Support functions of Clarke's generalized jacobian and of its plenary hull

Cyril Imbert (1)

1 MIP - Mathématiques pour l'Industrie et la Physique

This paper studies two important mathematical objects which are useful in tackling the first-order behaviour of vector-valued locally Lipschitz functions in a finite dimensional setting: the Clarke generalized jacobian and its plenary hull. We aim at giving analytical expressions of the support functions of these compact convex sets of matrices. Our study was motivated by earlier works by J.-B. Hiriart-Urruty and recent papers by Zs. Pales and V. Zeidan. The expressions of the support functions are applied, for instance, to provide a new proof of a chain rule on generalized jacobians of composed locally Lipschitz functions (without further assumptions). Applications of our results to the second-order behaviour of C^1 functions with locally Lipschitz gradients are considered.

Mots clés: vector-valued functions; generalized jacobian; support function; plenary hull

Domaines: Equations aux dérivées partielles [math.AP]

Cyril Imbert. Support functions of Clarke's generalized jacobian and of its plenary hull. Nonlinear Analysis: Theory, Methods and Applications, 2002, 29 (8), pp.1111-1125. ⟨hal-00176517⟩