Back to my homepage
Bipotentials / Hamiltonian inclusions / Majorization and quasiconvexity

  • A symplectic Brezis-Ekeland-Nayroles principle, with Géry de Saxcé,(2014)
    • We propose a modification of the hamiltonian formalism which can be used for dissipative systems. This work continues arXiv:0810.1419 and advances by the introduction of a symplectic version of the Brezis-Ekeland-Nayroles principle [Brezis Ekeland 1976] [Nayroles 1976]. As an application we show how standard plasticity can be treated in our formalism.
  • A variational formulation for constitutive laws described by bipotentials (with Géry de Saxcé and Claude Vallée),(2011)
    • Inspired by the algorithm of Berga and de Saxce for solving the discretisation in time of the evolution problem for an implicit standard material, we propose a general variational formulation in terms of bipotentials.
  • Blurred maximal cyclically monotone sets and bipotentials (with Géry de Saxcé and Claude Vallée), Analysis and Applications 8 (2010), no. 4, 1-14
    • 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 for a subgradient differential inclusion to be expressed as a implicit differential inclusion with the help of a bipotential.
  • Blurred constitutive laws and bipotential convex covers (with Géry de Saxcé and Claude Vallée), Mathematics and Mechanics of Solids 16(2), (2011), 161-171 , DOI
    • In many practical situations, incertitudes affect the mechanical behaviour that is given by a family of graphs instead of a single one. In this paper, we show how the bipotential method is able to capture such blurred constitutive laws, using bipotential convex covers.
  • Bipotentials for non monotone multivalued operators: fundamental results and applications (with Géry de Saxcé and Claude Vallée), Acta Applicandae Mathematicae, 110, 2(2010), 955-972 DOI
    • This is a survey of recent results about bipotentials representing multivalued operators. The notion of bipotential is based on an extension of Fenchel's inequality, with several interesting applications related to non associated constitutive laws in non smooth mechanics, such as Coulomb frictional contact or non-associated Drucker-Prager model in plasticity. Relations betweeen bipotentials and Fitzpatrick functions are described. Selfdual lagrangians, introduced and studied by Ghoussoub, can be seen as bipotentials representing maximal monotone operators. We show that bipotentials can represent some monotone but not maximal operators, as well as non monotone operators. Further we describe results concerning the construction of a bipotential which represents a given non monotone operator, by using convex lagrangian covers or bipotential convex covers.
  • Non maximal cyclically monotone graphs and construction of a bipotential for the Coulomb's dry friction law (with Géry de Saxcé and Claude Vallée), J. of Convex Analysis 17, No 1. (2010), 81-94
    • We show a surprising connexion between a property of the inf convolutions of a family of convex lower semicontinuous functions and the fact that intersections of maximal cyclically monotone graphs are the critical set of a bipotential. We then extend the results from arXiv:math/0608424 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 monotone.
  • Hamiltonian inclusions with convex dissipation with a view towards applications (2008), Mathematics and its Applications 1, 2 (2009), 228-251 (journal)
    • We propose a generalization of hamiltonian mechanics, as a hamiltonian inclusion with convex dissipation function. We obtain a dynamical version of the approach of Mielke to quasistatic rate-independent processes. Then we show that a class of models of dynamical brittle damage can be formulated in this setting.
  • Four applications of majorization to convexity in the calculus of variations(2007), Linear Algebra and its Applications 429, 2008, 1528-1545
    • Re-writed and updated version of math.FA/0105044. The applications are: (1) simple necessary and sufficient conditions for an isotropic objective function to be rank one convex on the set of matrices with positive determinant, (2) a very short proof of a theorem of Thompson and Freede, Ball, or Le Dret, concerning the convexity of a class of isotropic functions which appear in nonlinear elasticity, (3)a lower semicontinuity result for elastic energy functionals depending on Hencky's logarithmic strain, (4) a compact proof of Dacorogna-Marcellini-Tanteri theorem, explaining the resemblance of this theorem with the Horn-Thompson theorem.
  • Construction of bipotentials and a minimax theorem of Fan (2006) (with Géry de Saxcé and Claude Vallée)
    • 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 of the bipotentials. We prove here another reconstruction theorem for a bipotential from a convex lagrangian cover, this time using a convexity notion related to a minimax theorem of Fan.
  • Existence and construction of bipotentials for graphs of multivalued laws(2006)(with Géry de Saxcé and Claude Vallée), Journal of Convex Analysis, vol. 15, no. 1, 2008, 87-104.
    • 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.
  • The variational complex of a diffeomorphisms group
    • In this paper we propose a variational complex associated to a diffeomorphisms group with first order jet in a Lie group. We study the structure of null lagrangians and we prove some fundamental properties of them, as well as their connection to differential invariants of the group action.
  • Quasiconvexity versus group invariance This is the written version of a lecture held on Feb. 22 at the Mathematical Institute, Oxford, Applied Analysis and Mechanics Seminars,Hilary Term 1999.
    • The lower invariance under a given arbitrary group of diffeomorphisms extends the notion of quasiconvexity. The non-commutativity of the group operation (the function composition) modifies the classical equivalence between lower semicontinuity and quasiconvexity. In this context null lagrangians are particular cases of integral invariants of the group.
  • Lower semi-continuity of integrals with $G$-quasiconvex potential, Z. Angew. Math. Phys.,   53,  6 (2002), pp 949-961  
    • This paper introduces the proper notion of variational quasiconvexity associated to a group of bi-Lipschitz homeomorphisms. We prove a lower semicontinuity theorem connected to this notion, which improves a result of Dacorogna and Fusco. In the second part of the paper we apply this result to a class of functions, introduced in the next paper. Such functions are multiplicative quasiconvex, hence they induce lower semicontinuous integrals.
  • Majorisation with applications in Elasticity
    • Theorem 5.1 gives simple necessary and sufficient conditions for an isotropic objective function to be rank one convex on the set of matrices with positive determinant.
    • Theorem 6.2 describes a class of possible non-polyconvex but multiplicative quasiconvex isotropic functions. This class is not contained in a well known theorem of Ball (6.3 in this paper) which gives sufficient conditions for an isotropic and objective function to be polyconvex.
    • I show that there is a new way to prove directly the quasiconvexity (in the multiplicative form). Relevance of Schur convexity for the description of rank one convex hulls is explained.