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Calculus of variations with differential forms

Saugata Bandyopadhyay, Bernard Dacorogna, Swarnendu Sil (2015)

Journal of the European Mathematical Society

We study integrals of the form Ω f d ω , where 1 k n , f : Λ k is continuous and ω is a k - 1 -form. We introduce the appropriate notions of convexity, namely ext. one convexity, ext. quasiconvexity and ext. polyconvexity. We study their relations, give several examples and counterexamples. We finally conclude with an application to a minimization problem.

Complex calculus of variations

Michel Gondran, Rita Hoblos Saade (2003)

Kybernetika

In this article, we present a detailed study of the complex calculus of variations introduced in [M. Gondran: Calcul des variations complexe et solutions explicites d’équations d’Hamilton–Jacobi complexes. C.R. Acad. Sci., Paris 2001, t. 332, série I]. This calculus is analogous to the conventional calculus of variations, but is applied here to 𝐂 n functions in 𝐂 . It is based on new concepts involving the minimum and convexity of a complex function. Such an approach allows us to propose explicit solutions...

Continuity of solutions to a basic problem in the calculus of variations

Francis Clarke (2005)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We study the problem of minimizing Ω F ( D u ( x ) ) d x over the functions u W 1 , 1 ( Ω ) that assume given boundary values φ on Γ : = Ω . The lagrangian F and the domain Ω are assumed convex. A new type of hypothesis on the boundary function φ is introduced: thelower (or upper) bounded slope condition. This condition, which is less restrictive than the familiar bounded slope condition of Hartman, Nirenberg and Stampacchia, allows us to extend the classical Hilbert-Haar regularity theory to the case of semiconvex (or semiconcave) boundary...

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