Self-dual Lagrangians Research Articles

Extension of monotone operators and Lipschitz maps invariant for a group of isometries

Abstract We study monotone operators in reflexive Banach spaces that are invariant with respect to a group of suitable isometric isomorphisms, and we show that they always admit a maximal extension which preserves the same invariance. A similar result applies to Lipschitz maps in Hilbert spaces, thus providing an invariant version of Kirszbraun–Valentine extension theorem. We then provide a relevant application to the case of monotone operators in $L^{p}$ -spaces of random variables which are invariant with respect to measure-preserving isomorphisms, proving that they always admit maximal dissipative extensions which are still invariant by measure-preserving isomorphisms. We also show that such operators are law invariant, a much stronger property which is also inherited by their resolvents, the Moreau–Yosida approximations, and the associated semigroup of contractions. These results combine explicit representation formulae for the maximal extension of a monotone operator based on self-dual Lagrangians and a refined study of measure-preserving maps in standard Borel spaces endowed with a nonatomic measure, with applications to the approximation of arbitrary couplings between measures by sequences of maps.

Dual equivalence between self-dual and topologically massive B∧F models coupled to matter in 3+1 dimensions

In this work, we revisit the duality between a self-dual non-gauge invariant theory and a topological massive theory in $3+1$ dimensions. The self-dual Lagrangian is composed by a vector field and an antisymmetric field tensor whereas the topological massive Lagrangian is build using a $B \wedge F$ term. Though the Lagrangians are quite different, they yield to equations of motion that are connected by a simple dual mapping among the fields. We discuss this duality by analyzing the degrees of freedom in both theories and comparing their propagating modes at the classical level. Moreover, we employ the master action method to obtain a fundamental Lagrangian that interpolates between these two theories and makes evident the role of the topological $B \wedge F$ term in the duality relation. By coupling these theories with matter fields, we show that the duality holds provided a Thirring-like term is included. In addition, we use the master action in order to probe the duality upon the quantized fields. We carried out a functional integration of the fields and compared the resulting effective Lagrangians.

Metric selfduality and monotone vector fields on manifolds

We develop a “metrically selfdual” variational calculus for c-monotone vector fields between general manifolds X and Y, where c is a coupling on X×Y. Remarkably, many of the key properties of classical monotone operators known to hold in a linear context extend to this non-linear setting. This includes an integral representation of c-monotone vector fields in terms of c-convex selfdual Lagrangians, their characterization as a partial c-gradients of antisymmetric Hamiltonians, as well as the property that these vector fields are generically single-valued. We also use a symmetric Monge–Kantorovich transport to associate to any measurable map its closest possible c-monotone “rearrangement”. We also explore how this metrically selfdual representation can lead to a global variational approach to the problem of inverting c-monotone maps, an approach that has proved efficient for resolving non-linear equations and evolutions driven by monotone vector fields in a Hilbertian setting.

Non-convex self-dual Lagrangians and new variational principles of symmetric boundary value problems: Evolution case

In this work, we shall present some new variational principles for evolutionary equations by the virtue of the Non-convex self-dual (Nc-SD) Lagrangians. It is established that how lifting Nc-SD Lagrangians to path spaces allows one to associate to an evolution boundary value problem several potential functions, which can often be used with relative ease compared to other methods such as the use of Euler-Lagrange functionals. These Lagrangians, indeed provide new representations and formulations for the superposition of semi-convex functions and symmetric operators. They yield new variational resolutions for large class of hamiltonian partial differential equations with a variety of linear and nonlinear boundary conditions including many of the standard ones. They can be adapted to easily deal with both nonlinear and homogeneous boundary value problems and, in most cases, solutions generated using this new method have greater regularity than the solutions obtained using the standard Euler-Lagrange function.

New approach to duality-invariant nonlinear electrodynamics

We survey a new approach to the duality-invariant systems of nonlinear electrodynamics, based on introducing auxiliary bi-spinor fields. In this approach, the entire information about the given self-dual system is encoded in the U(1) invariant interaction of the auxiliary fields, while the standard self-dual Lagrangians appear on shell as a result of eliminating auxiliary fields by their equations of motion. Starting from the simplest U(1) duality, we show how this approach can be generalized to the U(N) duality (with N independent Maxwell field strengths), as well as to self-dual systems of = 1 supersymmetric electrodynamics. Also, it works perfectly for self-dual systems with higher derivatives in the action.

Bispinor auxiliary fields in duality-invariant electrodynamics revisited: TheU(N)case

We update and detail the formulation of the duality-invariant systems of $N$ interacting Abelian gauge fields with $N$ auxiliary bispinor fields added. In this setting, the self-duality amounts to $U(N)$ invariance of the nonlinear interaction of the auxiliary fields. The $U(N)$ self-dual Lagrangians arise after solving the nonlinear equations of motion for the auxiliary fields. We also elaborate on a new extended version of the bispinor field formulation involving some additional scalar auxiliary fields and study $U(N)$ invariant interactions with derivatives of the auxiliary bispinor fields. Such interactions generate higher-derivative $U(N)$ self-dual theories.

A MODIFIED LEAST ACTION PRINCIPLE ALLOWING MASS CONCENTRATIONS FOR THE EARLY UNIVERSE RECONSTRUCTION PROBLEM

We address the early universe reconstruction (EUR) problem (as considered by Frisch and coauthors in [26]), and the related Zeldovich approximate model [46]. By substituting the fully nonlinear Monge–Ampere equation for the linear Poisson equation to model gravitation, we introduce a modified mathematical model (Monge-Ampere gravitation/MAG), for which the Zeldovich approximation becomes exact. The MAG model enjoys a least action principle in which we can input mass concentration effects in a canonical way, based on the theory of gradient flows with convex potentials and somewhat related to the concept of self-dual Lagrangians developed by Ghoussoub [29]. A fully discrete algorithm is introduced for the EUR problem in one space dimension.

Homogenization of Maximal Monotone Vector Fields via Selfdual Variational Calculus

Abstract We use the theory of selfdual Lagrangians to give a variational approach to the homogenization of equations in divergence form, that are driven by a periodic family of maximal monotone vector fields. The approach has the advantage of using Γ-convergence methods for corresponding functionals just as in the classical case of convex potentials, as opposed to the graph convergence methods used in the absence of potentials. A new variational formulation for the homogenized equation is also given.

Non-convex self-dual Lagrangians and variational principles for certain PDEʼs

We study the concept and the calculus of non-convex self-dual ( Nc-SD) Lagrangians and their derived vector fields which are associated to many partial differential equations and evolution systems. They yield new variational resolutions for large class of partial differential equations with variety of linear and non-linear boundary conditions including many of the standard ones. This approach seems to offer several useful advantages: It associates to a boundary value problem several potential functions which can often be used with relative ease compared to other methods such as the use of Euler–Lagrange functions. These potential functions are quite flexible, and can be adapted to easily deal with both non-linear and homogeneous boundary value problems.

Non-convex self-dual Lagrangians: New variational principles of symmetric boundary value problems

We study the concept and the calculus of Non-convex self-dual ( Nc-SD) Lagrangians and their derived vector fields which are associated to many partial differential equations and evolution systems. They indeed provide new representations and formulations for the superposition of convex functions and symmetric operators. They yield new variational resolutions for large class of Hamiltonian partial differential equations with variety of linear and nonlinear boundary conditions including many of the standard ones. This approach seems to offer several useful advantages: It associates to a boundary value problem several potential functions which can often be used with relative ease compared to other methods such as the use of Euler–Lagrange functions. These potential functions are quite flexible, and can be adapted to easily deal with both nonlinear and homogeneous boundary value problems. Additionally, in most cases the solutions generated using this new method have greater regularity than the solutions obtained using the standard Euler–Lagrange function. Perhaps most remarkable, however, are the permanence properties of Nc-SD Lagrangians; their calculus is relatively manageable, and their applications are quite broad.

The kernel average for two convex functions and its application to the extension and representation of monotone operators

We provide and analyze an average for two convex functions, based on a kernel function. It covers several known averages such as the arithmetic average, epigraphical average, and the proximal average. When applied to the Fitzpatrick function and the conjugate of Fitzpatrick function associated with a monotone operator, our average produces an autoconjugate (also known as selfdual Lagrangian) which can be used for finding an explicit maximal monotone extension of the given monotone operator. This completely settles one of the open problems posed by Fitzpatrick in the setting of reflexive Banach spaces.

Bipotentials for Non-monotone Multivalued Operators: Fundamental Results and Applications

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.

Superposition of selfdual functionals in non-homogeneous boundary value problems and differential systems

Selfdual variational theory -- developed in [8] and [9] -- allows for the superposition of appropriate 'boundary' Lagrangians with 'interior' Lagrangians, leading to a variational formulation and resolution of problems with various linear and nonlinear boundary constraints that are not amenable to standard Euler-Lagrange theory. The superposition of several selfdual Lagrangians is also possible in many natural settings, leading to a variational resolution of certain differential systems. These results are applied to nonlinear transport equations with prescribed exit values, Lagrangian intersections of convex-concave Hamiltonian systems, initial-value problems of dissipative systems, as well as evolution equations with periodic and anti-periodic solutions.

Self-dual teleparallel formulation of general relativity and the positive energy theorem

A self-dual and anti-self-dual decomposition of the teleparallel formulation of Einstein's general relativity is carried out and the self-dual Lagrangian of the teleparallel formulation of Einstein's general relativity, which is equivalent to the Ashtekar Lagrangian in vacuum, is obtained. Its Hamiltonian formulation and the constraint analysis are developed. Starting from Witten's equation the gauge condition of Nester and co-workers is derived directly and a new expression of the boundary term is obtained. Using this expression and Witten's identity the proof of the positive energy theorem by Nester and co-workers is extended to a case including momentum.