Series of convex functions: subdifferential, conjugate and applications to entropy minimization

A formula for the sub-differential of the sum of a series of convex functions defined on a Banach space was provided by X. Y. Zheng in 1998. In this paper, besides a slight extension to locally convex spaces of Zhengs results, we provide a formula for the conjugate of a countable sum of convex functions. Then we use these results for calculating the sub-differentials and the conjugates in two situations related to entropy minimization, and we study a concrete example met in Statistical Physics.

A class of non--associated materials: n--monotone materials

Generalized Standard Materials are governed by maximal cyclically monotone operators and modeled by convex potentials. Gery de Saxces Implicit Standard Materials are modeled by biconvex bipotentials. We analyze the intermediate class of n-monotone materials governed by maximal n-monotone operators and modeled by Fitzpatricks functions. Revisiting the model of elastic materials initiated by Robert Hooke, and insisting on the linearity, coaxiality and monotonicity properties of the constitutive law, we illustrate that Fitzpatricks representation of n-monotone operators coming from convex analysis provides a constructive method to discover the best bipotential modeling a n-monotone material. Giving up the symmetry of the linear constitutive laws, we find out that n-monotonicity is a relevant criterion for the materials characterization and classification.