Bipotentials for non monotone multivalued operators: fundamental results and applications

Existence and construction of bipotentials for graphs of multivalued laws

This is a first paper in convex analysis dedicated to the bipotential theory, based on an extension of Fenchel's inequality. Introduced by the second author, bipotentials lead to a succesful new treatment of the constitutive laws of some dissipative materials: frictional contact, non-associated Drucker-Prager model, or Lemaitre plastic ductile damage law. We solve here the problems of existence and construction of a bipotential for a nonsmooth mechanics constitutive law.

Non maximal cyclically monotone graphs and construction of a bipotential for the Coulomb's dry friction law

We show a surprising connexion between a property of the inf convolution of a family of convex lower semicontinuous functions and the fact that the intersection of maximal cyclically monotone graphs is the critical set of a bipotential. We then extend the results from arXiv:math/0608424v4 to bipotentials convex covers, generalizing the notion of a bi-implicitly convex lagrangian cover. As an application we prove that the bipotential related to Coulomb's friction law is related to a specific bipotential convex cover with the property that any graph of the cover is non maximal cyclically monoton…

Construction of bipotentials and a minimax theorem of Fan

In Mechanics, the theory of standard materials is a well-known application of Convex Analysis. However, the so-called non-associated constitutive laws cannot be cast in the mould of the standard materials. From the mathematical viewpoint, a non associated constitutive law is a multivalued operator which is not supposed to be monotone. A possible way to study non-associated constitutive laws by using Convex Analysis, proposed first in [12], consists in constructing a "bipotential" function of two variables, which physically represents the dissipation. This is a second paper on the mathematics o…

Blurred maximal cyclically monotone sets and bipotentials

Let X be a reflexive Banach space and Y its dual. In this paper we find necessary and sufficient conditions for the existence of a bipotential for a blurred maximal cyclically monotone graph. Equivalently, we find a necessary and sufficient condition on $ϕ\in Γ_{0}(X)$ for that the differential inclusion $y \in \bar{B}(ε) + \partial ϕ(x)$ can be put in the form $y \in \partial b(\cdot, y)(x)$, with $b$ a bipotential.

On Variational Inequalities with Multivalued Operators with Semi-Bounded Variation

In this paper we explore solvability of steady-state variational inequalities with multivalued operators. Moreover, we are studying the connections between the class of radially semi-continuous operators with semi-bounded variation and classes of pseudo-monotone and monotone mappings, and some properties of this operators.

非二次代价最优传输中单调映射的精细性质

Fine properties of monotone maps arising in optimal transport for non-quadratic costs

本文研究了非二次代价最优传输中多值单调映射的精细性质，证明了其在几乎处处为单值，并推导了相关结论。