Advances in Differential Equations

A fractional fundamental lemma and a fractional integration by parts formula -- Applications to critical points of Bolza functionals and to linear boundary value problems

Loïc Bourdin and Dariusz Idczak

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Abstract

In this paper, we first identify some integrability and regularity issues that frequently occur in fractional calculus of variations. In particular, it is well-known that Riemann-Liouville derivatives make boundary singularities emerge. The major aim of this paper is to provide a framework ensuring the validity of the fractional Euler-Lagrange equation in the case of a Riemann-Liouville derivative of order $\alpha \in (0,1)$. For this purpose, we consider the set of functions possessing $p$-integrable Riemann-Liouville derivatives and we introduce a class of quasi-polynomially controlled growth Lagrangian. In the first part of the paper, we prove a new fractional fundamental (du Bois-Reymond) lemma and a new fractional integration by parts formula involving boundary terms. The proof of the second result is based on an integral representation of functions possessing Riemann-Liouville derivatives. In the second part of the paper, we give not only a necessary optimality condition of Euler-Lagrange type for fractional Bolza functionals, but also necessary optimality boundary conditions. Finally, we give an additional application of our results: we prove an existence result for solutions of linear fractional boundary value problems. This last result is based on a Hilbert structure and the classical Stampacchia theorem.

Article information

Source
Adv. Differential Equations Volume 20, Number 3/4 (2015), 213-232.

Dates
First available in Project Euclid: 4 February 2015

Permanent link to this document
https://projecteuclid.org/euclid.ade/1423055200

Mathematical Reviews number (MathSciNet)
MR3311433

Zentralblatt MATH identifier
1309.26007

Subjects
Primary: 26A33: Fractional derivatives and integrals 49K99: None of the above, but in this section 70H03: Lagrange's equations

Citation

Bourdin, Loïc; Idczak, Dariusz. A fractional fundamental lemma and a fractional integration by parts formula -- Applications to critical points of Bolza functionals and to linear boundary value problems. Adv. Differential Equations 20 (2015), no. 3/4, 213--232. https://projecteuclid.org/euclid.ade/1423055200.


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