Composition functionals in higher order calculus of variations and Noether's theorem

Abstract

In the present paper, we discuss the existence and uniqueness of solution for higher-order calculus of variations problems, involving composition of functionals. Also, higher-order DuBois-Reymond conditions in the Sobolev space ${\mathbb{W}}^{m,p}\left(\right[t_1,t_2];\mathbb{R})$ are proven, both in integral and differential form, and under additional constraints. We consider the higher-order Noether's theorem and discuss invariance conditions for the main problem.

Keywords:

• Higher-order Noether's theorem
• existence and uniqueness
• DuBois-Reymond conditions
• Euler–Lagrange equations

2010 Mathematics Subject Classifications:

• 49S05
• 70H03
• 49K15

Acknowledgments

J. Vanterler acknowledges the financial support of a PNPD-CAPES (88882.305834/2018-01) scholarship of the Postgraduate Program in Applied Mathematics of IMECC-Unicamp and Gasto S. F. Frederico acknowledges the financial support by Fundao Cearence de Apoio ao Desenvolvimento Cientfico e Tecnolgico (FUNCAP)-BOLSA BPI (n BP4-00172-00054.02.00/20).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

J. Vanterler acknowledges the financial support of a PNPD-CAPES (88882.305834/2018-01) scholarship of the Postgraduate Program in Applied Mathematics of IMECC-Unicamp and Gasto S. F. Frederico acknowledges the financial support by Fundao Cearence de Apoio ao Desenvolvimento Cientfico e Tecnolgico (FUNCAP)-BOLSA BPI (n BP4-00172-00054.02.00/20).