EXISTENCE OF A WEAK SOLUTION FOR FRACTIONAL EULER-LAGRANGE EQUATIONS. Loïc Bourdin

Transcription

1 EXISTENCE OF A WEAK SOLUTION FOR FRACTIONAL EULER-LAGRANGE EQUATIONS. by Loïc Bourdin. Abstrct. We derive sufficient conditions ensuring the existence of wek solution u for frctionl Euler-Lgrnge equtions of the type: ( ) x (u,dα u,t)+d+ α y (u,dα u,t) = 0, (EL α ) on rel intervl [,b] nd where D α nd D α + re the frctionl derivtives of Riemnn-Liouville of order 0 < α < 1. Keywords: Frctionl Euler-Lgrnge equtions; existence; frctionl vritionl clculus. AMS Clssifiction: 70H03; 26A Introduction 1.1. Context in the frctionl clculus. The mthemticl field tht dels with derivtives of ny rel order is clled frctionl clculus. For long time, it ws only considered s pure mthemticl brnch. Nevertheless, during the lst two decdes, frctionl clculus hs ttrcted the ttention of mny reserchers nd it hs been successfully pplied in vrious res like computtionl biology [21] or economy [9]. In prticulr, the first nd well-estblished ppliction of frctionl opertors ws in the physicl context of nomlous diffusion, see [34, 35] for exmple. Let us mention [23] proving tht frctionl equtions is complementry tool in the description of nomlous trnsport processes. We refer to [15] for generl review of the pplictions of frctionl clculus in severl fields of Physics. In more generl point of view, frctionl differentil equtions re even considered s n lterntive model to non-liner differentil equtions, see [5]. For the origin of the clculus of vritions with frctionl opertors, we should look bck to when Riewe used non-integer order derivtives to better describe non conservtive systems in mechnics [28, 29]. Since then, numerous works on the frctionl vritionl clculus hve been mde. For instnce, in the sme spirit, uthors of [10, 11] hve recently derived frctionl vritionl structures for non conservtive equtions. Furthermore, one cn find comprehensive literture regrding necessry optimlity conditions nd Noether s theorem, see [1, 3, 4, 6, 13, 25]. Concerning the stte of the rt on the frctionl clculus of vritions nd respective frctionl Euler-Lgrnge equtions, we refer the reder to the

2 2 LOÏC BOURDIN recent book [22]. In the whole pper, we consider < b two rels, d N nd the following Lgrngin functionl L(u) = where L is Lgrngin, i.e. mp of the form: L(u,D α u,t) dt, (1) L : R d R d [,b] R (x,y,t) L(x,y,t), where D α is the left frctionl derivtive of Riemnn-Liouville of order 0 < α < 1 nd where the vrible u is function defined lmost everywhere (shortly.e.) on (,b) with vlues in R d. The precise definitions of the frctionl opertors of Riemnn-Liouville will be reclled in Section 2.2. It is well-known tht criticl points of the functionl L re chrcterized by the solutions of the frctionl Euler-Lgrnge eqution: ( ) x (u,dα u,t)+dα + y (u,dα u,t) = 0, (EL α ) where D α + is the right frctionl derivtive of Riemnn-Liouville, see detiled proofs in [1, 4] for exmple. However, s fr s the uthor is wre nd despite prticulr results in [16, 19], no existence result of solution for (EL α ) exists in generl cse. The im of this pper is to derive sufficient conditions on L so tht (EL α ) dmits wek solution. Let us note tht, in more generl setting, existence results for frctionl equtions is n emerging field. For instnce, there re recent results bout existence nd uniqueness of solution for clss of frctionl evolution equtions in [32, 33] Min result. We denote by the Eucliden norm of R d nd C := C([,b];R d ) the spce of continuous functions endowed with its usul norm. Definition 1. A function u is sid to be wek solution of (EL α ) if u C nd if u stisfies (EL α ).e. on [,b]. Let us enuncite the min result of the pper: Theorem 1. Let L be Lgrngin of clss C 1 nd 0 < (1/p) < α < 1. If L stisfies the following hypotheses denoted by (H 1 ), (H 2 ), (H 3 ), (H 4 ) nd (H 5 ): there exist 0 d 1 p nd r 1, s 1 C(R d [,b],r + ) such tht: (x,y,t) R d R d [,b], L(x,y,t) L(x,0,t) r 1 (x,t) y d 1 +s 1 (x,t); (H 1 ) there exist 0 d 2 p nd r 2, s 2 C(R d [,b],r + ) such tht: (x,y,t) R d R d [,b], x (x,y,t) r 2(x,t) y d 2 +s 2 (x,t); (H 2 ) there exist 0 d 3 p 1 nd r 3, s 3 C(R d [,b],r + ) such tht: (x,v,t) R d R d [,b], y (x,y,t) r 3(x,t) y d 3 +s 3 (x,t); (H 3 ) (2)

3 EXISTENCE OF A WEAK SOLUTION FOR FRACTIONAL EULER-LAGRANGE EQUATIONS. 3 coercivity condition: there exist γ > 0, 1 d 4 < p, c 1 C(R d [,b],[γ, [), c 2, c 3 C([,b],R) such tht: (x,y,t) R d R d [,b], L(x,y,t) c 1 (x,t) y p +c 2 (t) x d 4 +c 3 (t); (H 4 ) convexity condition: then (EL α ) dmits wek solution. t [,b], L(,,t) is convex, (H 5 ) Hypotheses denoted by (H 1 ), (H 2 ), (H 3 ) re usully clled regulrity hypotheses, see [8, 12]. In Section 5, we prove tht Hypothesis (H 5 ) cn be replced by different convexity ssumptions Ide of the proof of Theorem 1. In the clssicl cse α = 1, D 1 = D 1 + = d/dt nd consequently (EL α ) is nothing else but the clssicl Euler-Lgrnge eqution formulted in the 1750 s. In this cse, lot of results of existence of solutions hve been lredy proved. Let us recll tht there exist different pproches: A first pproch is to develop the clssicl Euler-Lgrnge eqution in order to obtin n implicit second order differentil eqution, see [14]. Then, under hyper regulrity or non singulrity condition on the Lgrngin L, the eqution cn be written s n explicit second order differentil eqution nd the Cuchy-Lipschitz theorem gives the existence of locl or globl regulr solutions; A second pproch consists in using the vritionl structure of the eqution, see [12]. Indeed, under some ssumptions, the criticl points of L correspond to the solutions of the clssicl Euler-Lgrnge eqution. The ide is then to prove the existence of criticl points of L. In this wy, uthor mkes some ssumptions (like coercivity nd convexity of the Lgrngin L) ensuring the existence of extrem of L. With this second method, uthor hs to use reflexive spces of functions nd consequently, he dels with wek solutions (in specific sense). In order to prove Theorem 1, we extend the second pproch to the strict frctionl cse (i.e. 0 < α < 1). Indeed, lthough there exist frctionl versions of the Cuchy-Lipschitz theorem (see [17, 30]), there is no simple rules for the frctionl derivtive of composition nd consequently, we cn not write (EL α ) in simpler wy. Hence, in the strict frctionl cse, we cn not follow the first method. Theorem 1 is bsed on the following preliminries: The introduction in Section 3 of n pproprite reflexive seprble Bnch spce E α,p (see (15)); Assuming Hypotheses (H 1 ), (H 2 ) nd (H 3 ), Theorem 2 in Section 4 sttes tht if u is criticl point of L, then u is wek solution of (EL α ); Assuming dditionlly Hypotheses (H 4 ) nd (H 5 ), Theorem 3 in Section 5 sttes tht L dmits globl minimizer. Hence, the proof of Theorem 1 is complete. Let us note tht the method developed in this pper is inspired by: the reflexive seprble Bnch spce introduced in [16] llowing to prove the existence of wek solution for clss of frctionl boundry vlue problems;

4 4 LOÏC BOURDIN the suitble hypotheses of regulrity, coercivity nd convexity given in [12] proving the existence of wek solution for clssicl Euler-Lgrnge equtions (i.e. in the cse α = 1) Orgnistion of the pper. The pper is orgnized s follows. In Section 2, some usul nottions of spces of functions re given. We recll the definitions of the frctionl opertors of Riemnn-Liouville nd some of their properties. Section 3 is devoted to the introduction nd to the study of the pproprite reflexive seprble Bnch spce E α,p. In Section 4, the vritionl structure of (EL α ) is considered nd we prove Theorem 2. In Section 5, we prove Theorem 3. Then, Section 6 is devoted to some exmples. Finlly, conclusion ends this pper. 2. Reminder bout frctionl clculus 2.1. Some spces of functions. For ny p 1, L p := L p( (,b);r d) denotes the clssicl Lebesgue spce of p-integrble functions endowed with its usul norm L p. Let us give some usul nottions of spces of continuous functions defined on [,b] with vlues in R d : AC := AC([,b];R d ) the spce of bsolutely continuous functions; C := C ([,b];r d ) the spce of infinitely differentible functions; Cc := Cc ([,b];r d ) the spce of infinitely differentible functions nd compctly supported in ],b[. We remind tht function f is n element of AC if nd only if f L 1 nd the following equlity holds: t [,b], f(t) = f()+ t f(ξ) dξ, (3) where f denotes the derivtive of f. We refer to [20] for more detils concerning the bsolutely continuous functions. Finlly, we denote by C (resp. AC or C ) the spce of functions f C (resp. AC or C ) such tht f() = 0. In prticulr, C c C AC. Convention: in the whole pper, n equlity between functions must be understood s n equlity holding for lmost ll t (,b). When it is not the cse, the intervl on which the equlity is vlid will be specified Frctionl opertors of Riemnn-Liouville. Since 1695, numerous notions of frctionl opertors emerged over the yer, see [17, 27, 30]. In this pper, we only del with the frctionl opertors of Riemnn-Liouville (1847) whose definitions nd some bsic results re reminded in this section. We refer to [17, 30] for the omitted proofs. Let α > 0 nd f be function defined.e. on (,b) with vlues in R d. The left (resp. right) frctionl integrl in the sense of Riemnn-Liouville with inferior limit (resp. superior limit b) of order α of f is given by: t ],b], I α f(t) := 1 Γ(α) t (t ξ) α 1 f(ξ) dξ, (4)

5 EXISTENCE OF A WEAK SOLUTION FOR FRACTIONAL EULER-LAGRANGE EQUATIONS. 5 respectively: t [,b[, I+ α 1 f(t) := Γ(α) t (ξ t) α 1 f(ξ) dξ, (5) where Γ denotes the Euler s Gmm function. If f L 1, then I αf nd Iα + f re defined.e. on (,b). Now, let us consider 0 < α < 1. The left (resp. right) frctionl derivtive in the sense of Riemnn-Liouville with inferior limit (resp. superior limit b) of order α of f is given by: t ],b], D α f(t) := d ( ( I 1 α dt f) (t) resp. t [,b[, D+ α f(t) := d ( ) I 1 α + dt f) (t). (6) From [17, Corollry 2.2, p.73], if f AC, then D f α nd D+f α re defined.e. on (,b) nd stisfy: D α f = I1 α f + f() (t ) α Γ(1 α) In prticulr, if f AC, then D f α = I 1 α f. nd D+ α f = I1 α f(b) + f + (b t) α Γ(1 α). (7) 2.3. Some properties of the frctionl opertors. In this section, we provide some properties concerning the left frctionl opertors of Riemnn-Liouville. One cn esily derive the nlogous versions for the right ones. Properties 1, 2 nd 3 re well-known nd one cn find their proofs in the clssicl literture on the subject (see [17, Lemm 2.3, p.73], [17, Lemm 2.1, p.72] nd [17, Lemm 2.7, p.76] respectively). The first result yields the semi-group property of the left Riemnn-Liouville frctionl integrl: Property 1. For ny α, β > 0 nd ny function f L 1, the following equlity holds: I α I β f = Iα+β f. (8) From Property 1 nd Equlities (6) nd (7), one cn esily deduce the following results concerning the composition between frctionl integrl nd frctionl derivtive. For ny 0 < α < 1, the following equlities hold: f L 1, D α I α f = f nd f AC, I α D α f = f. (9) Another clssicl result is the boundedness of the left frctionl integrl from L p to L p : Property 2. For ny α > 0 nd ny p 1, I α is liner nd continuous from Lp to L p. Precisely, the following inequlity holds: f L p, I f α L p (b )α Γ(1+α) f Lp. (10) The following clssicl property concerns the integrtion of frctionl integrls. It is occsionlly clled frctionl integrtion by prts: Property 3. Let 0 < α < 1. Let f L p nd g L q where (1/p) + (1/q) 1 + α (nd p 1 q in the cse (1/p)+(1/q) = 1+α). Then, the following equlity holds: I α f g dt = f I α +g dt. (11)

6 6 LOÏC BOURDIN This chnge of side of the frctionl integrl (from I α to Iα + ) is responsible of the emergence of D+ α in (EL α ) lthough only D α is involved in the Lgrngin functionl L. We refer to Section 4.2 for more detils. The following Property 4 completes Property 2 in the cse 0 < (1/p) < α < 1: indeed, in this cse, I α is dditionlly bounded from L p to C : Property 4. Let 0 < (1/p) < α < 1 nd q = p/(p 1). Then, for ny f L p, we hve: I α f is Holdër continuous on ],b] with exponent α (1/p) > 0; limi α t f(t) = 0. Consequently, I αf cn be continuously extended by 0 in t =. Finlly, for ny f Lp, we hve I αf C. Moreover, the following inequlity holds: (b ) α (1/p) f L p, I α f Γ(α) ( (α 1)q +1 ) 1/q f Lp. (12) Proof. Let us note tht this result is minly proved in [16]. Let f L p. We first remind the following inequlity: ξ 1 ξ 2 0, (ξ 1 ξ 2 ) q ξ q 1 ξq 2. (13) Let us prove tht I αf is Holdër continuous on ],b]. For ny < t 1 < t 2 b, using the Hölder s inequlity, we hve: I f(t α 2 ) I f(t α 1 t2 t1 1 ) = Γ(α) (t 2 ξ) α 1 f(ξ) dξ (t 1 ξ) α 1 f(ξ) dξ 1 t2 Γ(α) (t 2 ξ) α 1 f(ξ) dξ t t1 ( Γ(α) (t2 ξ) α 1 (t 1 ξ) α 1) f(ξ) dξ f L p Γ(α) f L p Γ(α) ( t2 ( t2 t 1 (t 2 ξ) (α 1)q dξ + f L p Γ(α) ( t1 t 1 (t 2 ξ) (α 1)q dξ + f L p Γ(α) ( t1 ) 1/q ( (t1 ξ) α 1 (t 2 ξ) α 1) q dξ ) 1/q ) 1/q 2 f L p Γ(α) ( (α 1)q +1 ) 1/q (t 2 t 1 ) α (1/p). ) 1/q (t 1 ξ) (α 1)q (t 2 ξ) (α 1)q dξ The proof of the first point is complete. Let us consider the second point. For ny t ],b], we cn prove in the sme mnner tht: I α f(t) f L p Γ(α) ( (α 1)q +1 ) 1/q (t )α (1/p) t 0. (14)

7 EXISTENCE OF A WEAK SOLUTION FOR FRACTIONAL EULER-LAGRANGE EQUATIONS. 7 The proof is now complete. 3. Spce of functions E α,p In order to prove the existence of wek solution of (EL α ) using vritionl method, we need the introduction of n pproprite spce of functions. This spce hs to present some properties like reflexivity, see [12]. For ny 0 < α < 1 nd ny p 1, we define the following spce of functions: E α,p := {u L p stisfying D α u Lp nd I α Dα u = u.e.}. (15) We endow E α,p with the following norm: Let us note tht: α,p : E α,p R + u ( u p L p + D α u p L p ) 1/p. (16) α,p : E α,p R + u D α u L p (17) is n equivlent norm to α,p for E α,p. Indeed, Property 2 leds to: u E α,p, u L p = I α Dα u L p (b )α Γ(1+α) Dα u Lp. (18) The gol of this section is to prove the following proposition: Proposition 1. Assuming 0 < (1/p) < α < 1, E α,p is reflexive seprble Bnch spce nd the compct embedding E α,p C holds. Then, in the rest of the pper, we consider: 0 < (1/p) < α < 1 nd q = p/(p 1). (19) Let us detil the different points of Proposition 1 in the following subsections E α,p is reflexive seprble Bnch spce. Let us prove this property. Let us consider (L p ) 2 the set L p L p endowed with the norm (u,v) (L p ) 2 = ( u p L p + v p L p ) 1/p. Since p > 1, (L p, L p) is reflexive seprble Bnch spce nd therefore, ( (L p ) 2, (L p ) 2 ) is lso reflexive seprble Bnch spce. We define Ω := {(u,d α u), u E α,p }. Let us prove tht Ω is closed subspce of ( (L p ) 2, (L p ) 2 ). Let (un,v n ) n N Ω such tht: (u n,v n ) (Lp ) 2 (u,v). (20) Let us prove tht (u,v) Ω. For ny n N, (u n,v n ) Ω. Thus, u n E α,p nd v n = D u α n. Consequently, we hve: L u p n u nd D u α L p n v. (21) For ny n N, since u n E α,p nd I α is continuous from Lp to L p, we hve: u n = I α D α u n L p I α v. (22)

8 8 LOÏC BOURDIN Thus, u = I α v, Dα u = Dα Iα v = v Lp nd I α Dα u = Iα v = u. Hence, u E α,p nd (u,v) = (u,d α u) Ω. In conclusion, Ω is closed subspce of ( (L p ) 2, (L p ) 2 ) nd then Ω is reflexive seprble Bnch spce. Finlly, defining the following opertor: A : E α,p Ω u (u,d α u), we prove tht E α,p is isometric isomorphic to Ω. This completes the proof of Section The continuous embedding E α,p C. Let us prove this result. Let u E α,p nd then D α u Lp. Since 0 < (1/p) < α < 1, Property 4 leds to I α Dα u C. Furthermore, u = I α D α u nd consequently, u cn be identified to its continuous representtive. Finlly, Property 4 lso gives: (b ) α (1/p) (23) u E α,p, u = I α Dα u Γ(α) ( (α 1)q +1 ) 1/q u α,p. (24) Since α,p nd α,p re equivlent norms, the proof of Section 3.2 is complete The compct embedding E α,p C. Let us prove this property. Since E α,p is reflexive Bnch spce, we only hve to prove tht: Let (u n ) n N E α,p such tht: Since E α,p C, we hve: E α,p C (u n ) n N E α,p such tht u n u, then u n u. (25) E α,p u n u. (26) u n C u. (27) Since(u n ) n N convergesweklyine α,p, (u n ) n N isboundedine α,p. Consequently, (D α u n) n N is bounded in L p by constnt M 0. Let us prove tht (u n ) n N C is uniformly lipschitzin on [,b]. According to the proof of Property 4, we hve: n N, t 1 < t 2 b, u n (t 2 ) u n (t 1 ) I α D α u n (t 2 ) I α D α u n (t 1 ) 2 D u α n L p Γ(α) ( (α 1)q +1 ) 1/p (t 2 t 1 ) α (1/p) 2M Γ(α) ( (α 1)q +1 ) 1/p (t 2 t 1 ) α (1/p). Hence, from Ascoli s theorem, (u n ) n N is reltively compct in C. Consequently, there exists subsequence of (u n ) n N converging strongly in C nd the limit is u by uniqueness of the wek limit. Now, let us prove by contrdiction tht the whole sequence (u n ) n N converges strongly to u in C. If not, there exist ε > 0 nd subsequence (u nk ) k N such tht: k N, u nk u > ε > 0. (28) Nevertheless, since (u nk ) k N is subsequence of (u n ) n N, then it stisfies: E α,p u nk u. (29)

9 EXISTENCE OF A WEAK SOLUTION FOR FRACTIONAL EULER-LAGRANGE EQUATIONS. 9 In the sme wy (using Ascoli s theorem), we cn construct subsequence of (u nk ) k N converging strongly to u in C which is contrdiction to (28). The proof of Section 3.3 is now complete Remrks. Let us remind the following property: From this result, we get the two following results: ϕ C c, Iα ϕ C. (30) C is dense in E α,p. Indeed, let us first prove tht C E α,p. Let u C L p. Since u AC nd u L p, we hve D u α = I 1 α u Lp. Since u AC, we lso hve I α D u α = u. Finlly, u E α,p. Now, let us prove tht C is dense in E α,p. Let u E α,p, then D u α L p. Consequently, there exists (v n ) n N Cc such tht: L v p n D α u nd then Iα v n Lp I α Dα u = u, (31) since I α is continuous from L p to L p. Defining u n := I v α n C for ny n N, we obtin: L u p n u nd D α u n = D α Iα v L n = v p n D α u. (32) Finlly, (u n ) n N C nd converges to u in E α,p. The proof of this point is complete; In the cse (1/p) < min(α,1 α), E α,p = {u L p stisfying D α u L p }. Indeed, let u L p stisfying D α u Lp nd let us prove tht I α Dα u = u. Let ϕ C c L 1. Since D α u Lp, Property 3 leds to: I α Dα u ϕ dt = D α u Iα + ϕ dt = d dt (I1 α u) Iα + ϕ dt. (33) Then, n integrtion by prts gives: I α Dα u ϕ dt = I 1 α u D1 α + ϕ dt. (34) Indeed, I α +ϕ(b) = 0 since ϕ C c nd I 1 α u() = 0 since u Lp nd (1/p) < 1 α. Finlly, using Property 3 gin, we obtin: I α Dα u ϕ dt = u I+ 1 α D+ 1 α ϕ dt = u ϕ dt, (35) which concludes the proof of this second point. In this cse, let us note tht such definition of E α,p could led us to nme it frctionl Sobolev spce nd to denote it by W α,p. Nevertheless, these notion nd nottion re lredy used, see [7]. 4. Vritionl structure of (EL α ) In the rest of the pper, we ssume tht Lgrngin L is of clss C 1 nd we define the Lgrngin functionl L on E α,p (with 0 < (1/p) < α < 1). Precisely, we define: L : E α,p R u L(u,D α u,t) dt. (36)

10 10 LOÏC BOURDIN L is sid to be Gâteux-differentible in u E α,p if the mp: DL(u) : E α,p R v DL(u)(v) := lim h 0 L(u+hv) L(u) h is well-defined for ny v E α,p nd if it is liner nd continuous. A criticl point u E α,p of L is defined by DL(u) = Gâteux-differentibility of L. Let us prove the following lemm: Lemm 1. The following implictions hold: L stisfies (H 1 ) = for ny u E α,p, L(u,D α u,t) L 1 nd then L(u) exists in R; L stisfies (H 2 ) = for ny u E α,p, / x(u,d α u,t) L 1 ; L stisfies (H 3 ) = for ny u E α,p, / y(u,d α u,t) L q. Proof. Let us ssume tht L stisfies (H 1 ) nd let u E α,p C. Then, D α u d 1 L p/d 1 L 1 nd the three mps t r 1 ( u(t),t ), s1 ( u(t),t ), L ( u(t),0,t ) C([,b],R + ) L L 1. Hypothesis (H 1 ) implies for lmost ll t [,b]: L(u(t),D α u(t),t) r 1(u(t),t) D α u(t) d 1 +s 1 (u(t),t)+ L(u(t),0,t). (38) Hence, L(u,D α u,t) L1 nd then L(u) exists in R. We proceed in the sme mnner in order to prove the second point of Lemm 1. Now, ssuming tht L stisfies (H 3 ), we hve D α u d 3 L p/d 3 L q for ny u E α,p. An nlogous rgument gives the third point of Lemm 1. Let us prove the following result: Proposition 2. Assuming tht L stisfies Hypotheses (H 1 ), (H 2 ) nd (H 3 ), L is Gâteuxdifferentible in ny u E α,p nd: u,v E α,p, DL(u)(v) = (37) x (u,dα u,t) v + y (u,dα u,t) D α v dt. (39) Proof. Let u, v E α,p C. Let ψ u,v defined for ny h [ 1,1] nd for lmost ll t [,b] by: ψ u,v (t,h) := L ( u(t)+hv(t),d α u(t)+hdα v(t),t). (40) Then, we define the following mpping: φ u,v : [ 1,1] R h Our im is to prove tht the following term: L(u+hv,D α u+hd α v,t) dt = ψ u,v (t,h) dt. L(u+hv) L(u) φ u,v (h) φ u,v (0) DL(u)(v) = lim = lim = φ h 0 h h 0 h u,v(0) (42) exists in R. In order to differentite φ u,v, we use the theorem of differentition under the integrl sign. Indeed, we hve for lmost ll t [,b], ψ u,v (t, ) is differentible on [ 1,1] (41)

11 EXISTENCE OF A WEAK SOLUTION FOR FRACTIONAL EULER-LAGRANGE EQUATIONS. 11 with: h [ 1,1], ψ u,v h ( (t,h) = u(t)+hv(t),d α x u(t)+hd v(t),t α ) v(t) + ( u(t)+hv(t),d α y u(t)+hd α v(t),t) D α v(t). (43) Then, from Hypotheses (H 2 ) nd (H 3 ), we hve for ny h [ 1,1] nd for lmost ll t [,b]: ψ u,v h (t,h) [r ( ) 2 u(t)+hv(t),t D α u(t)+hd α ( ) ] v(t) d 2 +s 2 u(t)+hv(t),t v(t) [ ( ) + r 3 u(t)+hv(t),t D α u(t)+hd α v(t) d 3 ( ) ] +s 3 u(t)+hv(t),t D α v(t). (44) We define: r 2,0 := mx r ( ) 2 u(t)+hv(t),t (t,h) [,b] [ 1,1] nd we define similrly s 2,0, r 3,0, s 3,0. Finlly, it holds: h (t,h) 2d 2 r 2,0 ( D α u(t) d 2 + D α v(t) d 2) v(t) +s 2,0 v(t) }{{}}{{}}{{} L p/d 2 L 1 C L C L 1 ψ u,v +2 d 3 r 3,0 D α u(t) d 3 + D α v(t) d 3 }{{} L p/d 3 L q α ) D }{{ v(t) } L p (45) α +s 3,0 D }{{ v(t). (46) } L p L 1 The right term is then L 1 function independent of h. Consequently, pplying the theorem of differentition under the integrl sign, φ u,v is differentible with: Hence: DL(u)(v) = φ u,v(0) = From Lemm 1, it holds: h [ 1,1], φ u,v(h) = ψ u,v (t,h) dt. (47) h ψ b u,v (t,0) dt = h x (u,dα u,t) v + y (u,dα u,t) D v α dt. (48) x (u,dα u,t) L 1 nd y (u,dα u,t) L q. (49) Since v C L nd D v α L p, DL(u)(v) exists in R. Moreover, we hve: DL(u)(v) x (u,dα u,t) + L 1 v y (u,dα u,t) D α v L p L q ( (b ) α (1/p) Γ(α) ( (α 1)q +1 ) 1/q x (u,dα u,t) + L 1 y (u,dα u,t) Consequently, DL(u) is liner nd continuous from E α,p to R. The proof is complete. L q ) v α,p.

12 12 LOÏC BOURDIN 4.2. Sufficient condition for wek solution. In this section, we prove the following theorem: Theorem 2. Let us ssume tht L stisfies Hypotheses (H 1 ), (H 2 ) nd (H 3 ). Then: u is criticl point of L = u is wek solution of (EL α ). (50) Proof. Let u be criticl point of L. Then, we hve in prticulr: v C c, DL(u)(v) = x (u,dα u,t) v + y (u,dα u,t) Dα v dt = 0. (51) For ny v Cc AC, D αv = I1 α v C. Since / y(u,dα u,t) Lq, Property 3 gives: ( ) v Cc, x (u,dα u,t) v +I+ 1 α y (u,dα u,t) v dt = 0. (52) Finlly, we define: t t [,b], w u (t) = x (u,dα u,t) dt. (53) Since / x(u,d αu,t) L1, w u AC nd w u = / x(u,d α u,t). Then, n integrtion by prts leds to: ( ( ) v Cc, I+ 1 α y (u,dα ) w u,t) u v dt = 0. (54) Consequently, there exists constnt C R d such tht: ( ) I+ 1 α y (u,dα u,t) = C +w u AC. (55) By differentition, we obtin: D α + ( ) y (u,dα u,t) = x (u,dα u,t), (56) nd then u E α,p C stisfies (EL α ).e. on [,b]. The proof is complete. Let us note tht the use of Property 3 in the previous proof leds to the emergence of D α + in (EL α ) lthough L is only dependent of D α. This symmetry in (EL α ) is strong drwbck in order to solve it explicitly. However, from Theorem 1, the existence of wek solution for (EL α ) will be gurntee. 5. Existence of globl minimizer of L In this section, underssumptions (H 4 ) nd (H 5 ), we prove the existence of globl minimizer u of L, see Theorem 3. Then, u is criticl point of L nd then, ccording to Theorem 2, u is wek solution of (EL α ). This concludes the proof of Theorem 1. As usul in vritionl method, in order to prove the existence of globl minimizer of functionl, coercivity nd convexity hypotheses need to be dded on the Lgrngin. We hve lredy define Hypotheses (H 4 ) (coercivity) nd (H 5 ) (convexity) in Section 1.2. In this section, we introduce two different convexity hypotheses (H 5 ) nd (H 5 ) under which Theorem 1 is still vlid:

13 EXISTENCE OF A WEAK SOLUTION FOR FRACTIONAL EULER-LAGRANGE EQUATIONS. 13 Convexity hypothesis denoted by (H 5 ): (x,t) R d [,b], L(x,,t) is convex nd ( L(,y,t) ) (y,t) R d [,b] is uniformly equicontinuous on Rd. (H 5 ) We remind tht the uniform equicontinuity of ( L(,y,t) ) (y,t) R d [,b] hs to be understood s: ε > 0, δ > 0, (x 1,x 2 ) (R d ) 2, x 2 x 1 < δ = (y,t) R d [,b], L(x 2,y,t) L(x 1,y,t) < ε. (57) Let us note tht Hypotheses (H 5 ) nd (H 5 ) re independent. Convexity hypothesis denoted by (H 5 ): (x,t) R d [,b], L(x,,t) is convex. (H 5 ) Hypothesis (H 5 ) is the wekest. Nevertheless, in this cse, the detiled proof of Theorem 3 is more complicted. Consequently, in the cse of Hypothesis (H 5 ), we do not develop the proof nd we use strong result proved in [12]. Let us prove the following preliminry result: Lemm 2. Let us ssume tht L stisfies Hypothesis (H 4 ). Then, L is coercive in the sense tht: lim L(u) = +. (58) u α,p + Proof. Let u E α,p, we hve: L(u) = Eqution (18) implies tht: L(u,D α u,t) dt d 4p u d 4 L d 4 (b )1 d4 p u d 4 L (b )α+1 p Γ(α+1) Finlly, we conclude tht: c 1 (u,t) D α u p +c 2 (t) u d 4 +c 3 (t) dt. (59) D α u d 4 d 4p L = (b )α+1 p Γ(α+1) u d 4 α,p. (60) u E α,p, L(u) γ D α u p L p c 2 u d 4 L d 4 (b ) c 3 (61) γ u p α,p c 2 (b ) α+1 d 4p Γ(α+1) u d 4 α,p (b ) c 3. (62) Since d 4 < p nd since the norms α,p nd α,p re equivlent, the proof is complete. Now, we re redy to prove Theorem 3: Theorem 3. Let us ssume tht L stisfies Hypotheses (H 1 ), (H 2 ), (H 3 ), (H 4 ) nd one of Hypotheses (H 5 ), (H 5 ) or (H 5 ). Then, L dmits globl minimizer.

14 14 LOÏC BOURDIN Proof. Let (u n ) n N be sequence in E α,p stisfying: L(u n ) inf v E α,p L(v) =: K. (63) Since L stisfies Hypothesis (H 1 ), L(u) R for ny u E α,p. Hence, K < +. Let us prove by contrdiction tht (u n ) n N is bounded in E α,p. In the negtive cse, we cn construct subsequence(u nk ) k N stisfying u nk α,p +. Since L stisfies Hypothesis (H 4 ), Lemm 2 gives: K = lim k N L(u nk ) = +, (64) which is contrdiction. Hence, (u n ) n N is bounded in E α,p. Since E α,p is reflexive, there exists subsequencestill denoted by (u n ) n N converging wekly in E α,p to n element denoted by u E α,p. Let us prove tht u is globl minimizer of L. Since: we hve: Cse L stisfies (H 5 ): by convexity, it holds for ny n N: L(u n ) = L(u n,d α u n,t) dt E α,p u n u nd E α,p C, (65) C u n u nd D α L u p n D α u. (66) L(u,D α u,t) dt + x (u,dα u,t) (u n u) dt+ y (u,dα u,t) (D α u n D α u) dt. (67) Since L stisfies Hypotheses (H 2 ) nd (H 3 ), / x(u,d u,t) α L 1 nd / y(u,d u,t) α L q. Consequently, using (66) nd mking n tend to +, we obtin: K = inf v E α,p L(v) Consequently, u is globl minimizer of L. L(u,D α u,t) dt = L(u). (68) Cse L stisfies (H 5 ): let ε > 0. Since (u n) n N converges strongly in C to u, we hve: N N, n N, u n u < δ, (69) where δ is given in the definition of (H 5 ). In consequence, it holds.e. on [,b]: Moreover, for ny n N, we hve: L(u n ) = n N, L ( u n (t),d α u n (t),t ) L ( u(t),d α u n (t),t ) < ε. (70) L(u,D α u,t) dt+ L(u n,d α u n,t) L(u,D α u n,t) dt + L(u,D α u n,t) L(u,D α u,t) dt. (71)

15 EXISTENCE OF A WEAK SOLUTION FOR FRACTIONAL EULER-LAGRANGE EQUATIONS. 15 Then, for ny n N, it holds by convexity: L(u n ) L(u,D α u,t) dt L(u n,d α u n,t) L(u,D α u n,t) dt + y (u,dα u,t) (D u α n D u) α dt. (72) And, using Eqution (70), we obtin for ny n N: L(u n ) L(u,D α u,t) dt ε(b )+ y (u,dα u,t) (D α u n D α u) dt. (73) We remind tht / y(u,d αu,t) Lq since L stisfies (H 3 ). Since (D αu n) n N converges wekly in L p to D α u, we obtin by mking n tend to + nd then by mking ε tend to 0: K = inf v E α,p L(v) Consequently, u is globl minimizer of L. Cse L stisfies (H 5 ): we refer to Theorem 3.23 in [12]. L(u,D α u,t) dt = L(u). (74) Finlly, combining Theorems 2 nd 3, the proof of Theorem 1 is now complete. 6. Exmples Let us consider some exmples of Lgrngin L stisfying Hypotheses of Theorem 1. Consequently, the frctionl Euler-Lgrnge eqution (EL α ) ssocited dmits wek solution u E α,p. The most clssicl exmple is the Dirichlet integrl, i.e. the Lgrngin functionl ssocited to the Lgrngin L given by: L(x,y,t) = 1 2 y 2. (75) In this cse, L stisfies Hypotheses (H 1 ), (H 2 ), (H 3 ), (H 4 ) nd (H 5 ) for p = 2. Hence, the frctionl Euler-Lgrnge eqution (EL α ) ssocited dmits wek solution in E α,p for (1/2) < α < 1. In more generl cse, the following Lgrngin L: L(x,y,t) = 1 p y p +(x,t), (76) where p > 1 nd C 1 (R d [,b],r + ), stisfies Hypotheses (H 1 ), (H 2 ), (H 3 ), (H 4 ) nd (H 5 ). Consequently, the frctionl Euler-Lgrnge eqution (ELα ) ssocited to L dmits wek solution in E α,p for ny (1/p) < α < 1. Let us note tht if for ny t [,b], (,t) is convex, then L stisfies Hypothesis (H 5 ). In the unidimensionl cse d = 1, let us tke Lgrngin with second term liner in its first vrible, i.e.: L(x,y,t) = 1 p y p +f(t)x, (77)

16 16 LOÏC BOURDIN where p > 1 nd f C 1 ([,b],r). Then, L stisfies Hypotheses (H 1 ), (H 2 ), (H 3 ), (H 4 ) nd (H 5 ). Then, the frctionl Euler-Lgrnge eqution (EL α ) ssocited dmits wek solution in E α,p for ny (1/p) < α < 1. Theorem 1 is result bsed on strong conditions on Lgrngin L. Consequently, some Lgrngin do not stisfy ll hypotheses of Theorem 1. We cn cite the Bolz s exmple in dimension d = 1 given by: L(x,y,t) = (y 2 1) 2 +x 4. (78) L does not stisfy Hypothesis (H 4 ) neither Hypothesis (H 5 ). Nevertheless, s usul with vritionl methods, the conditions of regulrity, coercivity nd/or convexity cn often be replced by weker ssumptions specific to the studied problem. As n exmple, we cn cite [31] nd references therein bout higher-order integrls of the clculus of vritions. Indeed, in this pper, it is proved tht clculus of vritions is still vlid with weker regulrity ssumptions. Conclusion The method developed in this pper gives frmework in order to study the existence of wek solutions for frctionl Euler-Lgrnge equtions. In this pper, we hve studied the specil cse of Lgrngin functionl involving frctionl derivtives of Riemnn-Liouville. Nevertheless, such method cn lso be developed in the cse of frctionl derivtives of Cputo or Hdmrd. Indeed, these opertors stisfy similr properties thn Riemnn-Liouville s ones, see [17, 30]. In fct, the sme method cn be developed in the cse of generl liner opertors used in [2, 18, 24, 26]: this is the im of forthcoming pper. References [1] O.P. Agrwl. Formultion of Euler-Lgrnge equtions for frctionl vritionl problems. J. Mth. Anl. Appl., 272(1): , [2] O.P. Agrwl. Generlized vritionl problems nd Euler-Lgrnge equtions. Comput. Mth. Appl., 59(5): , [3] R. Almeid, A.B. Mlinowsk nd D.F.M. Torres. A frctionl clculus of vritions for multiple integrls with ppliction to vibrting string. J. Mth. Phys., 51(3):033503, 12, [4] D. Blenu nd S.I. Muslih. Lgrngin formultion of clssicl fields within Riemnn-Liouville frctionl derivtives. Phys. Script, 72(2-3): , [5] B. Bonill, M. Rivero, L. Rodríguez-Germá nd J. J. Trujillo. Frctionl differentil equtions s lterntive models to nonliner differentil equtions. Appl. Mth. Comput., 187(1):79 88, [6] L. Bourdin, J. Cresson nd I. Greff. A continuous/discrete frctionl Noether s theorem. Accepted in Communictions in Nonliner Science nd Numericl Simultion. Preprint rxiv: v1 [mth.ds]. [7] H. Brezis. Functionl nlysis, Sobolev spces nd prtil differentil equtions. Universitext. Springer, New York, [8] L. Cesri. Optimiztion theory nd pplictions, volume 17 of Applictions of Mthemtics (New York). Springer-Verlg, New York, Problems with ordinry differentil equtions. [9] F. Comte. Opérteurs frctionnires en économétrie et en finnce. Prépubliction MAP5, [10] J. Cresson, I. Greff nd P. Inizn. Lgrngin for the convection-diffusion eqution. Mthemticl Methods in the Applied Sciences, 2011.

17 EXISTENCE OF A WEAK SOLUTION FOR FRACTIONAL EULER-LAGRANGE EQUATIONS. 17 [11] J. Cresson nd P. Inizn. Vritionl formultions of differentil equtions nd symmetric frctionl embedding. J. Mth. Anl. Appl., 385(2): , [12] B. Dcorogn. Direct methods in the clculus of vritions, volume 78 of Applied Mthemticl Sciences. Springer, New York, second edition, [13] G.S.F. Frederico nd D.F.M. Torres. A formultion of Noether s theorem for frctionl problems of the clculus of vritions. J. Mth. Anl. Appl., 334(2): , [14] C. Godbillon. Géométrie différentielle et mécnique nlytique. Hermnn, Pris, [15] R. Hilfer. Applictions of frctionl clculus in physics. World Scientific, River Edge, New Jersey, [16] F. Jio nd Y. Zhou. Existence of solutions for clss of frctionl boundry vlue problems vi criticl point theory. Comput. Mth. Appl., 62(3): , [17] A.A. Kilbs, H.M. Srivstv nd J.J. Trujillo. Theory nd pplictions of frctionl differentil equtions, volume 204 of North-Hollnd Mthemtics Studies. Elsevier Science B.V., Amsterdm, [18] V. Kirykov. Generlized frctionl clculus nd pplictions, volume 301 of Pitmn Reserch Notes in Mthemtics Series. Longmn Scientific & Technicl, Hrlow, [19] M. Klimek. Existence - uniqueness result for certin eqution of motion in frctionl mechnics. Bulletin of the polish ccdemy of sciences, 58(4): [20] A. Kolmogorov, S. Fomine nd V.M. Tihomirov. Eléments de l théorie des fonctions et de l nlyse fonctionnelle. Éditions Mir, Moscow, Avec un complément sur les lgèbres de Bnch, pr V. M. Tikhomirov, Trduit du russe pr Michel Drgnev. [21] R.L. Mgin. Frctionl clculus models of complex dynmics in biologicl tissues. Comput. Mth. Appl., 59(5): , [22] A.B. Mlinowsk nd D.F.M. Torres. Introduction to the frctionl clculus of vritions. Imperil College Press. London, [23] R. Metzler nd J. Klfter. The rndom wlk s guide to nomlous diffusion: frctionl dynmics pproch. Phys. Rep., 339(1):77, [24] T. Odzijewicz, A.B. Mlinowsk nd D.F.M. Torres. Frctionl clculus of vritions in terms of generlized frctionl integrl with pplictions to Physics. Abstr. Appl. Anl., [25] T. Odzijewicz, A.B. Mlinowsk nd D.F.M. Torres. Frctionl vritionl clculus with clssicl nd combined Cputo derivtives. Nonliner Anl., 75(3): , [26] T. Odzijewicz, A.B. Mlinowsk nd D.F.M. Torres. Generlized frctionl clculus with pplictions to the clculus of vritions. Comput. Mth. Appl., [27] I. Podlubny. Frctionl differentil equtions, volume 198 of Mthemtics in Science nd Engineering. Acdemic Press Inc., Sn Diego, CA, An introduction to frctionl derivtives, frctionl differentil equtions, to methods of their solution nd some of their pplictions. [28] F. Riewe. Nonconservtive Lgrngin nd Hmiltonin mechnics. Phys. Rev. E (3), 53(2): , [29] F. Riewe. Mechnics with frctionl derivtives. Phys. Rev. E (3), 55(3, prt B): , [30] S.G. Smko, A.A. Kilbs nd O.I. Mrichev. Frctionl integrls nd derivtives. Gordon nd Brech Science Publishers, Yverdon, Theory nd pplictions, Trnslted from the 1987 Russin originl. [31] M.R. Sidi Ammi nd D.F.M. Torres. Regulrity of solutions to higher-order integrls of the clculus of vritions. Internt. J. Systems Sci., 39(9): , [32] J. Wng nd Y. Zhou. A clss of frctionl evolution equtions nd optiml controls. Nonliner Anl. Rel World Appl., 12(1): , [33] Y. Zhou nd F. Jio. Nonlocl Cuchy problem for frctionl evolution equtions. Nonliner Anl. Rel World Appl., 11(5): , 2010.

18 18 LOÏC BOURDIN [34] A. Zoi, M.-C. Néel nd A. Cortis. Continuous-time rndom-wlk model of trnsport in vribly sturted heterogeneous porous medi. Phys. Rev. E, 81(3):031104, Mr [35] A. Zoi, M.-C. Néel nd M. Joelson. Mss trnsport subject to time-dependent flow with nonuniform sorption in porous medi. Phys. Rev. E, 80:056301, Loïc Bourdin, Lbortoire de Mthémtiques et de leurs Applictions - Pu (LMAP). UMR CNRS Université de Pu et des Pys de l Adour. E-mil : bourdin.l@etud.univ-pu.fr