Th.--Sci. math.--Grenoble 1, 1984. N°: 69.

Neste trabalho, estudamos a cohomologia de de Rham e métodos para os seus cálculos. Finalizamos com aplicações da cohomologia de de Rham em teoremas da topologia / In this work we study the de Rham cohomology and methods for its calculations. We close it with applications of the Rham cohomology in theorems from topology

Neste trabalho, estudamos a cohomologia de de Rham e métodos para os seus cálculos. Finalizamos com aplicações da cohomologia de de Rham em teoremas da topologia / In this work we study the de Rham cohomology and methods for its calculations. We close it with applications of the Rham cohomology in theorems from topology

Começamos definindo a cohomologia clássica de De Rham e provamos alguns resultados que nos permitem calcular tal cohomologia de algumas variedades diferenciáveis. Com o intuito de provar o Teorema de De Rham, escolhemos fazer a demonstração utilizando a noção de feixes, que se mostra como uma generalização da ideia de cohomologia. Como a cohomologia de De Rham não é a única que se pode definir numa variedade, a questão da unicidade dá origem a teoria axiomática de feixes, que nos dará uma cohomologia para cada feixe dado. Mostraremos que a partir da teoria axiomática de feixes obtemos cohomologias, além das cohomologias clássicas de De Rham, a cohomologia clássica singular e a cohomologia clássica de Cech e mostraremos que essas cohomologias obtidas a partir da noção axiomática são isomorfas as definições clássicas. Concluiremos que se nos restringirmos a apenas variedades diferenciáveis, essas cohomologias são unicamente isomorfas e este será o teorema de De Rham. / We begin by defining De Rhams classical cohomology and we prove some results that allow us a calculation of the cohomology of some differentiable manifolds. In order to prove De Rhams Theorem, we chose to make a demonstration using a notion of sheaves, which is a generalization of the idea of cohomology. Since De Rhams cohomology is not a only one that can be made into a variety, the question of unicity gives rise to axiomatic theory of sheaves, which give us a cohomology for each sheaf given. We will show that from the axiomatic theory of sheaves we obtain cohomologies, besides the classical cohomologies of De Rham, a singular classical cohomology and a classical cohomology of Cech and we will show that cohomologies are obtained from the axiomatic notion are classic definitions. We will conclude that if we restrict ourselves to only differentiable manifolds, these cohomologies are uniquely isomorphic and this will be De Rhams theorem.

Começamos definindo a cohomologia clássica de De Rham e provamos alguns resultados que nos permitem calcular tal cohomologia de algumas variedades diferenciáveis. Com o intuito de provar o Teorema de De Rham, escolhemos fazer a demonstração utilizando a noção de feixes, que se mostra como uma generalização da ideia de cohomologia. Como a cohomologia de De Rham não é a única que se pode definir numa variedade, a questão da unicidade dá origem a teoria axiomática de feixes, que nos dará uma cohomologia para cada feixe dado. Mostraremos que a partir da teoria axiomática de feixes obtemos cohomologias, além das cohomologias clássicas de De Rham, a cohomologia clássica singular e a cohomologia clássica de Cech e mostraremos que essas cohomologias obtidas a partir da noção axiomática são isomorfas as definições clássicas. Concluiremos que se nos restringirmos a apenas variedades diferenciáveis, essas cohomologias são unicamente isomorfas e este será o teorema de De Rham. / We begin by defining De Rhams classical cohomology and we prove some results that allow us a calculation of the cohomology of some differentiable manifolds. In order to prove De Rhams Theorem, we chose to make a demonstration using a notion of sheaves, which is a generalization of the idea of cohomology. Since De Rhams cohomology is not a only one that can be made into a variety, the question of unicity gives rise to axiomatic theory of sheaves, which give us a cohomology for each sheaf given. We will show that from the axiomatic theory of sheaves we obtain cohomologies, besides the classical cohomologies of De Rham, a singular classical cohomology and a classical cohomology of Cech and we will show that cohomologies are obtained from the axiomatic notion are classic definitions. We will conclude that if we restrict ourselves to only differentiable manifolds, these cohomologies are uniquely isomorphic and this will be De Rhams theorem.

We study the Neumann problem for the de Rham complex in a bounded domain of Rn with singularities on the boundary. The singularities may be general enough, varying from Lipschitz domains to domains with cuspidal edges on the boundary. Following Lopatinskii we reduce the Neumann problem to a singular integral equation of the boundary. The Fredholm solvability of this equation is then equivalent to the Fredholm property of the Neumann problem in suitable function spaces. The boundary integral equation is explicitly written and may be treated in diverse methods. This way we obtain, in particular, asymptotic expansions of harmonic forms near singularities of the boundary.

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Dans cette thèse, nous généralisons les inégalités universelles de Yang etde Levitin et Parnovski, concernant les valeurs propres du laplacien de Dirichlet sur undomaine euclidien borné, au cas du laplacien de Hodge-de Rham sur une sous-variétéeuclidienne fermée. Cela permet une extension de l’inégalité de Reilly et de l’inégalitéd’Asada, concernant respectivement la première valeur propre du laplacien et celle dulaplacien de Hodge-de Rham, à toutes les valeurs propres de ces deux opérateurs. Ensuite,nous obtenons une nouvelle inégalité algébrique qui relie les valeurs propres d’un opérateurauto-adjoint sur un espace d’Hilbert à deux familles d’opérateurs symétriques et antisymétriqueset à leurs commutateurs. Cette inégalité permet d’unifier et d’améliorer denombreux résultats connus concernant le laplacien, le laplacien de Hodge-de Rham, lecarré de l’opérateur de Dirac et plus généralement le laplacien agissant sur les sections d’unfibré vectoriel riemannien au-dessus d’une sous-variété euclidienne, le laplacien de Kohn,les puissances du laplacien... Dans une dernière partie, nous montrons une majoration dela première valeur propre du problème de Steklov sur un domaine Ω d’une sous-variétéeuclidienne ou sphérique, en fonction des r-courbures moyennes de son bord ∂Ω. / In this thesis, we generalize the Yang and the Levitin and Parnovski universalinequalities, concerning the eigenvalues of the Dirichlet Laplacian on a Euclideanbounded domain, to the case of the Hodge-de Rham Laplacian on a Euclidean closed submanifold.This gives an extension of Reilly’s inequality and Asada’s inequality, concerningthe first eigenvalues of the Laplacian and the Hodge-de Rham Laplacian respectively, toall eigenvalues of these operators. We also obtain a new abstract inequality relating theeigenvalues of a self-adjoint operator on a Hilbert space to two families of symmetric andskew-symmetric operators and their commutators. This inequality is proved useful both forunifying and for improving numerous known results concerning the Laplacian, the Hodgede Rham Laplacian, the square of the Dirac operator and more generally the Laplacianacting on sections of a Riemannian vector bundle on a Euclidean submanifold, the KohnLaplacian, a power of the Laplacian...In the last part, we obtain an upper bound for thefirst eigenvalue of Steklov problem on a domain Ω of a Euclidean or a spherical submanifoldin terms of the r-th mean curvatures of ∂Ω

The aim of this dissertation lies in establishing an o-minimal de Rham cohomology theory for smooth abstract-definable manifolds in an o-minimal expansion of the real field which admits smooth cell decomposition and defines the exponential function, by following the classical de Rham cohomology. We can specify the o-minimal cohomology groups and attain some properties as the existence of Mayer-Vietoris sequence and the invariance under smooth abstract-definable diffeomorphisms. However, in order to obtain the invariance of our o-minimal cohomology under abstract-definable homotopy we must, working in a tame context that defines sufficiently many primitives, assume the validity of a statement related to Bröcker\'s problem. / O objetivo desta tese reside em estabelecer uma cohomologia de De Rham o-minimal para variedades definíveis abstratas lisas em uma expansão o-minimal do corpo ordenado dos reais, a qual admite decomposição celular lisa e define a função exponencial, seguindo a cohomologia de De Rham clássica. Além de especificarmos os grupos da cohomologia de Rham o-minimal, obtemos algumas propriedades, como a existência da sequência de Mayer-Vietoris e a invariância sob difeomorfismos definíveis abstratos lisos. Todavia, a fim de lograrmos a invariância de nossa cohomologia o-minimal sob homotopia definível abstrata devemos, além de trabalhar num contexto moderado no qual muitas primitivas são definidas, assumir a validade de uma asserção relacionada ao problema de Bröcker.