Acta Mech 223, 685–704 (2012) DOI 10.1007/s00707-011-0588-6
Teodor M. Atanackovic · Marko Janev · Stevan Pilipovic · Dusan Zorica
Complementary variational principles with fractional derivatives
Received: 29 April 2011 / Revised: 7 September 2011 / Published online: 20 December 2011 © Springer-Verlag 2011
Abstract Complementary variational principles for a class of fractional boundary value problems are formulated. They are used for the error estimates of solutions for a general mechanical problems, first Painlevé equation also given in the form with fractional derivatives and in the task of image regularization.
1 Introduction There exists a rich literature devoted to linear differential equations with non-integer derivatives, see [18, 22,25,27,29,33] and the references therein. This is not the case for equations with left and right fractional derivatives since such equations are much more complicated. The reason is that the Laplace transform, the main tool in solving fractional differential equations, cannot be used in this case, see [11,12,31]. However, the variational principles with the fractional derivatives in the Lagrangian function lead to Euler-Lagrange equations with both left and right fractional derivatives (see [2–4,8,14,19]). Recall, fractional derivatives with respect to time variable are used to model systems where the memory plays an important rule (see [24]), while fractional derivatives with respect to space variable model non-local phenomena (see [10,15]). It is, therefore, important to develop methods for solving such equations. Among such methods, we propose the Ritz approach adjusted to fractional order equations, see [6,7,35] for integer order equations. In general, direct methods of the variational calculus provide an important tool for solving variational problems, especially if the complementary functionals are known. In this case, an easy error estimate procedure can be accomplished. In this paper, we do not consider the existence theorems concerning the solutions to Euler-Lagrange equations, i.e., minimizers of the fractional variational problems. We assume that solutions exist. Our intention is to develop complementary variational principles for the fractional variational problems along the lines T. M. Atanackovic Department of Mechanics, Faculty of Technical Sciences, University of Novi Sad, Trg D. Obradovica, 6, 21000 Novi Sad, Serbia E-mail:
[email protected] M. Janev · D. Zorica (B) Mathematical Institute, Serbian Academy of Sciences and Arts, Beograd, Kneza Mihaila, 36, 11000 Beograd, Serbia E-mail:
[email protected] M. Janev E-mail:
[email protected] S. Pilipovic Department of Mathematics, Faculty of Natural Sciences and Mathematics, University of Novi Sad, Trg D. Obradovica, 4, 21000 Novi Sad, Serbia E-mail:
[email protected]
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of [6,7,9,34]. In Sects. 3 and 4, we shall formulate theorems that give sufficient conditions for the existence of the fractional complementary variational principles when the Lagrangians are of the form: 2 (i) L 1 t, u, a Dαt u = 21 a Dαt u +Π (t, u) , u = (u 1 , . . . , u n ) is a vector function with the scalar argument t, and a Dαt u = a Dαt u 1 , . . . , a Dαt u n (see 4); (ii) L 2 (u, α grad u) = Π (u) + R(α gradu), u is a scalar function with the vector argument x = (x1 , . . . , xn ) and α grad u = α Dx1 u, . . . , α Dxn u (see 7). The practical aim of these theorems is to obtain an error estimate of an approximate solution to the EulerLagrange equations that correspond to the mentioned Lagrangians without knowing their exact solutions. These theorems will be proved in Sect. 5. The form of Lagrangian L 1 is motivated by the applications in mechanics, as well as in the study of the first Painlevé equation, which we also formulate in terms of the fractional derivatives. Since we do not have the existence result for the solution to boundary value problem for fractional and classical first Painlevé equation, we made the necessary assumption on their existence in order to formulate the complementary principle and find the error estimate. The form of Lagrangian L 2 is motivated by the problems of the image regularization. Sections 3.1 and 3.2 are devoted to the application of the theory developed in Sect. 3 to general mechanical problems, where Lagrangian consist of two separate terms, representing kinetic and potential energy. These problems are also analyzed numerically, and the error estimate is obtained. In Sect. 4.1, we use the theory presented in Sect. 4 for the image regularization task and give the error estimate as well.
2 Notation The left Riemann–Liouville fractional derivative of order α ∈ [n − 1, n) , n ∈ N, is defined (see [33]) as α a Dt
f (t) =
d dt
n
1 f (t) = Γ (n − α)
n−α a It
d dt
n t (t − τ )n−α−1 f (τ ) dτ, t ∈ [a, b] .
(1)
a
Similarly, the right Riemann–Liouville fractional derivative of order α ∈ [n − 1, n) , n ∈ N, is defined as α t Db
b d n n−α 1 d n f (t) = − f (t) = − (τ − t)n−α−1 f (τ ) dτ, t ∈ [a, b] . t Ib dt Γ (n − α) dt
(2)
t
Function f in (1) and (2) belongs to the set of n-times absolutely continuous function, denoted by AC n [a, b] , i.e., f has n − 1 continuous derivatives, and the n-th derivative is an integrable function in [a, b] . We shall also need the following integration by parts formula, see [33], b a
α a Dt
b
f (t) g (t) dt =
f (t) t Dαb g (t) dt.
(3)
a
Here f ∈ a Itα (L p [a, b]) and g ∈ t Ibα (L q [a, b]) , 1p + q1 ≤ 1 + α, where a Itα (L p [a, b]) (and t Ibα (L q [a, b])) is the class of functions f that can be represented as f = a Itα ϕ f = t Ibα ϕ for some ϕ ∈ L p [a, b] , p ≥ 1. b Note that in the limit α → 1, (3) transforms into the classical formula, namely a dtd f (t) g (t) dt = b [ f (t) g (t)]ab − a f (t) dtd g (t) dt. n Let u (t) = (u 1 (t) , . . . , u n (t)) , t ∈ [a, b] , n ∈ N. We assume u ∈ AC 1 [a, b] and define the left and right Riemann–Liouville fractional derivative of u of order α ∈ (0, 1) as follows: α α α α α α (4) a Dt u := a Dt u 1 , . . . , a Dt u n , t Db u := t Db u 1 , . . . , t Db u n , where a Dαt u j , respectively, t Dαb u j , j ∈ {1, . . . , n} , are the left and right Riemann–Liouville fractional derivatives defined by (1) and (2).
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For the case of a function of a vector-valued argument, we shall need a generalization of a standard nabla operator ∇ = ∂∂x1 , . . . , ∂ ∂xn . Let f ∈ C 1 (Rn ) . The (standard) gradient of a function is grad f = ∇ f =
∂f ∂f ,..., ∂ x1 ∂ xn
.
Also in the case of the composition of functions f (x) = f (g (x)) , x ∈ Rn , g = (g1 , . . . , gn ) , we use notation ∂f ∂f grad g f = ∇ g f = ,..., . ∂g1 ∂gn In order to “fractionalize” the Nabla operator ∇, we recall the Fourier transform of u ∈ L 1 (Rn ) , uˆ (ξ ) ≡ F [u (x)] (ξ ) :=
u (x) e−iξ ·x dn x, ξ ∈ Rn ,
Rn
where the dot in ξ · x denotes the scalar product. The fractional Sobolev space H s (Rn ) , s > 0, is defined by
H s Rn = u ∈ L 2 Rn : u H s (Rn ) < ∞ , where ⎛ ⎞1/2 s 2 u H s (Rn ) = ⎝ 1 + |ξ |2 uˆ (ξ ) dn ξ ⎠ .
(5)
Rn
The left and right fractional Nabla operators are α
∇ :=
α
Dx1 , . . . , α Dxn , ∇ α := Dαx1 , . . . , Dαxn ,
(6)
where α
Dx j u := F −1
iξ j
α
uˆ (ξ ) , Dαx j u := (−1)α α Dx j u, j = 1, . . . , n.
We note that, like in the classical setting, we have the fractional gradients if the fractional Nabla operators (6) are applied to f ∈ H s (Rn ) as α
grad f := α ∇ f =
α
Dx1 f, . . . , α Dxn f , gradα f := ∇ α f = Dαx1 f, . . . , Dαxn f ,
(7)
and the fractional divergences of a vector function g = (g1 , . . . , gn ) , g ∈ (H s (Rn ))n , that are defined as α
div g := α ∇ · g =
n
α
Dx j g j , divα g := ∇ α · g =
j=1
n
Dαx j g j .
j=1
Note that in the previous expressions, we used dot to denote the scalar product g · h := nj=1 g j h j . Also, we use the notation g 2 ≡ g 2 = g · g = nj=1 g 2j . The integration by parts also holds, i.e., for f ∈ H s (Rn ) and g ∈ (H s (Rn ))n , s ≥ 1 Rn
α
grad f (x) · g (x) dn x =
Rn
f (x) divα g (x) dn x.
(8)
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3 Lagrangian depending on a vector function of scalar argument n We assume u = (u 1 , . . . , u n ) ∈ AC 2 [a, b] . Let us consider the Lagrangian 1 α 2 (9) L t, u, a Dαt u = a Dt u + Π (t, u) , 2 2 2 where a Dαt u = nj=1 a Dαt u j . We assume that [a, b] t → Π (t, ·) is a C 1 mapping from [a, b] into the space of function with the continuous derivatives up to order two, denoted by C 2(Rn ) , i.e., Π (t, ·) : x → Π (t, x) , Π (t, ·) ∈ C 2 (Rn ) . We shall denote this space by C 1 [a, b] ; C 2 (Rn ) . In formulating the Hamilton principle of least action, one is faced with the problem of finding a minimum of a functional J (u) ≡ I
u, a Dαt u
b =
L
t, u, a Dαt u
b dt =
a
1 α 2 a Dt u + Π (t, u) dt, 2
(10)
a
Un,
where u belongs to a space of admissible functions defined as n U n = u : u ∈ AC 2 [a, b] , u satisfies prescribed boundary conditions . We refer to [2,3,8] for the formulation of the Hamilton principle and Euler-Lagrange equations for the Lagrangian depending on a scalar function, as well as to [23] for the Lagrangian depending on a vector function. In the sequel, we shall also refer to the principle of the least action as the primal variational principle. Requiring that (10) attains a minimum over U n , we obtain the Euler-Lagrange equations α t Db grada Dαt u L + grad u L = 0. Since L has the form (9), we have grada Dαt u L = a Dαt u j , j ∈ {1, . . . , n} , and j
α α t Db a Dt u + grad u Π = 0.
(11)
Each of the Euler-Lagrange equations can be written as a system of two equations with independent functions u and p. These equations are called the canonical (or the Hamilton) equations (for the case of a scalar function see [31]). In order to obtain the Hamilton equations, like in the classical case, we define the generalized momentum and the corresponding Hamiltonian as p := grada Dαt u L = a Dαt u, 1 H (t, u, p) := p · a Dαt u − L t, u, a Dαt u = p 2 − Π (t, u) , 2
(12) (13)
where p ∈ P n = (AC 1 [a, b])n at least, since a Dαt u j = dtd a It1−α u j and a It1−α u j ∈ AC 2 [a, b] , j = 1, . . . , n. b We write (10) in terms of the Hamiltonian as a p · a Dαt u − H (t, u, p) dt, and use the integration by parts formula (3) to obtain b I (u, p) =
α t Db
p · u − H (t, u, p) dt =
a
b
1 2 α D p · u − + Π u) dt. p (t, t b 2
(14)
a
Note that I is a functional with independent functions u and p. By requiring that I attains its minimum at (u, p) ∈ U n × P n , we obtain the Hamilton equations α a Dt u
= grad p H = p,
α t Db
p = grad u H = −grad u Π.
(15)
In order to formulate the dual functional, we pose the following condition that enables us to solve uniquely the second Hamilton equation (15)2 with respect to u. Condition 1 (a) Π ∈ C 1 [a, b] ; C 2 (Rn ) ;
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(b) grad Π (t, ·) : Rn → D is a bijection of Rn to an open subset D of Rn for every t ∈ [a, b] ; 2 (t,x) n (c) det ∂ ∂Π xi ∂ x j = 0 for every t ∈ [a, b] and every x ∈ R . If Condition 1 is satisfied, then there exists Φ : [a, b] × D → Rn , of class C 1 , such that u (t) = Φ t, t Dαb p (t) , t ∈ [a, b]
(16)
and α t Db
p ≡ −gradΦ Π t, Φ t, t Dαb p , t ∈ [a, b] ,
(17)
i.e., the second Hamilton equation (15)2 is identically satisfied with Φ. Now, the dual functional G is obtained from (14) as ( p ∈ P n ) G ( p) ≡ I Φ t, t Dαb p , p =
b
α t Db
p ·Φ
1 2 α p − p + Π t, Φ t, t Db p dt. 2
t, t Dαb
(18)
a
Summarizing, we say that the complementary principle holds if for the primal functional J, (10), there exists the dual functional G, (18), so that if J attains a minimum at a function u ∈ U n , then G attains a maximum at a function p ∈ P n , where the connection between u and p is given by (12). Theorem 1 Assume that Π satisfies Condition 1. Additionally, assume that there exists a constant c > 0, such that n ∂ 2 Π (t, x) dxi dx j ≥ c (dx)2 , t ∈ [a, b] , x ∈ Rn . ∂ xi ∂ x j
(19)
i, j=1
Then the complementary principle holds for the functionals (10) and (18), i.e., it holds J (u) ≤ J (U) , G ( p) ≥ G ( P) ,
(20)
where U = u + δu and P = p + δ p for u, U ∈ U n and p, P ∈ P n . Moreover, U − u L 2 [a,b] ≤
2 (J (U) − G ( P)). c
(21)
The proof of Theorem is given in Sect. 5. For the estimates of type (21), where the Lagrangians in primal and dual functionals contain derivatives of integer order, see for example [6,9]. Remark 1 (i) Recall [16] that the necessary and sufficient condition for the positive definiteness of 2 (t,x) ∗ ∗ V (t, dx) = i,n j=1 ∂ ∂Π xi ∂ x j dx i dx j is that there exists a continuous function V (dx) > 0, V (0) = 0, such that V (t, dx) ≥ V ∗ (dx) for all (t, dx) ∈ [a, b] × Rn . Thus, (19) is the sufficient condition for the positive definiteness. b b 2 (ii) In certain cases, we can find k > 0, such that a a Dαt δu dt ≥ k a (δu)2 dt. Then we have the sharper form of (21): U − u L 2 [a,b] ≤ We shall show this in Proposition 1.
2 (J (U) − G ( P)). c+k
(22)
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3.1 An example with the quadratic potential Π In the following, we treat the Lagrangian (9) for the case of the one-dimensional function u = u (t) , t ∈ [0, 1] and the quadratic potential Π (t, u) = 21 w (t) u 2 (t) − q (t) u (t) , t ∈ [0, 1] , so that the Lagrangian (9) takes the form 1 α 2 1 2 L t, u, 0 Dαt u = (23) 0 Dt u + wu − qu, 2 2 where α ∈ (0, 1) , w ∈ C 1 [0, 1] , mint∈[0,1] w (t) = w0 > 0, q ∈ C 1 [0, 1] . We set the space of admissible functions to be
U = u : u ∈ C 1 [0, 1] , u (0) = 0, 0 Dαt u (t)t=1 = 0 . According to [26], if u ∈ C 1 [a, b] and α ∈ (0, 1) , then a Dαt u ∈ L r [a, b] t Dαb u ∈ L r [a, b] for 1 ≤ r < α1 . Since α ∈ (0, 1) , and u ∈ C 1 [0, 1] , u (0) = 0, we have that 0 Dαt u is continuous on [0, 1] . Note that (23) represents the fractionalized Lagrangian considered in [6], p. 8. The Euler-Lagrange equation (11), subject to specified boundary conditions, reads α α α (24) t D1 0 Dt u + wu − q = 0, u (0) = 0, 0 Dt u (t) t=1 = 0, while the generalized momentum, according to (12), is p = 0 Dαt u. Since the function Π satisfies Condition 1, the second canonical equation (15)2 solved with respect to u yields α tD p − q u = Φ t, t Dα1 p = − 1 . w Therefore, the complementary functionals J and G, given by (10) and (18), respectively, become
1 J (u) = 0
1 G ( p) = − 2
1 α 2 1 2 0 Dt u + wu − qu dt, 2 2 1
α t D1 p
−q w
(25)
2 +p
2
dt.
(26)
2 (J (U ) − G (P)), w0
(27)
0
The error estimate (21), given in Theorem 1, yields U − u L 2 [0,1] ≤
where U and P are approximate solutions to (24), since (19) yields ∂ 2 Π (t, u) = w (t) ≥ w0 , t ∈ [0, 1] . ∂u 2 Let us show that we can improve the error estimate (27) in the sense of Remark 1 (ii). Namely, we state the following proposition. α ∞ Proposition 1 Let f ∈ L 1 (0, 1) have an integrable fractional derivative 0 Dt f ∈ L (0, 1) of order α ∈ 1 1−α 2 f (t) = 0. Let s1 , s2 > 1, and s2 < 3−2α . Then it holds 2 , 1 , such that 0 It t=0
1
1 | f (τ )| dτ ≤ K 2
0
α 0 D f (τ )2 dτ, t
0
where K =
(28)
σ 2σ 1
(Γ (α))2 (σ + α − 1)2σ (ρs1 + 1) s1
, σ =
1 1 1 − , ρ=2 α+ − 3. s2 2 s2
(29)
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Proof See Theorem 5.5, p. 57, from [5] and assume: p = 2, r = 2, ω1 = ω2 = 1, l = 1, μ1 = 0, r1 = 2.
Therefore, by (28), (22) becomes U − u L 2 [a,b] ≤
2 w0 +
1 K
(J (U ) − G (P)).
(30)
Numerical analysis. In order to obtain an approximate solution to (24), we use the Ritz method. Thus, we assume U (t) =
N
ci Ui (t) , t ∈ [0, 1] ,
(31)
i=1
where ci are arbitrary constants and Ui , i = 1, . . . , N , are trial functions satisfying boundary conditions (24)2,3 . This leads to α 0 Dt Ui
1 Ui (0) + (t) = Γ (1 − α) t α Γ (1 − α) =
1 Γ (1 − α)
t 0
t 0
Ui (τ ) dτ (t − τ )α
Ui (τ ) dτ. (t − τ )α
Put t 0
Ui (τ ) dτ = gi (t) , t ∈ [0, 1] , (t − τ )α
(32)
where gi , i = 1, . . . , N , are functions that will be given bellow. Boundary condition (24)3 implies gi (1) = 0, i = 1, . . . , N . From [17], p. 572, we have that the solution of (32) reads Ui (t)
sin (απ) d = π dt
t 0
gi (τ ) (t − τ )1−α
dτ, t ∈ [0, 1] , i = 1, . . . , N .
(33)
Now, since gi (1) = 0, we chose gi in the form gi (t) = (1 − t)i , t ∈ [0, 1] , i = 1, . . . , N . Substituting this in (33), we obtain, up to an arbitrary constants Ci t Ui (t) = Ci 0
(1 − τ )i dτ, t ∈ [0, 1] , i = 1, . . . , N . (t − τ )1−α
We put Ci = 1, i = 1, . . . , N . The first three functions, defined for t ∈ [0, 1], are tα t α+1 − , α α (α + 1) tα t α+1 t α+2 −2 +2 , U2 (t) = α α (α + 1) α (α + 1) (α + 2) tα t α+1 t α+2 t α+3 −3 +6 −6 . U3 (t) = α α (α + 1) α (α + 1) (α + 2) α (α + 1) (α + 2) (α + 3) U1 (t) =
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Let Pi = Γ (α) (1 − t)i , i = 1, . . . , N , and let ki , i = 1, . . . , N , be arbitrary constants. We shall find P in the form N
P (t) =
ki Pi (t) , t ∈ [0, 1] .
(34)
i=1
For the numerical purposes, we take w = w0 = q = 1, α = 0.6, and N = 9. By substituting (31) into (25) and minimizing with respect to ci , i = 1, . . . , 9, and by substituting (34) into (26) and maximizing with respect to ki , i = 1, . . . , 9, we obtain J (U ) = −0.176499, G (P) = −0.176850. 2 By applying Proposition 1, we obtain s2 < 3−2α = 1.1111. We chose s1 = 2 and s2 = 1.1 so that from (29) we obtain K = 9.9771. The error estimate (30) yields 2 U − u L 2 [0,1] = (J (U ) − G (P)) = 0.0252692, w0 + K1
while from (27), we have U − u L 2 [0,1] ≤
2 (J (U ) − G (P)) = 0.0265053,
so that we improved the error estimate by 4.7%. Multiplying (24)1 by u, we obtain α α 2 t D1 0 Dt u u + u − u = 0.
(35)
Integration of (35) and the use of (3) lead to 1
α 2 0 Dt u u
1
+ u − u dt = 2J (u) + 2
0
u (t) dt = 0. 0
Therefore, 1 J (u) = − 2
1 u (t) dt 0
and 1 −0.176499 ≤ − 2
1 u (t) dt ≤ −0.176850. 0
In Fig. 1, we present the plot of an approximate solution U for (24), obtained as described above. 3.2 Complementary principles for the fractional and classical first Painlevé equation Our aim in this section is to formulate the complementary variational principles for the fractional and classical first Painlevé equation. In the case of the fractional equation, we formulate the complementary principle on a specific domain of admissible functions. In the case of the classical equation, we know the existence result for the Cauchy problem (see [20]) and define the space of admissible functions in accordance with the initial data for which solutions exist. In this way, we construct, in this set of admissible functions, a numerical solution and give the error estimate. Our analysis opens several questions and indicates the possibility of the use of the complementary principles in both cases. In both fractional and classical Painlevé equation, we set the space of admissible functions to be ! t 2 U = u : u ∈ C [0, 1] , u (0) = 0, u (1) = 1, u (t) ≥ , t ∈ [0, 1] . (36) 6
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U t 0.5
0.4
0.3
0.2
0.1
0.0
0.2
0.4
0.6
0.8
1.0
t
Fig. 1 Approximate solution U for (24)
The fractional first Painlevé equation. Consider the Lagrangian of the form (9) 1 α 2 3 L t, u, 0 Dαt u = 0 Dt u + Π (t, u) , Π (t, u) = 2u − tu. 2 The Euler-Lagrange equation (11) corresponding to this Lagrangian is α α 2 t D1 0 Dt u (t) = −6u (t) + t, t ∈ [0, 1] ,
(37)
where u ∈ U , given by (36). Actually, we can assume u ∈ AC 2 ([0, 1]). We note that we do not if the solution to (37) exists in U , and this is an open problem. If α = 1, then (37) is the first Painlevé equation on the finite time interval t ∈ [0, 1] . The primal functional (10) reads 1 J (u) =
1 α 2 3 0 Dt u + 2u − tu dt. 2
(38)
0
The generalized momentum (12) is p = 0 Dαt u, so that the Hamiltonian (13) takes the form H (t, u, p) =
1 2 p − 2u 3 + tu, t ∈ [0, 1] , u, p ∈ R. 2
The functional I , given by (14), reads I (u, p) =
1
α t D1 p
1 α 1 2 1 2 3 3 u − p + 2u − tu dt = D p u − + 2u − tu dt. p t 1 2 2
0
(39)
0
Canonical equations (also given by (15)) are obtained by the minimization of (39) as α 0 Dt u
= p,
α t D1 p
= −6u 2 + t, t ∈ [0, 1] , u, p ∈ R.
(40)
Since Π satisfies Condition 1 and u ∈ U , the second canonical equation (40)2 solved with respect to u yields −t Dα1 p + t α u = Φ t, t D1 p = , t ∈ [0, 1] , p ∈ R. (41) 6 Therefore, the dual functional follows from (18), and it takes the form 1 √ " 3 1 6 G ( p) = − (42) −t Dα1 p + t + p 2 dt. 9 2 0
Now, the error estimate follows from Theorem 1 and it is given by (21).
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The classical first Painlevé equation. It is obtained from (37) as u¨ (t) = 6u 2 − t, t ∈ [0, 1] ,
(43) α α α since limα→1 0 Dαt u (t) = u˙ (t) ≡ du(t) dt , lim α→1 t D1 u (t) = −u˙ (t) and therefore lim α→1 t D1 0 Dt u (t) = −u¨ (t) ≡
d2 u(t) . dt 2
We recall that the space of admissible functions U is given by (36).
Remark 2 We know [20, Lemma 1] that the Cauchy problem u¨ (t) = 6u 2 − t, u (t)|t=0 = u 0 , u˙ (t)|t=0 = v0 , t > 0,
(44)
has three √ sets of solutions denoted by A, B, and C. The set A consists of one-parameter family solutions that have t as asymptotics as t → ∞. Set B consists of two-parameter family of solutions oscillating around the √ asymptotics − t as t → ∞. Set C is also √ a two-parameter family; it consists of solutions (as long as they exist) that remain above the asymptotics t, or cross it from bellow (and therefore remain above). Also, from [20, Remark 4] we have that there exists γ > 0, such that all solutions to (44) with |u 0 | > γ and v0 = 0 in the set C blow up at some finite x0 . Numerical experiments in [30, Table 1] show that the solutions to (44) belong to set B if initial data u 0 and v0 have values in a certain domain. Thus, having in mind the previous remark and since we consider the variational problem, we consider the solutions belonging to either set A or set C (as long as the solution exists). Thus, we assume that initial and boundary data (36) are satisfied so that our solution coincides with the solution to the Cauchy problem (44) for suitably chosen initial data u 0 and v0 . This is the main hypothesis, and it will be tested in the subsequent numerical analysis. The primal functional, according to (38), reads 1 J (u) =
1 2 u˙ + 2u 3 − tu dt. 2
(45)
0
Canonical equations and the function Φ, by (40) and (41), are u˙ = p, p˙ = 6u − t, u = Φ 2
t, t Dα1 p
=
p˙ + t . 6
(46)
The dual functional is obtained from (42) as 1 √ 3 1 6 2 G ( p) = p (1) − p˙ + t + p dt. 9 2
(47)
0
We note that the first term in (47) is the consequence of integration by parts in (39), as described in Sect. 2, and the boundary conditions in (43)2,3 . Let U ≥ 0 and P be approximate solutions to (46)1,2 . According to the complementary principle, we have J (U ) − G (P) ≥ J (U ) − G ( p) = J (U ) − J (u) , so that 1 J (U ) − G (P) ≥ 2
1
1 ˙ 2L 2 [0,1] , ˙ 2 + 12Ψ (δu)2 dt ≥ δ u (δ u) 2
0
where Ψ = (1 − ε) u + εU > 0, ε ∈ (0, 1) . Let t ∗ ∈ [0, 1] be such that |M| := δu L ∞ [0,1] = sup δu (t) . t∈[0,1]
(48)
Complementary variational principles
695
Then, we have t ∗
t ∗ |δ u˙ (t)| dt.
δ u˙ (t) dt, i.e. |M| ≤
M= 0
(49)
0
On the other hand, 1
1 |δ u˙ (t)| dt.
δ u˙ (t) dt, i.e. |M| ≤
M =− t∗
(50)
t∗
Thus, (49) and (50) imply 1 |M| ≤ 2
1 |δ u˙ (t)| dt. 0
By the Chauchy-Schwartz inequality, we have ⎞1/2 ⎛ 1 ⎞1/2 ⎛ 1 1⎝ 1 |M| ≤ dt ⎠ ⎝ |δ u˙ (t)|2 dt ⎠ , i.e. δu L ∞ [0,1] ≤ δ u ˙ L 2 [0,1] . 2 2 0
0
From the previous inequality and (48), if the solution to (43) exists, then we have √ 2 δu L ∞ [0,1] = U − u L ∞ [0,1] ≤ J (U ) − G (P). 2
(51)
Numerical analysis. We tested numerically the procedure outlined above. Thus, we assume U (t) = c1 t c2 + (1 − c1 − c4 ) t c3 + c4 t c5 ,
(52)
where c1 , . . . , c5 are arbitrary constants. By substituting (52) into (45) and minimizing with respect to ci , i = 1, . . . , 5, we obtain U (t) = 0.858624 t 5.3256 − 0.657 24 t 5.3256 + 0.798615 t 0.994154 , J (U ) = 0.609162. Also, let P (t) = d1 + d2 t d3 + d4 t d5 ,
(53)
where d1 , . . . , d5 are arbitrary constants. By substituting (53) into (47) and maximizing with respect to di , i = 1, . . . , 5, we obtain P (t) = 0.802055 + 0.504117 t 3.96437 + 0.571215 t 4.92265 , G (P) = 0.608744. According to the formula for P, i.e., the approximate solution for u˙ (see (46)1 ), we have P (0) = 0.802055. Since, u 0 = 0 and (approximately) v0 = 0.802055, we see, according to [30, Table 1], that our solution belongs either to set A or to set C. From (51), if the solution to (43) exists, we have √ 2 U − u L ∞ [0,1] ≤ J (U ) − G (P) = 0.014457. 2 In Fig. 2, we present the plot of an approximate solution U for (43), obtained as described above.
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U t 1.0
0.8
0.6
0.4
0.2
0.2
0.4
0.6
0.8
1.0
t
Fig. 2 Approximate solution U for (43)
4 Lagrangian depending on a scalar function of vector argument We consider a scalar function u (x) = u (x1 , . . . , xn ) , such that u ∈ U = H s+α (Rn ) ∩ L ∞ (Rn ) , α > 0, s ≥ max {1, α} , where H s is the Sobolev space defined in Sect. 2. This implies that u(α) = α grad u ∈ (H s (Rn ) ∩ L ∞ (Rn ))n . We consider the Lagrangian L(u, u(α) ) = Π (u) + R(u(α) ),
(54)
d2 Π dx 2
≥ c, c > 0, for every x ∈ R and R ∈ C ∞ (Rn ) satisfies where Π ∈ C 2 (R) satisfies Π (x) = grad R (x)| x=0 = 0. This condition is motivated by the classical case, where R(u(α=1) ) is the kinetic energy and it is equal to zero when all velocities (components of u(1) ) are zero. This form of the Lagrangian is used in the image denoising, which will be demonstrated by an example in Sect. 4.1. The primal functional has the form J (u) ≡ I (u, u(α) ) = L(u, u(α) )dn x = Π (u) + R(u(α) ) dn x. (55) Rn
Rn
The Euler-Lagrange equation, obtained by the minimization of the primal functional J, is1 ∂L divα grad u(α) L + = 0, i.e. ∂u α n α α α ∂ R Dx1 u, . . . , Dxn u Π + div grad u(α) R = 0, or Π (u) + Dx j = 0. ∂ α Dx j u
(56)
j=1
The generalized momentum is defined by p := grad u(α) L = grad u(α) R.
(57)
In order to show that p ∈ P n = (H s (Rn ))n , s ≥ max {1, α} , we use: Theorem 2 ([32], Theorem 2, p. 368)Let G ∈ C ∞ (Rn ) and let G (0) = 0. Suppose s ≥ 1 and σ p < s < m ∈ N, where σ p = n · max 0,
1 p
− 1 . Then there exists a constant c, such that
G ( f 1 , . . . , f n ) F p,q s (R) ≤ c · max
j∈[1,n]
# # # f j#
s (R) F p,q
· max
j∈[1,n]
# #m−1 1 + # f j # L ∞ ( R) ,
n s (R) ∩ L ∞ (R) holds for all ( f 1 , . . . , f n ) ∈ F p,q . 1 The procedure of deriving (56) is standard one (see [2]) and it is omitted, although we have not seen equation of the type (56) before.
Complementary variational principles
697
s , so According to [32], (vi) on p. 13, Proposition (vii) on p. 14 and (5), we have that H s = H2s = F2,2 α α
∂ R Dx1 u,..., Dxn u is such that p j ∈ H s (Rn ) ∩ that σ2 = n · max 0, − 21 = 0. We also have that p j = ∂ α Dx j u ∂ R α Dx1 u,...,α Dxn u L ∞ (Rn ) , j = 1, . . . , n, since α Dx1 u, . . . , α Dxn u ∈ (H s (Rn ) ∩ L ∞ (Rn ))n , ∈ α ∂ Dx j u n) C ∞ (Rn ) and ∂ R(x∂1x,...,x = 0. j (x1 ,...,xn )=(0,...,0) Note that we assumed s ≥ max {1, α} because in (56), we have divα grad u(α) R = divα p. We define the Hamiltonian as
H (u, p) := p · u(α) − L(u, u(α) ) = p · u(α) − Π (u) − R(u(α) ).
(58)
The Hamiltonian depends on both canonical variables (u, p); hence, we use (57) and the following condition in order to obtain u(α) as a function of p. Condition 2 (a) R ∈ C ∞ (Rn ) ; (b) grad R : Rn → D is a bijection on open set D ⊂ Rn ; 2 (c) det ∂∂ xR(x) = 0 for every x ∈ Rn . i ∂x j Then, by the implicit function theorem, there exists ϒ : D → Rn of class (C ∞ )n , such that (57) can be inverted and solved for u(α) , i.e., u(α) = ϒ ( p) . Therefore, (58) becomes H (u, p) = p · ϒ ( p) − Π (u) − R(ϒ ( p)).
(59)
We write the Lagrangian in (55) by using (58), i.e., we express the Lagrangian in terms of the Hamiltonian, so that I (u, u(α) ) = p · u(α) − H (u, p) dn x. Rn
Next we use the integration by parts formula (8) to obtain I (u, p) = u divα p − H (u, p) dn x Rn
=
u divα p − p · ϒ ( p) + Π (u) + R(ϒ ( p)) dn x.
(60)
Rn
We treat (60) as a variational problem with independent functions u and p. By minimizing (60) with respect to (u, p) ∈ U × P n , we obtain the Hamilton equations u(α) = grad p H = ϒ ( p) , divα p =
∂H = −Π . ∂u
(61)
In order to formulate the dual functional, we solve the second Hamilton equation (61)2 with respect to u. Since Π ∈ C 2 (R) satisfies Π (x) ≥ c, c > 0, for every x ∈ R, there exists θ : R → R of class C 1 , such that (61)2 can be inverted and solved for u, i.e., u = θ divα p . (62) Finally, by (60) and (62), we obtain the dual functional as α G ( p) = θ div p divα p − p · ϒ ( p) + Π θ divα p + R(ϒ ( p)) dn x. Rn
We formulate the following theorem, which we prove in Sect. 5.
(63)
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Theorem 3 Assume that Π ∈ C 2 (R) satisfies Π (x) ≥ c, c > 0, for every x ∈ R. Also assume that R ∈ C ∞ (Rn ) satisfies grad R (x)| x=0 = 0 and Condition 2. Additionally, assume n ∂ 2 R (x) dxi dx j ≥ 0, x ∈ Rn . ∂ xi ∂ x j
(64)
i, j=1
Then the complementary principle holds for the functionals (55) and (63), i.e., it holds J (u) ≤ J (U ) , G ( p) ≥ G ( P) , where U = u + δu and P = p + δ p for u, U ∈ U and p, P ∈ P n . Moreover, 2 U − u L 2 (Rn ) ≤ (J (U ) − G( P)). c
(65)
(66)
Again, the corresponding results for the Lagrangians containing integer order derivatives are given in [6]. 4.1 Example in the image regularization In this section, as an example of the general theory presented in Sect. 4, we shall specify the Lagrangian, given by (54), in the form that is intensively used the image denoising and regularization tasks (see [1,13,28]). Namely, 1 in the Lagrangian (54) we chose Π (u) = 2λ (u − u 0 )2 , u 0 ∈ H s+α (R2 ), α > 0, s ≥ max {1, α} , λ > 0, and R(u(α) ) = 1 + (u(α) )2 , u(α) = α grad u. This form of the function R is called the “minimal surface” edge stopping function, see [1]. Therefore, the Lagrangian takes the form L(u, u(α) ) =
1 (u − u 0 )2 + 1 + (u(α) )2 , u ∈ H s+α (R2 ). 2λ
The primal functional (55) becomes 1 J (u) = (u − u 0 )2 + 1 + (u(α) )2 d2 x, 2λ
(67)
R2
while the Euler-Lagrange equation, calculated by (56), is u − u0 u(α) = 0. + divα 2 (α) λ 1 + (u ) 2 The generalized momentum p ∈ H s R2 , given by (57), becomes
Since R(x) =
u(α) . p := 1 + (u(α) )2
(68)
p . u(α) = ϒ ( p) = 1 − p2
(69)
√ 1 + x 2 satisfies Condition 2, we have that (68) can be solved with respect to u(α) as
Note that from (69), it follows that p 2 < 1. The Hamiltonian (59) takes the form " 1 H (u, p) := − (u − u 0 )2 − 1 − p 2 , 2λ while the canonical equations, given by (61), are p u0 − u u(α) = , divα p = . 2 λ 1− p
(70)
Complementary variational principles
699
1 Since Π (u) = 2λ (u − u 0 )2 satisfies Π (u) = λ1 = c ≥ 0, we have that the second canonical equation (70)2 can be solved with respect to u as u = θ divα p = u 0 − λ divα p. (71)
The dual functional, according to (63), with (69) and (71), reads " 1 α 2 α 2 G ( p) = u 0 div p − λ div p + 1 − p d2 x. 2
(72)
R2
According to Theorem 3, the primal functional (67) attains its minimum, while the dual functional (72) attains its maximum. Thus, by (66) we have δu L 2 (Rn ) ≤ 2λ (J (U ) − G( P)), (73) since (64) is also satisfied. In [21], we calculate u by a numerical approach and obtain an approximate solution for which we use (73) in order to achieve the required precision of u. 5 Proofs of Theorems 1 and 3 In this section, we prove Theorems 1 and 3. Note that the assumptions of Sect. 3, respectively, Sect. 4, imply the existence of the functionals J and G, given in the form (10) and (18), respectively, (55) and (63). Proof of Theorem 1 By the Taylor expansion formula for Π and U = u + δu, we have J (U) − J (u) =
b 1 2
2 α a Dt U
−
α 2 a Dt u
+ Π (t, U) − Π (t, u) dt
a
b =
α a Dt u
α · a Dt δu + δu · grad u Π (t, u) dt
a
1 + 2
b
2 α a Dt δu
+ δu · [δu · ∇u ] grad u Π (t, u)
u=ξ
dt
a
1 = 2
b a
≥
c 2
⎤ ⎡ n 2 Π (t, u) α 2 ∂ ⎣ a Dt δu + (δu i ) δu j ⎦ ∂u i ∂u j i, j=1
b (δu)2 dt =
c δu2L 2 [a,b] , 2
dt u=ξ
(74)
a
where ξ = u + η (U − u) , η ∈ (0, 1) . In obtaining (74), we used integration by parts formula (3) in order to use the Euler-Lagrange equations (11). We also used (19). Therefore, J (U) ≥ J (u) . Further, consider G ( p) and assume that p ∈ P n is a solution to (15). Let P = p + δ p be an arbitrary element of P n . We use the Taylor expansion formula for Π and Φ and consider the difference G ( P) − G ( p) b α α α α = t Db P · Φ t, t Db P − t Db p · Φ t, t Db p a
−
1 2 dt P − p 2 + Π t, Φ t, t Dαb P − Π t, Φ t, t Dαb p 2
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b
= a
α t Db δ p
α t Db δ p
+
· Φ t, t Dαb p − p · δ p
·∇
α t Db
p
Φ
t, t Dαb
α α p · t Db p + gradΦ Π t, Φ t, t Db p dt
b 1 α α α 2 t Db δ p · − (δ p)2 + t Db δ p · ∇t Dαb p Φ t, t Db p 2 a 2 α α α α + D δ p · ∇ Φ t, D p · t Db p + gradΦ Π t, Φ t, t Dαb p t b t b t Db p α α + t Db δ p · ∇t Dαb p Φ t, t Db p α α α · ∇Φ gradΦ Π t, Φ t, t Db p dt, × t Db δ p · ∇t Dαb p Φ t, t Db p p=ζ
where ζ = p + η ( P − p) , η ∈ (0, 1) . By (17), we have α α t Db p + gradΦ Π t, Φ t, t Db p = 0, while, according to (3), (16), and (15)1 , we have b
α t Db δ p
·Φ
t, t Dαb
b
p − p·δp =
a
α a Dt Φ
t, t Dαb p − p · δ p = 0.
a
Thus, we obtain 1 G ( P) − G ( p) = − 2
b
(δ p)2 − (2w1 + w2 ) p=ζ dt,
(75)
a
where
α α · t Db δ p · ∇t Dαb p Φ t, t Db p n ∂Φ j t, t Dαb p α α = δ t Db pi δ t Db p j , ∂ t Dαb pi i, j=1 α α w2 := t Db δ p · ∇t Dαb p Φ t, t Db p α α α α Φ t, grad × D δ p · ∇ D p · ∇ Π t, Φ t, D p t b t b Φ t b Φ t Db p n α α α 2 ∂ Π t, Φ t, t D p ∂Φi t, t D p ∂Φ j t, t D p b b b = δ t Dαb pk δ t Dαb pl . α α ∂Φi ∂Φ j ∂ t Db pk ∂ t Db pl w1 :=
α t Db δ p
(76)
i, j,k,l=1
(77) Let us write (17) as α t Db p j
∂Π t, Φ t, t Dαb p =− , j = 1, . . . , n, ∂Φ j
and differentiate it with respect to t Dαb pk . Then we obtain n ∂ 2 Π t, Φ t, t Dαb p ∂Φi t, t Dαb p δ j,k = − , ∂Φ j ∂Φi ∂ t Dαb pk i=1
(78)
Complementary variational principles
701
where δ j,k denotes the Kronecker delta. According to (76) and by the use of (78) in (77), we have n ∂Φk t, t Dαb p α α w2 = − δ t Db pk δ t Db pl = −w1 . ∂ t Dαb pl k,l=1
Now, (75) becomes ∂ 2 Π t, Φ t, t Dαb p (δ p) + ∂Φi ∂Φ j i, j,k,l=1 a ∂Φi t, t Dαb p ∂Φ j t, t Dαb p α α × δ t Db pk δ t Db pl dt ≥ 0. ∂ t Dαb pk ∂ t Dαb pl p=ζ
1 G ( P) − G ( p) = − 2
b
2
n
We conclude that G ( p) ≥ G ( P) . Let U and P be two approximate solutions to (15). Since J (U) ≥ J (u) and G ( p) ≥ G ( P) , we have that (20) holds. Moreover, J (U) − G ( P) ≥ J (U) − G ( p) = J (U) − J (u) , so that by (74) it holds J (U) − G ( P) ≥
c δu2L 2 [a,b] . 2
Proof of Theorem 3 By the use of the Taylor formula, U = u + δu and we have
α grad
(δu) =
δ (α grad u)
=
δu(α) ,
J (U ) − J (u) = Π (U ) − Π (u) + R(U (α) ) + R(u(α) ) dn x Rn
=
Π (u)δu + δu(α) · grad u(α) R(u(α) ) dn x
Rn
+ 1 = 2 c ≥ 2
1 2
Π (u)(δu)2 + δu(α) · δu(α) · ∇u(α) grad u(α) R(u(α) )
u=ξ
Rn
⎤ 2 R(u(α) ) ∂ ⎣Π (u)(δu)2 + δ α Dxi u δ α Dx j u ⎦ αD u ∂ αD u ∂ x x i j i, j=1 ⎡
Rn
dn x
n
u=ξ
c (δu) d x = δu2L 2 (Rn ) , 2 2 n
Rn
dn x
(79)
where ξ = u + η (U − u) , η ∈ (0, 1) . In obtaining (79), we used integration by parts formula (8) in order to use the Euler-Lagrange equations (56). We also used (64). Therefore, J (U ) ≥ J (u) . Now consider (63). It follows G( P) − G( p) := Δ1 + Δ2 , where
Δ1 := Rn
Δ2 := Rn
(80)
P (α) θ (P (α) ) − p (α) θ ( p (α) ) + Π (θ (P (α) )) − Π (θ ( p (α) )) dn x,
(81)
R (ϒ ( P)) − R (ϒ ( p)) − P · ϒ( P) + p · ϒ( p) dn x.
(82)
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In (81), we used the notation P (α) = divα P and p (α) = divα p. By the use of the Taylor formula, and P = p + δ p, (81) becomes dθ ( p (α) ) dΠ (θ ) (α) (α) Δ1 = δp (α) dn x θ(p ) + p + dθ d p (α) Rn 2 ( dθ ( p (α) ) d2 Π (θ ) dθ ( p (α) ) 1 2 + + 2 dθ 2 d p (α) d p (α) +
Rn d2 θ ( p (α) )
d( p (α) )2
p
(α)
dΠ (θ ) + dθ
(δp
(α) 2
)
p (α) =divα ξ
dn x,
(83)
where ξ = p + ( P − p), for some ∈ (0, 1). According to (61)2 , we have p (α) +
dΠ (θ ) = 0. dθ
Having in mind that θ = θ ( p (α) ), see (62), differentiation of the previous expression with respect to p (α) yields 1+
d2 Π (θ ) dθ ( p (α) ) = 0, dθ 2 d p (α)
so that (83) becomes Δ1 =
θ(p
(α)
)δp
1 d x− 2
(α) n
Rn
Rn
d2 Π (θ ) dθ 2
−1 (δp
(α) 2
)
dn x. p (α) =divα
(84)
ξ
Now, by the use of the Taylor formula in (82), we obtain δ p · ∇ p ϒ( p) · gradϒ R (ϒ) − p − ϒ( p) · δ p dn x Δ2 = Rn
1 + 2
δ p · ∇p
2
ϒ( p) · gradϒ R (ϒ) − p − 2δ p · δ p · ∇ p ϒ( p)
Rn
+ δ p · ∇ p ϒ( p) · δ p · ∇ p ϒ( p) · ∇ϒ gradϒ R (ϒ)
dn x,
(85)
p=ξ
where ξ = p + ( P − p), for some ∈ (0, 1). According to (57) and (61), we have p = gradϒ R (ϒ) , so that (85) yields
Δ2 = −
ϒ( p) · δ p dn x −
Rn
1 2
(86)
[2q1 − q2 ] p=ξ dn x,
(87)
Rn
where n ∂ϒk δ p · ∇ p ϒ( p) = δpk δpl , ∂ pl k,l=1 q2 := δ p · ∇ p ϒ( p) · δ p · ∇ p ϒ( p) · ∇ϒ gradϒ R (ϒ)
q1 := δ p ·
=
n i, j,k,l=1
∂ 2 R (ϒ) ∂ϒi ∂ϒ j δpk δpl . ∂ϒi ∂ϒ j ∂ pk ∂ pl
(88)
(89)
Complementary variational principles
703
We write (86) as ∂ R (ϒ ( p)) , j = 1, . . . , n, ∂ϒ j
pj =
and differentiate it with respect to pk . Then we obtain δ j,k =
n ∂ 2 R (ϒ ( p)) ∂ϒi
∂ϒ j ∂ϒi
i=1
∂ pk
.
(90)
According to (88) and by the use of (90) in (89), we have q2 =
n ∂ϒk δpk δpl = q1 . ∂ pl
k,l=1
Now, (87) becomes Δ2 = −
1 2
ϒ( p) · δ p dn x −
Rn
⎡
⎤
n
⎣
i, j,k,l=1
Rn
∂ϒi ∂ϒ j δpk δpl ⎦ ∂ϒi ∂ϒ j ∂ pk ∂ pl ∂ 2 R (ϒ)
dn x. p=ξ
Using the previous expression and (84) in (80), we obtain G( P) − G( p) = θ (divα p) divα δ p − ϒ( p) · δ p dn x Rn
1 − 2 +
Rn n
i, j,k,l=1
By (8), (61)1 , (62), and
u(α)
=
d2 Π (θ ) dθ 2
−1
divα δ p
2
∂ 2 R (ϒ) ∂ϒi ∂ϒ j δpk δpl ∂ϒi ∂ϒ j ∂ pk ∂ pl
dn x.
(91)
p=ξ
α grad u,
we have α α θ div p div δ p − ϒ ( p) · δ p dn x
Rn
=
α
grad θ divα p − ϒ ( p) · δ p dn x = 0,
Rn
so that (91) becomes G( P) − G( p) = −
+
1 2
Rn n
i, j,k,l=1
d2 Π (θ ) dθ 2
−1
divα δ p
2
∂ 2 R (ϒ) ∂ϒi ∂ϒ j δpk δpl ∂ϒi ∂ϒ j ∂ pk ∂ pl
dn x ≤ 0. p=ξ
We conclude that G ( p) ≥ G ( P) . Let U and P be two approximate solutions to (61). Since J (U ) ≥ J (u) and G ( p) ≥ G ( P) , we have that (65) holds. Moreover, we have J (U ) − G ( P) ≥ J (U ) − G ( p) = J (U ) − J (u) . Thus, by (79), J (U ) − G ( P) ≥
c δu2L 2 (Rn ) . 2
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Acknowledgments This research was supported by projects 174005 (T.M.A. and D.Z.), TR32035 (M.J.), and 174024, III44006 (S.P.) of the Serbian Ministry of Science.
References 1. Aubert, G., Kornprobst, P.: Mathematical Problems in Image Processing, Partial Differential Equations and the Calculus of Variation. Springer Science + Business media, LLC, New York (2006) 2. Agrawal, O.P.: Formulation of Euler-Lagrange equations for fractional variational problems. J. Math. Anal. Appl. 272, 368–379 (2002) 3. Agrawal, O.P.: Fractional variational calculus and the transversality conditions. J. Phys. A Math. Gen. 39, 10375– 10384 (2006) 4. Agrawal, O.P., Muslih, S.I., Baleanu, D.: Generalized variational calculus in terms of multi-parameters fractional derivatives. Commun. Nonlinear Sci. Numer. Simulat. 16, 4756–4767 (2011) 5. Anastassiou, G.A.: Fractional Differentiation Inequalities. Springer Science + Business Media, LLC, New York (2009) 6. Arthurs, A.M.: Complementary Variational Principles. Clarendon Press, Oxford (1980) 7. Arthurs, A.M.: Dual extremum principles and error bounds for a class of boundary value problems. J. Math. Anal. Appl. 41, 781–795 (1973) 8. Atanackovic, T.M., Konjik, S., Pilipovic, S.: Variational problems with fractional derivatives: Euler-Lagrange equations. J. Phys. A Math. Theor. 41, 095201 (12 p) (2008) 9. Atanackovic, T.M., Djukic, Dj.S.: An extremum variational principle for a class of boundary value problems. J. Math. Anal. Appl. 93, 344–362 (1983) 10. Atanackovic, T.M., Stankovic, B.: Generalized wave equation in nonlocal elasticity. Acta Mech. 208, 1–10 (2009) 11. Atanackovic, T.M., Stankovic, B.: On a class of differential equations with left and right fractional derivatives. Z. Angew. Math. Mech. 87, 537–546 (2007) 12. Atanackovic, T.M., Stankovic, B.: On a differential equation with left and right fractional derivatives. Fract. Calc. Appl. Anal. 10, 139–150 (2007) 13. Bai, J., Feng, X.C.: Fractional order anisotropic diffusion for image denoising. IEEE T. Image Process. 16, 2492–2502 (2007) 14. Baleanu, D.: Fractional variational principles in action. Phys. Scr. T 136, 014006–014011 (2009) 15. Carpinteri, A., Cornetti, P., Sapora, A.: A fractional calculus approach to nonlocal elasticity. Eur. Phys. J. Special Top. 193, 193–204 (2011) 16. Demidovic, B.P.: Lectures on Mathematical Theory of Stability. Nauka, Moscow (1967) (in Russian) 17. Gahov, F.D.: Boundary Value Problems. Nauka, Moscow (1977) (in Russian) 18. Gorenflo, R., Vessella, S.: Abel Integral Equations. Lecture Notes in Mathematics 1461. Springer, Berlin (1991) 19. Herzallah, M.A.E., Baleanu, D.: Fractional-order Euler-Lagrange equations and formulation of Hamiltonian equations. Nonlinear Dyn. 58, 385–391 (2009) 20. Holmes, P., Spence, D.: On a Painlevé-type boundary-value problem. Q. J. Mech. Appl. Math. 37, 525–538 (1984) 21. Janev, M., Atanackovic, T., Pilipovic, S., Obradovic, R.: Image denoising by a direct variational minimization. EURASIP J. Adv. Sig. Pr. 2011:8 (2011) 22. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006) 23. Klimek, M.: On Solutions of Linear Fractional Differential Equations of a Variational Type. Czestochowa University of Technology, Czestochowa (2009) 24. Mainardi, F.: Fractional Calculus and Waves in Linear Viscoelasticity. Imperial College Press, London (2010) 25. Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993) 26. Mishura, Y.S.: Stochastic Calculus for Fractional Brownian Motion and Related Processes. Springer, Berlin (2008) 27. Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, New York (1974) 28. Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE T. Pattern Anal. 12, 629–639 (1990) 29. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999) 30. Qin, HZ., Lu, YM.: A note on an open problem about the first Painlevé equation. Acta Math. Appl. Sin-E. 24, 203–210 (2008) 31. Rabei, E.M., Nawafleh, K.I., Hijjawi, R.S., Muslih, S.I., Baleanu, D.: The Hamilton formalism with fractional derivatives. J. Math. Anal. Appl. 327, 891–897 (2007) 32. Runst, T., Sickel, W.: Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations. Walter de Gruyter & Co, Berlin (1996) 33. Samko, S.G., Kilbas, A.A., Marichev, I.I.: Fractional Integrals and Derivatives. Gordon and Breach, Amsterdam (1993) 34. Sewell, M.J.: Maximum and Minimum Principles. Cambridge University Press, Cambridge (1987) 35. Vujanovic, B.D., Jones, S.E.: Variational Methods in Nonconservative Phenomena. Academic press, London (1989)
