Fractional Sobolev Spaces via Riemann-Liouville Derivatives

Using Riemann-Liouville derivatives, we introduce fractional Sobolev spaces, characterize them, define weak fractional derivatives, and show that they coincide with the Riemann-Liouville ones. Next, we prove equivalence of some norms in the introduced spaces and derive their completeness, reflexivity, separability, and compactness of some imbeddings. An application to boundary value problems is given as well.

In the last years, many papers and books on fractional calculus and its applications have appeared.Most of them concern fractional differential equations, including calculus of variations and optimal control.In the classical (positive integer) case the fundamental role in this field is played by the mentioned Sobolev spaces.To our best knowledge, there are no "fractional" Sobolev spaces based on the notion of fractional derivative in Riemann-Liouville sense, which seems to be the most used in the theory of fractional differential equations.Our aim is to give some systematic basics for applications of fractional calculus to differential equations.More precisely, we extend the above definitions, with the aid of the Riemann-Liouville derivatives, to the case of noninteger positive (fractional) order , derive a fractional counterpart of Theorem 1, and prove the basic properties of the introduced spaces.When  =  ∈ N, the obtained results reduce to the classical ones.In the literature, some generalizations of Sobolev spaces to noninteger orders, on a domain Ω ⊂ R  , are known (cf.[2]): Gagliardo spaces  , (Ω), Besov spaces  , (Ω), and Nikolskii spaces  , (Ω).They have been introduced with the aid of approaches different from ours and their comparison with our spaces (in the case of Ω = (, )) is an open problem.Let us point that only Gagliardo spaces coincide with the classical Sobolev spaces when  = .
The paper is organized as follows.In the second section, we recall some basic notions and facts from the fractional calculus including a characterization of functions possessing the left (right) Riemann-Liouville derivatives.In the third section, we derive some special cases of the fractional theorem on the integration by parts.In the fourth section, we define the fractional Sobolev spaces of any order  > 0 and characterize them.In the fifth section, we derive a fractional counterpart of Theorem 1, define the weak fractional derivatives of order  > 0, and show that they coincide with the Riemann-Liouville derivatives.In the sixth section, we introduce two norms in the fractional Sobolev spaces and prove their equivalence.In the seventh section, we derive completeness, reflexivity, and separability of the introduced spaces.In the eighth section, we prove compactness of some imbeddings.In the ninth section, we present some applications of the obtained results to fractional boundary value problems using a variational approach.
In the paper, we limit ourselves to the left fractional Sobolev spaces, but in an analogous way one can define right spaces and derive their appropriate properties.
In the case of  =  it reduces to the theorem on the integral representation of type (6) of functions belonging to  ,1 .Theorem 7. If  − 1 <  ≤ ,  ∈ N, and  ∈   (11) with  ∈   .
Let  > 0. By the right Riemann-Liouville fractional integral of ℎ ∈  1 on the interval [, ] we mean a function Similarly, as in the case of left integral, we put Remark 9. Clearly,   −  has the properties analogous to those described in Remarks 5 and 6.
Proof.If  ∈   and Proof.Let  ∈  ,1 + be of the form (11) with  0 ,  1 , . . .,  −1 ∈ R  and  ∈  1 .We have (Lemma 14 and Remark 9) Existence of the first integral and the first equality follows from Lemma 14; in the third equality we used an integral form of a classical fractional theorem on the integration by parts (cf.[3, formula (2.20)]), in the fourth equality we used continuity of    and in the fifth equality we used the equalities The proof is completed.
The next theorem will be used in the last section of the paper.Proof of this theorem is contained in [4] and in [5] its extension to the case of fractional derivatives of higher order is derived (the method of the proof is the same in both papers).

Fractional Sobolev Spaces
Let 0 <  ≤ 1 and let 1 ≤  < ∞.By left Sobolev space of order  we will mean the set A function  given above will be called the weak left fractional derivative of order  ∈ (0, 1] of ; let us denote it by   + .Uniqueness of this weak derivative follows from [1, Lemma IV. Remark 18.Since  1 −  = − 1  = − (1) for  ∈  ∞  , we see that the weak left fractional derivative  1  + of  coincides with the classical weak derivative   =  1  of .Consequently, for  ∈ N.
We have the following characterization of  , + .
where  0 ,  1 , . . .,  −1 ∈ R  and  ∈   .If  = 1, from the above formula it follows (cf.( 40)) that If  ≥ 2, from Theorem 13 (b) and (40) it follows that This means that  −1 +  ∈  , + and, consequently, So, in both cases ( = 1,  ≥ 2) there exist   ∈ R  and  ∈   such that To show that  ∈  , + ∩   it is sufficient to check that Indeed, we have (below, we use the following elementary formula for ] > 0,  > 0) Theorem 13(b) implies the equality where with  ∈   .So (cf.Theorem 12),  ∈ . From Theorem 13 (b) it also follows that From the first part of the above proof (case of  = 1) and from the uniqueness of the weak fractional derivative the following theorem follows (cf.also [4,

Weak Fractional Derivatives
Now, we will prove an extension of Theorem 1.  Proof.Case of  =  follows from Theorem 1. So, let us consider the case of −1 <  < .We will apply the induction with respect to  ∈ N.
When  = 1, it is sufficient to use the definition of
Indeed, if  ∈  , + with a fixed  ∈ (,  + 1), then The proof is completed.Proof.We will use the induction with respect to  ∈ N.

𝛼,𝑝 𝑎+
Now, we are in a position to prove some basic properties of the introduced spaces.
In the proofs of the next two theorems we use the method presented in [1].
The above theorem implies the following.
Corollary 32.The imbedding converges to the function where The proof is completed.
The above theorem implies the following.

Application to Boundary Value Problems
In this section, we will demonstrate an application of the obtained results to fractional boundary value problems.
Namely, let us fix  ∈ (1/2, 1) and consider the following problem: where  ∈ where   is the gradient (in ) of a potential  = (, ), with the above boundary conditions can be studied using the direct method of calculus of variations (cf.[11] for the case of (144)).
Remark 38.From Corollary 32 it follows that the imbedding  1, ⊂   is compact for any 1 ≤  < ∞.Corollary 34 implies the compactness of the imbedding  ,1 ⊂   for any  ∈ N,  ≥ 2, and 1 ≤  < ∞.The following problem is open: is it possible to strengthen Theorem 31 or Theorem 33 to deduce the compactness of the imbedding  1,1 ⊂   for any 1 ≤  < ∞?
2. By a solution to this problem we mean a function  ∈ ,2+ such that   +  and   −   +  exist, and satisfying the above equation and boundary conditions.Under the assumption on  the boundary conditions make sense ([10, Property 4]).It is easy to see that  is the closed subspace of the Hilbert space ( ,2 + , ‖ ⋅ ‖  , + ); it is sufficient to observe that it is closed in ,2+ with respect to ‖ ⋅ ‖ , , + and to use Theorem 24.Of course,  is a scalar product in  and the norm generated by  is simply the norm ‖ ⋅ ‖ ,2+ restricted to .Clearly,  is continuous and coercive; that is, there exists a constant  > 0 such that (, ) ≥ ‖‖2 [1]r  ∈ .So, Lax-Milgram theorem[1]or simply To our best knowledge, fractional Sobolev spaces via Caputo derivatives have not been investigated up to now and are an open problem.Remark 40.From the condition (136) it follows that one can search approximate solutions to (132) using numerical methods, for example, the gradient or projection of gradient methods applied to the functional