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.

Cette thèse est composée de deux parties indépendantes. La première est consacrée à l'étude de quelques problèmes elliptiques de type de Kirchhoff de la forme suivante : -M(ʃΩNul² dx) Δu = f(x, u) xЄΩ ; u(x) = o xЄƋΩ où Ω cRN, N ≥ 2, f une fonction de Carathéodory et M une fonction strictement positive et continue sur R+. Dans le cas où la fonction f est asymptotiquement linéaire à l’infini par rapport à l'inconnue u, on montre, en combinant une technique de troncature et la méthode variationnelle, que le problème admet au moins une solution positive quand la fonction M est non décroissante. Et si f(x, u) = |u|p-1 u + λg(x), où p >0, λ un paramètre réel et g une fonction de classe C1 et changeant de signe sur Ω, alors sous certaines hypothèses sur M, il existe deux réels positifs λ. et λ. tels que le problème admet des solutions positives si 0 < λ <λ. et n'admet pas de solutions positives si λ > λ.. Dans la deuxième partie, on étudie deux problèmes soulevés en dynamique des fluides. Le premier est une généralisation d'un modèle décrivant la propagation unidirectionnelle dispersive des ondes longues dans un milieu à deux fluides. En écrivant le problème sous la forme d'une équation de point fixe, on montre l'existence d'au moins une solution positive. On montre ensuite sa symétrie et son unicité. Le deuxième problème consiste à prouver l'existence de la vitesse, la pression et la température d'un fluide non newtonien, incompressible et non isotherme, occupant un domaine borné, en prenant en compte un terme de convection. L’originalité dans ce travail est que la viscosité du fluide ne dépend pas seulement de la vitesse mais aussi de la température et du module du tenseur des taux de déformations. En se basant sur la notion des opérateurs pseudo-monotones, le théorème de De Rham et celui de point fixe de Schauder, l'existence du triplet, (vitesse, pression, température) est démontré / This thesis consists of two independent parts. The first is devoted to the study of some elliptic problems of Kirchhoff-type in the following form : -M(ʃΩNul² dx) Δu = f(x, u) xЄΩ ; u(x) = o xЄƋΩ where Ω cRN, N ≥ 2, f is a Caratheodory function and M is a strictly positive and continuous function on R+. In the case where the function f is asymptotically linear at infinity with respect to the unknown u, we show, by combining a truncation technique and the variational method, that the problem admits a positive solution when the function M is nondecreasing. And if f(x, u) = |u|p-1 u + λg(x) where p> 0, λ a real parameter and g is a function of class C1 and changes the sign in Ω, then under some assumptions on M, there exist two positive real λ. and λ. such that the problem admits positive solutions if 0 < λ <λ., and no positive solutions if λ > λ.. In the second part, we study two problems arising in fluid dynamics. The first is a generalization of a model describing the unidirectional propagation of long waves in dispersive medium with two fluids. By writing the problem as a fixed point equation, we prove the existence of at least one positive solution. We then show its symmetry and uniqueness. The second problem is to prove the existence of the velocity, pressure and temperature of a non-Newtonian, incompressible and isothermal fluid, occupying a bounded domain, taking into account a convection term. The originality in this work is that the fluid viscosity depends not only on the velocity but also on the temperature and the modulus of deformation rate tensor. Based on the notion of pseudo-monotone operators, the De Rham theorem and the Schauder fixed point theorem, the existence of the triplet, (velocity, pressure, temperature) is shown