∗ if N > 2, 1 ≤ p < +∞ if N = 2, u ∈ H01 (Ω ), and V0 is the first zero of the function
√
V (r ) = J0 2 r =
+∞ −
(−1)k
k=0
rk
(k!)2
.
We remark that inequality (1.3) can be seen as an Hardy–Sobolev inequality whose remainder term is the norm of u in the
Lorentz space L
2N N −1
N1 , 2 . The constant |ωΩN| V0 is not achieved.
A related Lp version of (1.1) and its improvements are also well known. In [10] Gazzola et al. show that an extra Lp -norm of u can be added to (1.1) for every 1 < p < N ; in other words they prove that there exists a constant C (p, |Ω |) > 0 such that
∫
|∇ u|p dx −
N −p
p ∫
p
Ω
up Ω
p
dx ≥ C (p, |Ω |) ‖u‖Lp ,
|x|p
∀u ∈ W01,p (Ω ) .
(1.4)
In particular, they study the asymptotic behavior of C (p, |Ω |), not its sharp value. Closely related questions are studied simultaneously and independently by Adimurthi et al. in [11]. Inequality (1.1) can be also improved by adding an Lq norm of the gradient of u; see [6,9] for questions related to optimality. Finally, for the sake of completeness, we refer to the following papers, which deal with Hardy inequalities with different type of remainder terms. See [12–16]. In this paper we are interested in finding the expression of the best constant in inequalities of the type
∫
|∇ u|p dx −
N −p
p ∫
p
Ω
up
|x|p
Ω
dx ≥ C [Q (u)]p ,
∀u ∈ W01,p (Ω ) ,
that involve a Lorentz or a Lebesgue norm of u as remainder term. Here, the definition of the best constant must be under1,p stood as follows. For any u ∈ W0 (Ω ), u ̸≡ 0, let
F (u) =
Ω
|∇ u|p dx −
N −p p
p
up
Ω | x| p
[Q (u)]p
,
and set
λ(Ω ) =
inf
1,p u∈W0 (Ω ) u̸≡0
F (u).
The best constant C is then defined as C = inf λ(Ω ′ ), where Ω ′ varies in the class of all bounded, open subset of RN containing the origin, such that Ω ′ = |Ω | . The first part of this paper is devoted to get the best value of the constantC (α, |Ω |) in a generalized version of (1.3), obtained by adding to the right-hand side the norm of u in the Lorentz space L N2N , 2 with α < 2 −α
∫
|∇ u|2 dx −
(N − 2) 2 4
Ω
u2
∫ Ω
|x|2
dx ≥ C (α, |Ω |) ‖u‖2
L N2N −α ,2
,
∀u ∈ H01 (Ω ) .
(1.5)
Then we compute the optimal constant C (p, |Ω |) in (1.4) when p = 3, that is
∫
|∇ u|3 dx − Ω
N −3 3
3 ∫
u3 Ω
|x|3
dx ≥ C (|Ω |) ‖u‖3L3 ,
∀u ∈ W01,3 (Ω ) .
(1.6)
In order to reach both results, as usual we firstly reduce the study to spherically symmetric functions defined on Ω # , simply by replacing u with u# . The Pólya–Szego principle (see [17]) and Hardy–Littlewood inequality (see [2]) ensure that the previous assumption is not restrictive. Then, the optimal values of C (α, |Ω |) and C (|Ω |) are obtained as minima of timely
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S. Cuomo, A. Perrotta / Nonlinear Analysis 74 (2011) 5784–5792
functionals associated respectively to inequalities (1.5) and (1.6). To get such minima, we have to apply classical arguments of Calculus of Variations. So, we simplify the inequalities with a suitable change of variables (see [3]) and we try to solve the Euler equations linked to such inequalities. This is possible for what it concerns the Euler equation of (1.5) but cannot be obtained for the second one. To overcome this difficulty, we study only the qualitative behaviour of such a solution. We point out that the properties of the solution to the Euler equation we found with the qualitative analysis are sufficient to apply classical arguments of Calculus of Variations to (1.6), as already done for inequality (1.5). 2. Best constant in Hardy–Sobolev inequalities with a remainder term In this section we are interested in determining the optimal value of the constants C (α, |Ω |) in (1.5) with α < 2 and C (|Ω |) in (1.6). To this aim, we firstly recall the definition of the spherically decreasing rearrangement of a function u and some related properties. Let Ω be a measurable subset of RN and u : Ω → R a measurable function in Ω . The distribution function of u is the decreasing map µ from [0, +∞[ into [0, +∞[ defined at any point t ≥ 0 as the measure of a level set of u, {x ∈ Ω : |u(x)| ≥ t }. The decreasing rearrangement u∗ of u is the distribution function of µ u∗ (s) = sup {t ≥ 0 : µ(t ) > s} ,
s ∈ (0, |Ω |) .
The main property of rearrangements is the fact that the distribution of u∗ is µ, in other words u and u∗ are equidistributed. Let us by ωN the measure of the unit ball of RN and by Ω # the ball of RN centered at the origin such that denote |Ω | = Ω # . For every x ∈ Ω # , the spherically decreasing rearrangement of u is defined as the decreasing rearrangement u∗ valued in ωN |x|N
u# (x) = u∗ ωN |x|N ,
x ∈ Ω #.
# Obviously, and spherically symmetric; moreover u and u# are equidistributed, and the level set u is decreasing # # x ∈ Ω : u (x) ≥ t is the ball centered at the origin and whose measure is µ(t ). For an exhaustive treatment of rearrangements see, for example, [18–21]. Here we just recall the Hardy–Littlewood inequality
∫ Ω#
∫
u# (x) v# (x) dx ≤
Ω
|u (x) v (x)| dx ≤
∫ Ω#
u# (x) v # (x) dx,
(2.1)
with u, v measurable functions (see [2]), and the Pólya–Szego principle (see [17])
∫ RN
# p ∇ u (x) dx ≤ 1,p
∫
|∇ u (x)|p dx
RN
(2.2)
where u ∈ W0 RN , 1 < p < N . Now let Ω be a bounded open set of RN containing the origin, N ≥ 2, 1 ≤ p < 2∗ if N > 2, 1 ≤ p < +∞ if N = 2 and u ∈ H01 (Ω ). We prove the following
Theorem 2.1. Let Ω be a bounded open set of RN , with N ≥ 2. There exists a constant C (α, |Ω |) such that inequality (1.5) holds for every α < 2 and u ∈ H01 (Ω ). The optimal value of C (α, |Ω |) is given by 2
ωNN
C (α, |Ω |) = w
2−α 0
|Ω |
2−α N
,
where w0 is the first zero of the function
W (r ) = J0
α
r 1− 2
=
1 − α2
+∞ −
(−1)k
k=0
r (2−α)k
(2 − α)2k (k!)2
.
Here J0 denotes, as usual, the Bessel function of zeroth order. Proof. The first step consists into the reduction of the problem to a spherically symmetric one. We replace u with u# and Ω with Ω # . By Hardy–Littlewood inequality (2.1) and Pólya–Szego principle (2.2), the left hand side of (1.5) decreases, while the right side increases. This implies that it is enough to prove the theorem only in the radial case. By using a suitable homothety, which transforms Ω # in the ball BR centered at the origin and with radius C (α, |Ω |)α−2 R, inequality (1.5) can be written as
∫
|∇ u|2 dx − BR
(N − 2) 2 4
u2
∫ BR
dx ≥ 2
|x|
u2
∫ BR
|x|α
dx,
∀u ∈ H01 (BR ) ,
with R to be determined. We first prove our result in dimension two.
(2.3)
S. Cuomo, A. Perrotta / Nonlinear Analysis 74 (2011) 5784–5792
5787
Fig. 1. Closed path.
If N = 2 inequality (2.3) reads as R
∫
′ 2 u
R
∫
u2 r 1−α dr ,
rdr ≥
0
∀u ∈ H01 (0, R) .
(2.4)
0
Let us consider the functional associated to (2.4) J (u) =
R
∫
′ 2 u
r − u2 r 1−α dr ,
(2.5)
0
and its Euler equation u′′ r + u′ + ur 1−α = 0.
(2.6)
As well known, the functions for which equality in (2.4) holds satisfy (2.6). Through a power expansion, we obtain the following solution of (2.6) α
W ( r ) = J0
r 1− 2
1 − α2
=
+∞ − k=0
(−1)k
r (2−α)k
(2 − α)2k (k!)2
.
The properties of the Bessel function of zeroth order, J0 , imply that W (r ) is decreasing, satisfies conditions W (0) = 1, W ′ (0) = 0 and has a zero, denoted with w0 . Our target is to show that the one parameter family of extremals u(r ) = cW (r ), with c ∈ R, minimizes the functional (2.5). We take R = w0 and for instance u ∈ C 1 (0, R); we embed W (r ) in the extremal field defined by the vector (1, Φ (r , u)), where Φ (r , u) =
u(r )W ′ (r ) W (r )
(see [22] Lemma 3.1). Denoted with f (r , u, u′ ) the integrand in (2.5), the differential form
ζ = P (r , u)dr + Q (r , u)du, whose coefficients are P (r , u) = f (r , u, Φ (r , u)) − Φ (r , u)fu′ (r , u, Φ (r , u)), Q (r , u) = fu′ (r , u, Φ (r , u)), is exact (see [22]), so the integral of ζ along the closed path represented in Fig. 1 is equal to zero. This ensures that, for every ε > 0 and for every w ∈ C 1 (0, R) with w(R) = 0 J (w) =
R
∫
f (r , w, w ′ ) − f (r , w, Φ (r , w)) + (Φ (r , w) − w ′ )fu′ (r , w, Φ (r , w)) dr
0
ε
∫ +
[f (r , W (ε), Φ (r , W (ε))) − Φ (r , W (ε))fu′ (r , W (ε), Φ (r , W (ε)))] dr +
0
R
∫ ε
f (r , W , W ′ )dr .
It can be shown that
∫ lim
ε→0 0
ε
[f (r , W (ε), Φ (r , W (ε))) − Φ (r , W (ε))fu′ (r , W (ε), Φ (r , W (ε)))] dr +
R
∫ ε
f (r , W , W ′ )dr = 0.
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S. Cuomo, A. Perrotta / Nonlinear Analysis 74 (2011) 5784–5792
Moreover we want to prove that f (r , w, ξ ) − f (r , w, ξ1 ) + (ξ1 − ξ ) fu′ (r , w, ξ ) ≥ 0, for any ξ1 , ξ in R (Weierstrass condition). We have f (r , w, ξ ) − f (r , w, ξ1 ) + (ξ1 − ξ ) fu′ (r , w, ξ )
[ =
′ 2 (N − 2)2 w2 w2 2 (N − 2)2 w12 w2 − 2 − ξ1′ + + 21 − 2 (ξ − ξ1 ) ξ1 ξ − 2 2 4
r
r
4
r
]
r
2 2 2 = ξ ′ − ξ1′ . The fact that W (r ) can be embedded in (1, Φ (r , u)) and the Weierstrass condition imply that W (r ) yields a minimum for J (u) (see [22], Theorem 3.3.1). Now let us consider the case N ≥ 3. If we perform the following change of variable (see [3]),
v(r ) = r −
N −2 2
u(r ),
r = |x| ,
with u(0) = 0 (this condition ensures that v ∈ H01 (Ω )), the Hardy–Sobolev inequality (2.3) can be read as the Sobolev inequality in the plane (2.4) with u replaced by v . Taking for instance u ∈ C01 (BR ) and using again the previous change of variable, we go back to u and obtain (2.3) with R = w0 . The restriction u ∈ C01 (BR ) can be removed by a density argument. A dimensional analysis on the constant shows that the optimal value of C (α, |Ω |) in (1.5) is 2
ωNN
C (α, |Ω |) = w
2−α 0
|Ω |
2−α N
.
Finally, we observe that such a constant is not attained; this is due to the fact that it corresponds to equality in (2.4), which is realized when v(r ) = cW (r ) and so u(|x|) = c |x|−
N −2 2
W (|x|),
but u(|x|) is not in H because W (0) = 1. 1
The result obtained in Theorem 2.1 suggested us the idea of using the same procedure to get the optimal value of the constant C (|Ω |) in (1.6) too. The strong difference between the two results consists in the fact that we cannot solve the Euler equation associated to (1.6). To overcome this difficulty we analyze such equation in a different way; through a numerical approach we don’t get the explicit expression but the qualitative behavior of its solution. In Theorem 2.2 we shall show how to use these information to get the best value of C (|Ω |). Theorem 2.2. Let Ω be a bounded open set of RN , with N > 3. There exists a constant C (|Ω |) such that inequality (1.6) holds 1 ,3 for every u ∈ W0 (Ω ). The optimal value of the constant C (|Ω |) is given by C (|Ω |) = v
3 0
ωN |Ω |
N3
,
where v0 is the first zero of the solution of the Euler equation of the functional associated to (1.6). Proof. As in Theorem 2.1, the first step consists in the reduction of the problem to a spherically symmetric one. To this aim, we replace u with u# and Ω with Ω # . By Hardy–Littlewood inequality (2.1) and Pólya–Szego principle (2.2), the left hand side of (1.6) decreases, while the right side does not change. This implies that it is enough to prove the theorem only in the radial case. Using a suitable homothety, which transforms Ω # in the ball BR centered at the origin and with radius C (|Ω |)3 R, inequality (1.6) can be written as
∫
|∇ u|3 dx −
N −3
BR
3
3 ∫
u3 BR
∫
dx ≥ 3
| x|
u3 dx,
∀u ∈ W01,3 (Ω ) ,
(2.7)
BR
with radius R to be determined. If we perform the following change of variable N −3 u(r ) = r − 3 v(r ),
r = |x| ,
with v(0) = 0 and take for instance u ∈
(2.8) C01
(BR ), the Hardy–Sobolev inequality (2.7) becomes
S. Cuomo, A. Perrotta / Nonlinear Analysis 74 (2011) 5784–5792 R
∫
2 3 dr ≥ r 2 v ′ + (N − 3)r v v ′
5789
R
∫
r 2 v 3 dr .
(2.9)
0
0
Let us consider the functional associated to (2.9) J (v) =
∫
R
2 3 r 2 v ′ + (N − 3)r v v ′ − r 2 v 3 dr ,
(2.10)
0
and its Euler equation 2r v v − 2 ′ ′′
2 3
(N − 3)r vv + ′′
9−N
3
2
r v′
2
− (N − 3)vv ′ − r 2 v 2 = 0. 3
(2.11)
We have to solve such equation; unfortunately, a power expansion does not work in this case because it gives only a recursive relation between its coefficients. For this reason, we reformulate (2.11) in terms of the initial value problem
′ y1 = y2 y21 ′ − 9−3 N y22 + 23 (N − 3) y1ry2 r y2 = 2ry2 − 32 (N − 3)y1 y1 (0) = 1 y2 (0) = 0
in BR in BR ,
(2.12)
where v(r ) := y1 (r ), v ′ (r ) := y2 (r ) and the initial data represent the first two coefficients of the series. In Section 3 we show that (2.12) has a unique solution; obviously this ensures that there exists a solution V (r ) to (2.11). Moreover, thanks to a numerical analysis, we can deduce that V (r ) has a zero, v0 , and that V (r ) is decreasing in the interval [0, v0 ] (see Section 3.1). Even without the explicit expression of V (r ), the previous properties are enough to apply classical arguments of the Calculus of Variations to get a minimum of (2.10). Firstly we choose R = v0 ; proceeding as in Theorem 2.1 is possible to prove that V (r ) can be embedded in the extremal field defined by the vector (1, Φ (r , v)), where Φ (r , v) = in (2.10), we get
v(r )V ′ (r ) V (r )
(see [22] Lemma 3.1). Denoting by f (r , v, v ′ ) the integrand
f (r , w, ξ ) − f (r , w, ξ1 ) + (ξ1 − ξ ) fv ′ (r , w, ξ ) ≥ 0, for any ξ1 , ξ in R (Weierstrass condition). In fact f (r , w, ξ ) − f (r , w, ξ1 ) + (ξ1 − ξ ) fv ′ (r , w, ξ ) = ξ13 − ξ 3 − 3ξ12 (ξ1 − ξ ). The Weierstrass condition follows from the fact that V (r ) is decreasing in [0, R]. We conclude that V (r ) yields a minimum for J (v) (see [22] Theorem 3.3.1), and so we obtain (2.9) with R = v0 . Performing again the change of variables (2.8) and coming back to u we obtain (2.7) with R = v0 . The restriction u ∈ C01 (BR ) can be removed by density. A dimensional analysis of the constant shows that the optimal value of C (|Ω |) in (1.6) is C (|Ω |) = v03
ωN |Ω |
N3
.
Finally, we observe that such a constant is not attained; the reason of this is the fact that it corresponds to equality in (1.4), which happens for v(r ) = cV (r ) and so u(|x|) = c |x|−
N −3 3
1,3
but u(|x|) is not in W0
V (|x|), because V (0) = 1.
3. Numerical issues on the Euler equation The main problem in the proof of Theorem 2.2 is to solve the Euler equation (2.11). As pointed out in the previous section, it is not possible to get the explicit expression of such a solution V (r ). To overcome this difficulty, we rewrite the Eq. (2.11) in terms of the initial value problem (2.12) and analyze it from a numerical point of view. This kind of approach ensures not only that there exists a solution V (r ) to (2.11), but also gives us its qualitative behavior. This section is organized as follows: in Section 3.1 we get the solution of the problem (2.12) through a Numerical Method; in Section 3.2 we show the qualitative behavior of V (r ).
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Fig. 2. Lax–Richtmyer scheme.
3.1. Numerical solution of the Euler equation In this subsection we show how to apply the scheme just described to analyze the initial value problem (2.12). To this aim we first prove that (2.12) has a unique solution; then, fixed any consistentnumerical method for the discrete problem, if this satisfies the Lipschitz condition, then the method is also zero-stable. From the Lax Scheme we deduce that the method is convergent and hence the solution to discrete problem associated to (2.12) converges to the continuous one (see [23], Fig. 2). Let us denote by g (r , y1 , y2 ) the right hand side of (2.12),
g (r , y1 , y2 ) = y2 ,
y21 r
−
9−N 3
y22 + 32 (N − 3)
y1 y2 r
2ry2 − 23 (N − 3)y1
,
the Jacobian matrix of g is
0
Jg = 18t y1 y2 − 3t (N − 3) 2
1
+ (N − 3) 2 [3ty2 − (N − 3)y1 ]2 y21
2 2 y2
3t (N − 9) 2
y22
− 2t (N − 9)(N − 3)y1 y2 − 2(N − 3) + 9t
2
3
y21
.
2t [3ty2 − (N − 3)y1 ]
2
Extending by continuity at the origin both g and Jg , we deduce that the problem (2.12) satisfies the Cauchy Theorem, so there exists a unique solution to (2.12). To describe the behavior of V (r ) we choose, for example, the explicit Euler Method and we get the following discrete version of (2.12)
zn+1 = zn + hqn zn2 − 9−3 N q2n + 32 (N − 3) znrnqn rn qn+1 = qn + h , 2rn wn − 32 (N − 3)zn z0 = 1 q0 = 0 where ri = r0 + ih, i = 1, . . . , n, h = Rn , zi = y1 (ri ) and qi = y2 (ri ). The sequences zn and qn represent an approximation of V (r ) := y1 (r ) and V ′ (r ) := y2 (r ), respectively. From a direct computation we deduce (i) the sequence qn < 0, i.e. V (r ) is decreasing; (ii) z0 = V (0) = 1 and moreover there exists an r ∈ [0, R] such that zn (r ) = V (r ) < 0. Because of the convergence of the Euler Method and of the continuity of the solution V (r ), we trivially conclude that there exists a v0 ∈ [0, R] such that zn (v0 ) = V (v0 ) = 0. This procedure works for every N > 3, so the solution V (r ) to the Euler equation (2.11) has a zero v0 and is decreasing in [0, v0 ] in any dimension N > 3. Fig. 3 and Table 1 refer to the use of the Euler method to analyze problem (2.12) in the model case N = 11. 3.2. More experiments In this subsection we want experimentally confirm that the qualitative analysis described in the previous is independent of the election of the numerical method used to solve the problem (2.12) and from dimension N. The numerical simulations are carried out on a Intel(R) Pentium(R) M with 1.7 GHz processor by means the ODE suite numerical solvers of initial value problems for ordinary differential equations of Matlab R14 [24]. We remark that the expression ‘‘V (r ) has a zero in r’’ means that the function V (r ) assumes in r a value that is close to zero, with an |v −r | approximation which depends on the accurancy required; moreover 0v < 10−k . 0
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Fig. 3. Numerical solution and derivative behavior.
Table 1 Sequences of Euler method. rn
Numerical solution zn
Derivative qn
0.0010 0.3010 0.9010 1.2010 2.4010 2.7010 3.9010 4.2010 4.5010 5.1010
1.0000 0.9994 0.9849 0.9644 0.7415 0.6472 0.1655 0.0319 −0.1048 −0.4078
−0.0057 −0.0501 −0.0876 −0.2899 −0.3374 −0.4410 −0.4496 −0.4633 −0.5751
0
Table 2 ODE numerical integration. Numerical solver
Zero approximation
Number of steps
ode23 ode45 ode113
(3.671, 1.574e−02) (3.677, 1.279e−02) (3.706, 2.854e−05)
29 45 34
Table 3 v0 localization. N
v0
Zero approximation
7 9 13 15
3.706 3.795 4.147 4.810
2.854e−05 9.634e−02 1.566e−01 3.403e−02
In a first test we fix the value of N = 7, the Absolute error on the computed solution as AbsTol = 1e − 6 and ODE interval of integration is [0, 5]. Table 2 refer to the use of different numerical methods as explicit Runge–Kutta (4,5), AdamsBashforth-Moulton, Gear’s method (see [23]). In the second column, the first term represents an approximation of r and the second one corresponds to the value V (r ). We observe that a general purpose solver as ode23 is more efficient in terms of number of points used to reach the accuracy. On the contrary, ode113 is to prefer in terms of accuracy. In the following test (see Table 3 and Fig. 4) we fix ode113 as numerical solver, the Absolute error on the computed solution as AbsTol = 1e − 6. We numerically show that the location of the zero v0 of the function V (r ) depends on the dimension N.
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Fig. 4. V (r ) behavior for different values of N.
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