Blot, Joël Calculus of variations in mean and convex Lagrangians. II. (English) Zbl 0679.49022 Bull. Aust. Math. Soc. 40, No. 3, 457-463 (1989). Summary: [For part I, see the author, J. Math. Anal. Appl. 134, No.2, 312-321 (1988; Zbl 0655.49011).] We prove the Legendre necessary conditions of the calculus of variations in mean in an arbtrary finite dimension. When the Lagrangian is convex, we establish that if the Euler-Lagrange equation possesses an almost periodic solution then it possesses periodic and constant solutions. We deduce from this fact various consequences on the structure of the set of almost periodic solutions. Cited in 1 ReviewCited in 9 Documents MSC: 49K05 Optimality conditions for free problems in one independent variable 49J10 Existence theories for free problems in two or more independent variables 42A75 Classical almost periodic functions, mean periodic functions 34C25 Periodic solutions to ordinary differential equations 34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations Keywords:Legendre necessary conditions; Euler-Lagrange equation; periodic solution; almost periodic solutions Citations:Zbl 0655.49011 PDFBibTeX XMLCite \textit{J. Blot}, Bull. Aust. Math. Soc. 40, No. 3, 457--463 (1989; Zbl 0679.49022) Full Text: DOI References: [1] Dunford, Linear operators, Part I: General theory (1958) · Zbl 0088.32102 [2] Besicovitch, Almost periodic functions (1932) [3] DOI: 10.1016/0022-247X(88)90025-X · Zbl 0655.49011 [4] Blot, C.R. Acad. Sci. Paris Sér I Math t.306 pp 809– (1988) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.