- Research
- Open access
- Published:
Linear and nonlinear convolution elliptic equations
Boundary Value Problems volume 2013, Article number: 211 (2013)
Abstract
In this paper, the separability properties of elliptic convolution operator equations are investigated. It is obtained that the corresponding convolution-elliptic operator is positive and also is a generator of an analytic semigroup. By using these results, the existence and uniqueness of maximal regular solution of the nonlinear convolution equation is obtained in spaces. In application, maximal regularity properties of anisotropic elliptic convolution equations are studied.
MSC:34G10, 45J05, 45K05.
1 Introduction
In recent years, maximal regularity properties for differential operator equations, especially parabolic and elliptic-type, have been studied extensively, e.g., in [1–13] and the references therein (for comprehensive references, see [13]). Moreover, in [14, 15], on embedding theorems and maximal regular differential operator equations in Banach-valued function spaces have been studied. Also, in [16, 17], on theorems on the multiplicators of Fourier integrals obtained, which were used in studying isotropic as well as anisotropic spaces of differentiable functions of many variables. In addition, multiplicators of Fourier integrals for the spaces of Banach valued functions were studied. On the basis of these results, embedding theorems are proved.
Moreover, convolution-differential equations (CDEs) have been treated, e.g., in [1, 18–22] and [23]. Convolution operators in vector valued spaces are studied, e.g., in [24–26] and [27]. However, the convolution-differential operator equations (CDOEs) are a relatively less investigated subject (see [13]). The main aim of the present paper is to establish the separability properties of the linear CDOE
and the existence and uniqueness of the following nonlinear CDOE
in E-valued spaces, where is a possible unbounded operator in a Banach space E, and are complex-valued functions, and λ is a complex parameter. We prove that the problem (1.1) has a unique solution u, and the following coercive uniform estimate holds
for all , and . The methods are based on operator-valued multiplier theorems, theory of elliptic operators, vector-valued convolution integrals, operator theory and etc. Maximal regularity properties for parabolic CDEs with bounded operator coefficients were investigated in [1].
2 Notations and background
Let denote the space of all strongly measurable E-valued functions that are defined on the measurable subset with the norm
Let C be the set of complex numbers, and let
A linear operator , is said to be uniformly positive in a Banach space E if is dense in E, does not depend on x, and there is a positive constant M so that
for every and , , where I is an identity operator in E, and is the space of all bounded linear operators in E, equipped with the usual uniform operator topology. Sometimes, instead of , we write and denote it by . It is known (see [28], §1.14.1) that there exist fractional powers of the positive operator A. Let denote the space with the graphical norm
Let denote Schwartz class, i.e., the space of E-valued rapidly decreasing smooth functions on , equipped with its usual topology generated by semi-norms. denoted by just S. Let denote the space of all continuous linear operators , equipped with the bounded convergence topology. Recall is norm dense in when .
Let , where are integers. An E-valued generalized function is called a generalized derivative in the sense of Schwartz distributions of the function if the equality
holds for all .
Let F denote the Fourier transform. Through this section, the Fourier transformation of a function f will be denoted by . It is known that
for all .
Let Ω be a domain in . and will denote the spaces of E-valued bounded uniformly strongly continuous and m-times continuously differentiable functions on Ω, respectively. For the space will be denoted by . Suppose and are two Banach spaces. A function is called a multiplier from to if the map , is well defined and extends to a bounded linear operator
Let Q denotes a set of some parameters. Let be a collection of multipliers in . We say that is a collection of uniformly bounded multipliers (UBM) if there exists a positive constant M independent on such that
for all and .
A Banach space E is called an UMD-space [29, 30] if the Hilbert operator
is bounded in , [29]. The UMD spaces include, e.g., , spaces and Lorentz spaces , .
A set is called R-bounded (see [5, 6, 12]) if there is a positive constant C such that
for all and , , where is a sequence of independent symmetric -valued random variables on . The smallest C, for which the above estimate holds, is called an R-bound of the collection W and denoted by .
A set , dependent on parameters , is called uniformly R-bounded with respect to h if there is a positive constant C, independent of , such that for all and ,
This implies that .
Definition 2.1 A Banach space E is said to be a space, satisfying the multiplier condition, if for any the R-boundedness of the set
implies that Ψ is a Fourier multiplier, i.e., for any .
The uniform R-boundedness of the set
i.e.,
implies that is a uniformly bounded collection of Fourier multipliers (UBM) in .
Remark 2.2 Note that if E is UMD space, then by virtue of [5, 7, 12, 25], it satisfies the multiplier condition. The UMD spaces satisfy the uniform multiplier condition (see Proposition 2.4).
Definition 2.3 A positive operator A is said to be a uniformly R-positive in a Banach space E if there exists such that the set
is uniformly R-bounded.
Note that every norm bounded set in Hilbert spaces is R-bounded. Therefore, all sectorial operators in Hilbert spaces are R-positive.
Let , and , be standard unit vectors of ,
and let , be a closed linear operator in E with domain independent of x. The Fourier transformation of is a linear operator with the same domain defined as
(For details see [[2], p.7].) Let be a closed linear operator in E with domain independent of x. Then, it is differentiable if there is the limit
in the sense of E-norm.
Let , be closed linear operator in E with domain independent of x and . We can define the convolution in the distribution sense by
(see [2]).
Let and E be two Banach spaces, where is continuously and densely embedded into E. Let l be a integer number. denote the space of all functions from such that and the generalized derivatives with the following norm
It is clearly seen that
A function satisfying the equation (1.1) a.e. on , is called a solution of equation (1.1).
The elliptic CDOE (1.1) is said to be separable in if for the equation (1.1) has a unique solution u, and the following coercive estimate holds
where the constant C do not depend on f.
In a similar way as Theorem in [31], Theorem and by reasoning as Theorem 3.7 in [7], we obtain the following.
Proposition 2.4 Let E be UMD space, and suppose there is a positive constant K such that
Then is UBM in for .
Proof Really, some steps of proof trivially work for the parameter dependent case (see [7]). Other steps can be easily shown by setting
instead of
and by using uniformly R-boundedness of set . However, parameter depended analog of Proposition 3.4 in [7] is not straightforward. Let and be Fourier multipliers in . Let converge to in , and let be uniformly bounded with respect to h and N. Then by reasoning as Proposition 3.4 in [7], we obtain that the operator function is uniformly bounded with respect to h. Hence, by using steps above, in a similar way as Theorem 3.7 in [7], we obtain the assertion.
Let and be two Banach spaces. Suppose that and . Then will denote operator for and . □
In a similar way as Proposition 2.11 in [12], we have
Proposition 2.5 Let . If is R-bounded, then the collection is also R-bounded.
From [11], we obtain the following.
Theorem 2.6 Let the following conditions be satisfied
-
1.
E is a Banach space satisfying the uniform multiplier condition, and are certain parameters;
-
2.
l is a positive integer, and are n-tuples of nonnegative integer numbers such that , ;
-
3.
A is an R-positive operator in E with .
Then the embedding is continuous, and there exists a positive constant such that
Theorem 2.7 Let the following conditions be satisfied
-
1.
E is a Banach space satisfying the uniform multiplier condition, and are certain parameters;
-
2.
l is a positive integer, and are n-tuples of nonnegative integer numbers such that , ;
-
3.
A is an R-positive operator in E with .
Then the embedding is continuous, and there exists a positive constant such that
for all .
3 Elliptic CDOE
Condition 3.1 Assume that and the following hold
where , .
In the following, we denote the operator functions by for .
Lemma 3.2 Assume Condition 3.1 holds, and is a uniformly φ-positive operator in E with . Then, the following operator functions
are uniformly bounded, where .
Proof By virtue of Lemma 2.3 in [4] for , and there is a positive constant C such that
Since , in view of (3.1) and resolvent properties of positive operators, we get that is invertible and
Next, let us consider . It is clearly seen that
Since A is uniformly φ-positive and , then setting in the following well-known inequality
we obtain
Taking into account the Condition 3.1 and (3.1)-(3.3), we get
□
Lemma 3.3 Assume Condition 3.1 holds, and . Let be a uniformly φ-positive operator in a Banach space E with , and let
Then, operator functions are uniformly bounded.
Proof Let us first prove that is uniformly bounded. Really,
where
and
By using (3.1) and (3.5), we get
Due to positivity of A, by using (3.1) and (3.5), we obtain
Since, is uniformly φ-positive, by using (3.1), (3.3) and (3.4) for and , we get
In a similar way, the uniform boundedness of is proved. Next, we shall prove is uniformly bounded. Similarly,
where
Let us first show that is uniformly bounded. It is clear that
Due to positivity of A, by virtue of (3.1) and (3.3)-(3.5), we obtain . In a similar way, we have . Hence, operator functions , are uniformly bounded. From the representations of , it easy to see that operator functions contain similar terms as , namely, the functions will be represented as combinations of principal terms
where . Therefore, by using similar arguments as above and in view of (3.6), one can easily check that
□
Lemma 3.4 Let all conditions of the Lemma 3.2 hold. Suppose that E is a Banach space satisfying the uniform multiplier condition, and is a uniformly R positive operator in E. Then, the following sets
are uniformly R-bounded for and .
Proof Due to R-positivity of A we obtain that the set
is R bounded. Since
the set is R -bounded. Moreover, in view of Condition 3.1 and (3.1), there is a positive constant M such that
Then, by virtue of Kahane’s contraction principle, Lemma 3.5 in [5], we obtain that the set is uniformly R-bounded. Then by Lemma 3.2, we obtain the uniform R-boundedness of sets , i.e,
Moreover, due to boundedness of , in view of Condition 3.1 and by virtue of (3.1) and (3.3), we obtain
In view of representation (3.6) and estimate (3.8), we need to show uniform R-boundedness of the following sets
for . By virtue of Kahane’s contraction principle, additional and product properties of R-bounded operators, see, e.g., Lemma 3.5, Proposition 3.4 in [5], and in view of (3.7), it is sufficient to prove uniform R-boundedness of the following set
Since
thanks to R-boundedness of , we have
for all , , , , where is a sequence of independent symmetric -valued random variables on . Thus, in view of Kahane’s contraction principle, additional and product properties of R-bounded operators and (3.9), we obtain
The estimate (3.10) implies R-boundedness of the set . Moreover, from Lemma 3.2, we get
i.e., we obtain the assertion. □
The following result is the corollary of Lemma 3.4 and Proposition 2.4.
Result 3.5 Suppose that all conditions of Lemma 3.3 are satisfied, E is UMD space, and is a uniformly R-positive operator in E. Then the sets , are uniformly R-bounded.
Now, we are ready to present our main results. We find sufficient conditions that guarantee separability of problem (1.1).
Condition 3.6 Suppose that the following are satisfied
-
1.
For and , , ;
-
2.
and , , ;
-
3.
For and ,
Theorem 3.7 Suppose that Condition 3.6 holds, and E is a Banach space satisfying the uniform multiplier condition. Let be a uniformly R-positive in E with . Then, problem (1.1) has a unique solution u, and the following coercive uniform estimate holds
for all , and .
Proof By applying the Fourier transform to equation (1.1), we get
Hence, the solution of equation (1.1) can be represented as . Then there are positive constants and , so that
where are operator functions defined in Lemma 3.3. Therefore, it is sufficient to show that the operator-functions are UBM in . However, these follow from Lemma 3.4. Thus, from (3.13), we obtain
for all . Hence, we get assertion.
Let O be an operator in that is generated by the problem (1.1) for , i.e.,
□
Result 3.8 Theorem 2.6 implies that the operator O is separable in X, i.e., for all , all terms of equation (1.1) also are from X, and for solution u of equation (1.1), there are positive constants and so that
Condition 3.9 Let for . Moreover, there are positive constants and so that for ,
Remark 3.10 Condition 3.9 is checked for the regular elliptic operators with smooth coefficients on sufficiently smooth domains considered in the Banach space , (see Theorem 5.1).
Theorem 3.11 Assume that all conditions of Theorem 3.7 and Condition 3.9 are satisfied. Let E be a Banach space satisfying the uniform multiplier condition. Then, problem (1.1) has a unique solution , and the following coercive uniform estimate holds
for all , and .
Proof By applying the Fourier transform to equation (1.1), we obtain , where
So, we obtain that the solution of equation (1.1) can be represented as . Moreover, by Condition 3.9, we have
Hence, by using estimates (3.12), it is sufficient to show that the operator functions and are UBM in . Really, in view of Condition 3.9, and uniformly R-positivity of , these are proved by reasoning as in Lemma 3.4. □
Condition 3.12 There are positive constants and such that
for and
in cases, where , for and .
Theorem 3.13 Let all conditions of Theorem 3.11 and Condition 3.12 hold. Then for , there are positive constants and , so that
Proof The left part of the inequality above is derived from Theorem 3.11. So, it remains to prove the right side of the estimate. Really, from Condition 3.12 for we have
Hence, applying the Fourier transform to equation (1.1), and by reasoning as Theorem 3.11, it is sufficient to prove that the function
is a multiplier in . In fact, by using Condition 3.12 and the proof of Lemma 3.2, we get desired result. □
Result 3.14 Theorem 3.13 implies that for all , there are positive constants and , so that
From Theorem 3.7, we have the following.
Result 3.15 Assume all conditions of Theorem 3.7 hold. Then, for all , the resolvent of operator O exists, and the following sharp estimate holds
Result 3.16 Theorem 3.7 particularly implies that the operator for is positive in , i.e., if is uniformly R-positive for , then (see, e.g., [28], § 1.14.5) the operator is a generator of an analytic semigroup in .
From Theorems 3.7, 3.11, 3.13 and Proposition 2.4, we obtain the following.
Result 3.17 Let conditions of Theorems 3.7, 3.11, 3.13 hold for Banach spaces , respectively. Then assertions of Theorems 3.7, 3.11, 3.13 are valid.
4 The quasilinear CDOE
Consider the equations
in E-valued spaces, where is a possible unbounded operator in Banach space E, are complex-valued functions, and denote all differential operators that . Let
Remark 4.1 By using Theorem 2.7, we obtain that the embedding is continuous, and by trace theorem [32] (or [19]) for , , , ,
Let denote by . Consider the linear CDOE
From Theorem 3.7, we conclude that problem (4.2) has a unique solution , and the coercive uniform estimate holds
for all , .
Condition 4.2 Assume that all conditions of Theorem 3.11 are satisfied for and . Suppose that
-
1.
The function: is a Lipschitz function from to , i.e.,
for all ;
-
2.
is a measurable function for each u, , , , , and is continuous with respect to , . Moreover, there exists such that
for all , , and , .
Theorem 4.3 Let Condition 4.2 hold. Then, there exist a radius and such that for each with there exists a unique with satisfying equation (3.13).
Proof We want to to solve problem (4.1) locally by means of maximal regularity of the linear problem (4.2) via the contraction mapping theorem. For this purpose, let w be a solution of the linear BVP (4.2). Consider the following ball
Let such that . Let , .
Define a map G on by
where u is a solution of problem (4.1). We want to show that , and that L is a contraction operator in Y. Consider the function
We claim that , moreover, δ and can be chosen such that . In fact, since by Theorem 2.7, , and one has
Thus, Q is measurable and
Now, by Remark 4.1, , by choosing and , it follows that
Moreover, by Theorem 3.11 and by embedding Theorem 2.6, we get
Thus, G maps the set to . Let us show that G is a strict contraction. Let
It is clearly seen that is a solution of the linear problem (4.2) for
Then, by using estimate (4.3) and reasoning as above, we get
Choose , so that , we obtain that G is a strict contraction. Then by virtue of contraction mapping principle, we obtain that problem (4.1) has a unique solution . □
5 Boundary value problems for integro-differential equations
In this section, by applying Theorem 3.7, the BVP for the anisotropic type convolution equations is studied. The maximal regularity of this problem in mixed norms is derived. In this direction, we can mention, e.g., the works [2, 18, 21] and [33].
Let , where is an open connected set with a compact -boundary ∂ Ω. Consider the BVP for integro-differential equation
where
In general, , so equation (4.4) is anisotropic. For , we get isotropic equation. If , , will denote the space of all p-summable scalar-valued functions with a mixed norm (see, e.g., [34]), i.e., the space of all measurable functions f defined on , for which
Analogously, denotes the Sobolev space with a corresponding mixed norm [34]. Let Q denote the operator, generated by problem (4.4) and (5.1). In this section, we present the following result.
Theorem 5.1 Let the following conditions be satisfied
-
1.
for each and for each with , and , ;
-
2.
for each , , , ;
-
3.
For , , , , let ;
-
4.
For each local BVP in local coordinates corresponding to
has a unique solution for all and for with ;
-
5.
The (1) part of Condition 3.6 is satisfied, , and there are positive constants , , so that
Then, for and problems (4.4) and (5.1) have a unique solution , and the following coercive uniform estimate holds
Proof Let . It is known [29] that is UMD space for . Consider the operator A in , defined by
Therefore, problems (4.4) and (5.1) can be rewritten in the form of (1.1), where , are functions with values in . It is easy to see that and are operators in defined by
In view of conditions and by [[5], Theorem 8.2] operators and for sufficiently large , are uniformly R-positive in . Moreover, by (3.3), the problems
for and for sufficiently large μ, have unique solutions that belong to , and the coercive estimates hold
for solutions of problems (5.4) and (5.5). Then in view of (5) condition and by virtue of embedding theorems [34], we obtain
Moreover by using (5) condition for we have
i.e., all conditions of Theorem 3.7 hold, and we obtain the assertion. □
6 Infinite system of IDEs
Consider the following infinity system of a convolution equation
for and .
Condition 6.1 There are positive constants and , so that for for all and some ,
Suppose that , and there are positive constants , , so that
Let
Let Q be a differential operator in , generated by problem (5.7) and . Applying Theorem 3.7, we have the following.
Theorem 6.2 Suppose that (1) part of Condition 3.6 and Condition 6.1 are satisfied. Then
-
1.
For all , for , the equation (6.1) has a unique solution that belongs to , and the coercive uniform estimate holds
-
2.
For , there exists a resolvent of operator Q and
Proof Really, let and , . Then
It is easy to see that is uniformly R-positive in , and all conditions of Theorem 3.7 are hold. Therefore, by virtue of Theorem 3.7 and Result 4.1, we obtain the assertions. □
Remark 6.3 There are a lot of positive operators in concrete Banach spaces. Therefore, putting concrete Banach spaces instead of E and concrete positive differential, pseudo differential operators, or finite, infinite matrices, etc. instead of operator A in (1.1) and (4.1), we can obtain the maximal regularity of different class of convolution equations, Cauchy problems for parabolic CDEs or it’s systems, by virtue of Theorem 3.7 and Theorem 3.11, respectively.
References
Amann H 1. In Linear and Quasi-Linear Equations. Birkhäuser, Basel; 1995.
Amann H: Operator-valued Fourier multipliers, vector-valued Besov spaces, and applications. Math. Nachr. 1997, 186: 5-56.
Agarwal R, Bohner P, Shakhmurov R: Maximal regular boundary value problems in Banach-valued weighted spaces. Bound. Value Probl. 2005, 1: 9-42.
Dore G, Yakubov S: Semigroup estimates and non coercive boundary value problems. Semigroup Forum 2000, 60: 93-121. 10.1007/s002330010005
Denk R, Hieber M: R -boundedness, Fourier multipliers and problems of elliptic and parabolic type. Mem. Am. Math. Soc. 2003., 166: Article ID 788
Gorbachuk VI, Gorbachuk ML: Boundary Value Problems for Differential-Operator Equations. Naukova Dumka, Kiev; 1984.
Haller R, Heck H, Noll A: Mikhlin’s theorem for operator-valued Fourier multipliers in n variables. Math. Nachr. 2002, 244: 110-130. 10.1002/1522-2616(200210)244:1<110::AID-MANA110>3.0.CO;2-S
Lunardi A: Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkhäuser, Basel; 2003.
Ragusa MA: Homogeneous Herz spaces and regularity results. Nonlinear Anal., Theory Methods Appl. 2009, 71(12):e1909-e1914. 10.1016/j.na.2009.02.075
Sobolevskii PE: Inequalities coerciveness for abstract parabolic equations. Dokl. Akad. Nauk SSSR 1964, 57(1):27-40.
Shakhmurov VB: Embedding and maximal regular differential operators in Banach-valued weighted spaces. Acta Math. Sin. 2006, 22(5):1493-1508. 10.1007/s10114-005-0764-5
Weis L:Operator-valued Fourier multiplier theorems and maximal regularity. Math. Ann. 2001, 319: 735-758. 10.1007/PL00004457
Yakubov S, Yakubov Y: Differential-Operator Equations. Ordinary and Partial Differential Equations. Chapman & Hall/CRC, Boca Raton; 2000.
Shakhmurov VB: Embedding theorems and maximal regular differential operator equations in Banach-valued function spaces. J. Inequal. Appl. 2005, 4: 605-620.
Shakhmurov VB: Embedding and separable differential operators in Sobolev-Lions type spaces. Mathematical Notes 2008, 84(6):906-926.
Guliyev VS: Embeding theorems for spaces of UMD-valued functions. Dokl. Akad. Nauk SSSR 1993, 329(4):408-410. (in Russian)
Guliyev VS: On the theory of multipliers of Fourier integrals for Banach spaces valued functions. Tr. Math. Inst. Steklova 214. Investigations in the Theory of Differentiable Functions of Many Variables and Its Applications 1997. (in Russian)
Engler H: Strong solutions of quasilinear integro-differential equations with singular kernels in several space dimension. Electron. J. Differ. Equ. 1995, 1995(02):1-16.
Keyantuo V, Lizama C: Maximal regularity for a class of integro-differential equations with infinite delay in Banach spaces. Stud. Math. 2005, 168: 25-50. 10.4064/sm168-1-3
Prüss J: Evolutionary Integral Equations and Applications. Birkhäuser, Basel; 1993.
Poblete V: Solutions of second-order integro-differential equations on periodic Besov spaces. Proc. Edinb. Math. Soc. 2007, 50: 477-492. 10.1017/S0013091505001057
Vergara V: Maximal regularity and global well-posedness for a phase field system with memory. J. Integral Equ. Appl. 2007, 19: 93-115. 10.1216/jiea/1181075424
Shakhmurov VB: Coercive boundary value problems for regular degenerate differential-operator equations. J. Math. Anal. Appl. 2004, 292(2):605-620. 10.1016/j.jmaa.2003.12.032
Arendt W, Bu S: Tools for maximal regularity. Math. Proc. Camb. Philos. Soc. 2003, 134: 317-336. 10.1017/S0305004102006345
Girardi M, Weis L:Operator-valued multiplier theorems on and geometry of Banach spaces. J. Funct. Anal. 2003, 204(2):320-354. 10.1016/S0022-1236(03)00185-X
Hytönen T, Weis L: Singular convolution integrals with operator-valued kernels. Math. Z. 2007, 255: 393-425.
Shakhmurov VB, Shahmurov R: Sectorial operators with convolution term. Math. Inequal. Appl. 2010, 13(2):387-404.
Triebel H: Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Amsterdam; 1978.
Burkholder DL: A geometrical conditions that implies the existence certain singular integral of Banach space-valued functions. In Proc. Conf. Harmonic Analysis in Honor of Antonu Zigmund. Wadsworth, Belmont; 1983:270-286. Chicago, 1981
Bourgain J: Some remarks on Banach spaces in which martingale differences are unconditional. Ark. Mat. 1983, 21: 163-168. 10.1007/BF02384306
Shakhmurov VB: Maximal B -regular boundary value problems with parameters. J. Math. Anal. Appl. 2006, 320: 1-19. 10.1016/j.jmaa.2005.05.083
Lions JL, Peetre J: Sur une classe d’espaces d’interpolation. Publ. Math. Inst. Hautes Études Sci. 1964, 19: 5-68. 10.1007/BF02684796
Shakhmurov VB, Shahmurov R: Maximal B -regular integro-differential equations. Chin. Ann. Math., Ser. B 2009, 30(1):39-50. 10.1007/s11401-007-0553-9
Besov OV, Ilin VP, Nikolskii SM: Integrals Representations of Functions and Embedding Theorem. Nauka, Moscow; 1975.
Acknowledgements
The authors would like to thank the referees for valuable comments and suggestions in improving this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Shakhmurov, V.B., Ekincioglu, I. Linear and nonlinear convolution elliptic equations. Bound Value Probl 2013, 211 (2013). https://doi.org/10.1186/1687-2770-2013-211
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-2770-2013-211