A chain rule involving vector functions of bounded variation

Abstract

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 → R . Two situations are investigated where the function γ ∘ f: t → γ ( f( t)) belongs to lbv( I R ) 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.