Analyse vectorielle

L'analyse vectorielle est une branche des mathématiques qui étudie les champs de scalaires et de vecteurs suffisamment réguliers des espaces euclidiens, c'est-à-dire les applications différentiables d'un ouvert d'un espace euclidien à valeurs respectivement dans \R et dans . Du point de vue du mathématicien, l'analyse vectorielle est donc une branche de la géométrie différentielle. Cette dernière inclut l'analyse tensorielle qui apporte des outils plus puissants et une analyse plus concise entre autres des champs de vecteurs. Mais l'importance de l'analyse vectorielle provient de son utilisation intensive en physique et dans les sciences de l'ingénieur. C'est de ce point de vue qu'elle sera présentée dans l'article, et c'est pourquoi elle sera le plus souvent limitée au cas où E = \R^3 est l'espace usuel à trois dimensions. Dans ce cadre, un champ de vecteurs associe à chaque point de l'espace un vecteur (à trois composantes réelles), tandis qu'un champ de s

Calculus of Variations for Differential Forms

Swarnendu Sil

In this thesis we study calculus of variations for differential forms. In the first part we develop the framework of direct methods of calculus of variations in the context of minimization problems for functionals of one or several differential forms of the type,

\int_{\Omega} f(d\omega), \quad \int_{\Omega} f(d\omega_{1}, \ldots, d\omega_{m}) \quad \text{ and } \int_{\Omega} f(d\omega, \delta\omega).

We introduce the appropriate convexity notions in each case, called \emph{ext. polyconvexity}, \emph{ext. quasiconvexity} and \emph{ext. one convexity} for functionals of the type

\int_{\Omega} f(d\omega),

\emph{vectorial ext. polyconvexity}, \emph{vectorial ext. quasiconvexity} and \emph{vectorial ext. one convexity} for functionals of the type

\int_{\Omega} f(d\omega_{1}, \ldots, d\omega_{m})

and \emph{ext-int. polyconvexity}, \emph{ext-int. quasiconvexity} and \emph{ext-int. one convexity} for functionals of the type

\int_{\Omega} f(d\omega, \delta\omega).

We study their interrelationships and the connections of these convexity notions with the classical notion of polyconvexity, quasiconvexity and rank one convexity in classical vectorial calculus of variations. We also study weak lower semicontinuity and weak continuity of these functionals in appropriate spaces, address coercivity issues and obtain existence theorems for minimization problems for functionals of one differential forms.\smallskip In the second part we study different boundary value problems for linear, semilinear and quasilinear Maxwell type operator for differential forms. We study existence and derive interior regularity and

L^{2}

boundary regularity estimates for the linear Maxwell operator

\delta (A(x)d\omega) = f

with different boundary conditions and the related Hodge Laplacian type system

\delta (A(x)d\omega) + d\delta\omega = f,

with appropriate boundary data. We also deduce, as a corollary, some existence and regularity results for div-curl type first order systems. We also deduce existence results for semilinear boundary value problems \begin{align*} \left\lbrace \begin{gathered} \delta ( A (x) ( d\omega ) ) + f( \omega ) = \lambda\omega \text{ in } \Omega, \ \nu \wedge \omega = 0 \text{ on } \partial\Omega. \end{gathered} \right. \end{align*} Lastly, we briefly discuss existence results for quasilinear Maxwell operator \begin{align*} \delta ( A ( x, d \omega ) ) = f , \end{align*} with different boundary data.

EPFL2016

On a class of mean field solutions of the Monge problem for perfect and self-interacting systems

Philippe Choquard

The Monge problem [23], [27], as reformulated by Kantorovich [19], [20] is that of the transportation, at a minimum "cost", of a given mass distribu- tion from an initial to a final position during a given time interval. It is an optimal transport problem [28, sects. 1, 2]. Following the fluid mechanical solution provided by Benamou and Brenier for quadratic cost functions [4] ,[28, sects. 5.4, 8.1] and, by analogy with the fixed end problem in Analytical Mechanics, Lagrangian formulations are needed to solve this boundary value problem in time. They are also needed to determine the Actions as time in- tegral of Lagrangians, that are measures of the "cost"of the transportations [4, proposition 1.1]. Four versions of explicit constructions of Lagrangians are proposed in section 3. They are associated to the Hamiltonians of perfect and self-interacting systems presented in section 2. These Hamiltonians are ex- pressed in function of pairs of the well known canonically conjugated Clebsch variables, namely mass densities and velocity potentials [14], [15]. The first version consists in the elimination of the velocity potentials as a function of the densities and their time derivatives by inversion of the continuity equations de- rived from given Hamiltonians. The second version consists in the elimination of the gradient of the velocity potentials from the continuity equations thanks to the introduction of vector valued applications such that their divergences give the mass densities. It turns out that, up to a sign factor, these vector elds are canonically conjugated to Euler velocity elds. The third version is a generalization in nD of Gelfand mass coordinate, a constant of the motion in 1 D [17], by the introduction of n-dimensional vector valued applications that enable to determine the mass densities as the determinant of their Jaco- bian matrices. Comparison of this set of mass coordinates with other sets of constants of the motion familiar in Fluid Dynamics is made in sub-section 3.3. Note that version two and three are identical for one-dimensional problems. The fourth version is based on the introduction of the Lagrangian coordinates that describe the characteristics of the different models and are parametrized by the former auxiliary vector fields. As illustrations, weak solutions of several models of Coulombian and Newtonian systems known in Plasma Physics and in Cosmology, respectively, with spherically symmetric boundary densities are given in section 4. However, and up to one exception given in the sub-section 3.3, calculations of the actions associated to these illustrations are not reported in this paper, nor the important analysis of the convexity-concavity properties of our Lagrangians. Lastly, and for the same models as those evoked above, a survey of past work concerning weak solutions of the Cauchy problem obeying the Hopf-Lax variational principle extended to negative time and having cor- related initial conditions is given in the Introduction as well as the derivation of the continuum fluid limit from many particle Hamiltonians.

EPFL2010

Calculus of Variations for Differential Forms

Swarnendu Sil

\int_{\Omega} f(d\omega), \quad \int_{\Omega} f(d\omega_{1}, \ldots, d\omega_{m}) \quad \text{ and } \int_{\Omega} f(d\omega, \delta\omega).

We introduce the appropriate convexity notions in each case, called \emph{ext. polyconvexity}, \emph{ext. quasiconvexity} and \emph{ext. one convexity} for functionals of the type

\int_{\Omega} f(d\omega),

\emph{vectorial ext. polyconvexity}, \emph{vectorial ext. quasiconvexity} and \emph{vectorial ext. one convexity} for functionals of the type

\int_{\Omega} f(d\omega_{1}, \ldots, d\omega_{m})

and \emph{ext-int. polyconvexity}, \emph{ext-int. quasiconvexity} and \emph{ext-int. one convexity} for functionals of the type

\int_{\Omega} f(d\omega, \delta\omega).

L^{2}

boundary regularity estimates for the linear Maxwell operator

\delta (A(x)d\omega) = f

with different boundary conditions and the related Hodge Laplacian type system

\delta (A(x)d\omega) + d\delta\omega = f,

EPFL2016

On a class of mean field solutions of the Monge problem for perfect and self-interacting systems

Philippe Choquard

EPFL2010

Calculus of Variations for Differential Forms

Calculus of variations: A differential form approach

On a class of mean field solutions of the Monge problem for perfect and self-interacting systems

Calculus of Variations for Differential Forms

Calculus of variations: A differential form approach

On a class of mean field solutions of the Monge problem for perfect and self-interacting systems

Analyse vectorielle

Calculus of Variations for Differential Forms

Calculus of variations: A differential form approach

On a class of mean fi eld solutions of the Monge problem for perfect and self-interacting systems

Calculus of Variations for Differential Forms

Calculus of variations: A differential form approach

On a class of mean fi eld solutions of the Monge problem for perfect and self-interacting systems

On a class of mean field solutions of the Monge problem for perfect and self-interacting systems

On a class of mean field solutions of the Monge problem for perfect and self-interacting systems