Higher order problems in the calculus of variations: Du Bois-Reymond condition and regularity of minimizers - Université de Bretagne Occidentale Access content directly
Journal Articles Journal of Convex Analysis Year : 2020

Higher order problems in the calculus of variations: Du Bois-Reymond condition and regularity of minimizers

Abstract

Summary: This paper concerns an N-order problem in the calculus of variations of minimizing the functional ∫baΛ(t,x(t),…,x(N)(t))dt, in which the Lagrangian Λ is a Borel measurable, non autonomous, and possibly extended valued function. Imposing some additional assumptions on the Lagrangian, such as an integrable boundedness of the partial proximal subgradients (up to the (N−2)-order variable), a growth condition (more general than superlinearity w.r.t. the last variable) and, when the Lagrangian is extended valued, the lower semicontinuity, we prove that the N-th derivative of a reference minimizer is essentially bounded. We also provide necessary optimality conditions in the Euler-Lagrange form and, for the first time for higher order problems, in the Erdmann-Du Bois-Reymond form. The latter can be also expressed in terms of a (generalized) convex subdifferential, and is valid even without requiring neither a particular growth condition nor convexity in any variable.
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Dates and versions

hal-04315917 , version 1 (30-11-2023)

Identifiers

  • HAL Id : hal-04315917 , version 1

Cite

Piernicola Bettiol, Julien Bernis, Carlo Mariconda. Higher order problems in the calculus of variations: Du Bois-Reymond condition and regularity of minimizers. Journal of Convex Analysis, 2020, 27 (1), pp.179-204. ⟨hal-04315917⟩
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