Research Article | Open Access
Dariusz Idczak, Stanisław Walczak, "Fractional Sobolev Spaces via Riemann-Liouville Derivatives", Journal of Function Spaces, vol. 2013, Article ID 128043, 15 pages, 2013. https://doi.org/10.1155/2013/128043
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.
Let . Classical Sobolev space of order one on an open bounded interval is defined by (cf. ) where is the space of functions that are integrable with power , is the set of smooth functions with compact support , and is the classical derivative of . The function satisfying the above condition is denoted by and called the weak derivative of of order one.
Sobolev space of order is defined by (cf. ) The spaces can be characterized as follows.
Theorem 1. Let . Then, if and only if there exist functions such that for and , where denotes the classical derivative of of order . In such a case there exists an absolutely continuous function such that a.e. on , which has absolutely continuous classical derivatives , the derivative , and
Remark 2. One shows that , , ( times).
Remark 3. In our paper, we will identify functions defined on and (, , resp.) that are equal a.e. on .
Each of the functions , , is denoted by and called the weak derivative of of order .
The space endowed with a norm where , has many useful properties such as completeness, reflexivity (for ), and separability. Moreover, some imbeddings of these spaces are compact (cf. ).
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 , the obtained results reduce to the classical ones. In the literature, some generalizations of Sobolev spaces to noninteger orders, on a domain , are known (cf. ): Gagliardo spaces , Besov spaces , and Nikolskii spaces . They have been introduced with the aid of approaches different from ours and their comparison with our spaces (in the case of ) 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 and characterize them. In the fifth section, we derive a fractional counterpart of Theorem 1, define the weak fractional derivatives of order , 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.
By , where , we denote the set of all functions that have a representative (a.e. on ) which is absolutely continuous together with its classical derivatives of orders . Of course, such a function possesses also the classical derivative of order , existing a.e. on and belonging to . These classical derivatives are the weak derivatives of . It is known (cf. [3, Lemma 2.4]) that if and only if there exist and such that In such a case,
By we denote the set of all functions that have representation (6) with and . It is known that for .
Let , and let . By the left Riemann-Liouville fractional integral of on the interval we mean (cf. ) a function given by
Remark 4. The above integral exists and is finite a.e. on .
In view of the convergence (cf. [3, Theorem 2.7]) it is natural to put
Remark 5. If , the right side of (8) exists (and is finite) everywhere on . So, can be defined everywhere on . In such a case it is continuous on .
Remark 6. It is easy to see that if is essentially bounded on and , then the right side of (8) is defined and bounded everywhere on . So, in this case can be defined everywhere on . From [3, Theorem 3.6] it follows that in such a case is equal everywhere on to a continuous function on . It is also known (cf. [3, Theorem 2.6]) that if with , then .
Let , where . By we denote the set of all functions that have the representation with and . Of course, .
We say that possesses the left Riemann-Liouville derivative of order , , on the interval if . By this derivative we mean the derivative . Of course, .
The next theorem can be deduced from [3, Corollary 2.1, Lemma 2.5 (b), Lemma 2.6 (b)] but, to our best knowledge, it has not been formulated by other authors. In  we give a direct proof of it in the case of , in —when . In the case of it reduces to the theorem on the integral representation of type (6) of functions belonging to .
Theorem 7. If , , and , then has the left Riemann-Liouville derivative of order on the interval if and only if ; that is, has the representation (11). In such a case,
By () we denote the set of all functions possessing representation (11) with , .
Remark 8. It is easy to see that the above theorem implies the following one (for any ): has the left Riemann-Liouville derivative if and only if ; that is, has the representation (11) with .
Let . By the right Riemann-Liouville fractional integral of on the interval we mean a function Similarly, as in the case of left integral, we put
By we denote the set of all functions that have the representation with and . It is easy to see that .
We say that possesses the right Riemann-Liouville derivative of order , , on the interval if . By this derivative we mean the function . Of course, .
We also have the following.
Theorem 10. If , , and , then has the right Riemann-Liouville derivative of order on the interval if and only if ; that is, has the representation (15). In such a case,
By , , we denote the set of all functions possessing representation (15) with , .
Remark 11. As in the case of left Riemann-Liouville derivative, the following theorem holds true for any : has the right Riemann-Liouville derivative if and only if ; that is, has the representation (15) with .
Theorem 12. If , then if and only if or
3. Some Special Cases of Integration by Parts
Below, is the set of continuous functions and is the set of functions such that , (by we mean the function ). Of course, these classical derivatives coincide with the weak ones .
Lemma 14. If , , , and then , Consequently, .
Remark 15. Analogous lemma can be obtained for the left integral and derivative (without the term ).
We have the following special case of the theorem on integration by parts.
Theorem 16. If , , then for any and satisfying boundary conditions
Proof. Let be of the form (11) with and . 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  and in  its extension to the case of fractional derivatives of higher order is derived (the method of the proof is the same in both papers).
Theorem 17. If , , , , and , then for and .
4. Fractional Sobolev Spaces
Let and let . By left Sobolev space of order we will mean the set given by A function given above will be called the weak left fractional derivative of order of ; let us denote it by . Uniqueness of this weak derivative follows from [1, Lemma IV.2 and Propositions IV.18, IV.21].
Let , , . By the left Sobolev space of order we mean the set given by
Remark 18. Since for , we see that the weak left fractional derivative of coincides with the classical weak derivative of . Consequently, for .
We have the following characterization of .
Theorem 19. If , , and , then
Proof. Case of follows from (37) and from the fact that . So, let us consider the case of . We will apply the induction with respect to .
Let . If , then from Theorem 7 it follows that has the derivative . Theorem 16 implies that for any . So, with
Now, let us assume that , that is, , and there exists a function such that for any . To show that it is sufficient to check (cf. Theorem 7 and definition of ) that possesses the left Riemann-Liouville derivative of order , belonging to , that is, that is absolutely continuous on and its classical derivative of the first order (existing a.e. on ) belongs to .
If , then (cf. Lemma 14) and . From the differential form of the classical fractional theorem on the integration by parts (cf. [3, formula (2.64)]) it follows that This means (cf. (41) and (42)) that for any . So, . Consequently, is absolutely continuous and its classical derivative is equal a.e. on to .
Now, let and assume that for any . We will show that this equality holds true for any .
Indeed, let . If , then and . So, where and . If , from the above formula it follows (cf. (40)) that If , from Theorem 13 (b) and (40) it follows that This means that and, consequently, So, in both cases (, ) there exist and such that To show that it is sufficient to check that Indeed, we have (below, we use the following elementary formula for , ) for a.e. Thus, .
Conversely, let . To show that we have to check that Theorem 13(b) implies the equality where with . So (cf. Theorem 12), . From Theorem 13 (b) it also follows that (existence of the weak fractional derivative and equality follow from the relation and (40)).
From the first part of the above proof (case of ) and from the uniqueness of the weak fractional derivative the following theorem follows (cf. also [4, fractional fundamental lemma]).
Theorem 20. If , then the weak left fractional derivative of a function coincides with its left Riemann-Liouville fractional derivative a.e. on .
Remark 21. If and , then and, consequently, . If , then is the set of all functions belonging to that satisfy the condition .
5. Weak Fractional Derivatives
Now, we will prove an extension of Theorem 1.
Theorem 22. Let , , , and . Then, if and only if there exist functions such that for any . In such a case there exist the left Riemann-Liouville derivatives of and
Proof. Case of follows from Theorem 1. So, let us consider the case of . We will apply the induction with respect to .
When , it is sufficient to use the definition of with and Theorem 20.
So, let and assume that the theorem is true for any . We will show that it is true for any .
Indeed, if with a fixed , then Relation (59) implies (by the induction assumption) the existence of functions such that (of course, ) for any and . Moreover, From the existence of and from (60) it follows that . In particular, there exists derivative . If we put we see that and for any (the second equality follows from Theorem 16).
Now, let us assume that there exist functions such that for any . Since , the above condition for and the induction assumption mean that
Moreover, condition (64) for , Lemma 14, and the integral form of the classical fractional theorem on the integration by parts (cf. [3, formula (2.20)]) imply that for any . This means that . So, if we fix and use once again Lemma 14 and the integral form of the classical fractional theorem on the integration by parts, we obtain Thus, from Theorem 1 it follows that Using (65) and (68) we assert that .
Functions will be called the weak left fractional derivatives of of orders , respectively. Their uniqueness follows from [1, Lemma IV.2 and Propositions IV.18,IV.21]. From the above theorem it follows that they coincide with the appropriate Riemann-Liouville derivatives.
We have the following counterpart of Remark 2.
Theorem 23. If , , and , then
Proof. If , then and from the definition of the Riemann-Liouville derivative and Theorem 22 it follows that The proof is completed.
6. Norms in
Let us fix where and consider in the space a norm given by (here denotes the classical norm in ).
We have the following theorem.
Theorem 24. If and , then the norm is equivalent to a norm given by
Proof. We will use the induction with respect to .
Assume that and . Then, for given by with and , we have where (cf. [3, formula (2.72)]). Thus (of course, and ), where
Consequently, where .
Now, we will prove that there exists a constant such that Indeed, let and consider a coordinate function of with a fixed . The mean value theorem implies the existence of such that From the absolute continuity of it follows that for any . Consequently, for . In particular, So, where . Thus, and, consequently, where .
When , then (cf. Remark 21) is the set of all functions belonging to that satisfy the condition . Consequently, in the same way as in the case of (putting ) we obtain the inequality with some . The inequality with some is obvious (it is sufficient to put and use the fact that ).
Now, let us assume that the assertion holds true for some . We will prove that it is true for .
Let . We know that there exist constants and such that