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Article Dans Une Revue Journal of Functional Analysis Année : 1987

A chain rule involving vector functions of bounded variation

Résumé

By f ϵ lbv(I, X), we mean that f is a function of a real interval I to a Banach space X, with bounded variation on every compact subinterval of I; to such f, an X-valued measure df, called its differential measure, classically corresponds. Let Ω be an open convex subset of X and γ: Ω → . Two situations are investigated where the function γ ∘ f: t → γ (f(t)) belongs to lbv(I) and some properties of the real measure d(γ ∘ f) are established. In the first case, γ is supposed convex and continuous in Ω. The subdifferential δγ is invoked in the sense of Convex Analysis; under the ordering of real measures, d(γ ∘ f) is shown to satisfy some inequalities. This generalizes previous results of one of the authors, aimed at deriving energy-like inequalities in nonsmooth mechanical evolution problems. In the second case, γ is supposed Lipschitz on every bounded subset of Ω and Clarke's generalized gradient of γ is used. In both situations, if γ happens to be Gâteaux-differentiable, and f ϵ lbv(I, X) continuous, a chain rule of the familiar form is found to hold. Finally, for γ Fréchet-differentiable, an expression of d(γ ∘ f) is obtained.
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Dates et versions

hal-01788917 , version 1 (09-05-2018)

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Jean Jacques Moreau, Michel Valadier. A chain rule involving vector functions of bounded variation. Journal of Functional Analysis, 1987, 74 (2), pp.333 - 345. ⟨10.1016/0022-1236(87)90029-2⟩. ⟨hal-01788917⟩

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