Cette thèse est dédiée a l''etude de certaines propriétés des équations d' évolutions non-autonomes $u'(t)+A(t)u(t)=f(t), u(0)=x.$ Il s'agit précisément de la propriété de la régularité maximale $L^p$: étant donnée $fin L^{p}(0,tau;H)$, montrer l'existence et unicité de la solution $u in W^{1,p}(0,tau;H)$. Ce problème a 'et'e intensivement étudie dans le cas autonome, i.e., $A(t)=A$ pour tout $t$. Dans le cas non-autonome, le problème a été considéré par J.L.Lions en 1960. Nous montrons divers résultats qui étendent tout ce qui est connu sur ce problème. On suppose ici que la famille des opérateurs $(mathcal{A}(t))_{tin [0,tau]}$ est associée à des formes quasi-coercives, non autonomes $(fra(t))_{t in [0,tau]}.$ Nous considérons également le problème de régularité maximale pour les d'ordre 2 (équations des ondes). Plusieurs exemples et applications sont considérés. / This Thesis is devoted to certain properties of non-autonomous evolution equations $u'(t)+A(t)u(t)=f(t), u(0)=x.$ More precisely, we are interested in the maximal $L^p$-regularity: given $fin L^{p}(0,tau;H),$ prove existence and uniqueness of the solution $u in W^{1,p}(0,tau;H)$. This problem was intensively studied in the autonomous cas, i.e., $A(t)=A$ for all $t.$ In the non-autonomous cas, the problem was considered by J.L.Lions in 1960. We prove serval results which extend all previously known ones on this problem. Here we assume that the familly of the operators $(mathcal{A}(t))_{tin [0,tau]}$ is associated with quasi-coercive, non-autonomous forms $(fra(t))_{t in [0,tau]}.$ We also consider the problem of maximal regularity for second order equations (the wave equation). Serval examples and applications are given in this Thesis.

Cette thèse comporte deux sujets d’étude mêlés. Le premier concerne l’étude de la bornitude sur Lp de la transformée de Riesz d∆-½ , où ∆ désigne l’opérateur de Laplace-Beltrami (positif). Le second traite de la régularité de Sobolev W1,p de la solution de l’équation de la chaleur non perturbée. Nous établissons également quelques résultats concernant les transformées de Riesz d’opérateurs de Schrödinger avec un potentiel comportant éventuellement une partie négative.Dans le cadre de ces travaux, nous nous plaçons sur une variété riemanienne (M, g) complète et non compacte. Nous supposons que M satisfait la propriété de doublement de volume (de constante de doublement égale à D) ainsi qu’une estimation gaussienne supérieure pour son noyau de la chaleur (celui associé à l’opérateur ∆). Nous travaillons avec le laplacien de Hodge-de Rham, noté ∆, agissant sur les 1-formes différentielles de M. En s’appuyant sur la formule de Bochner, liant ∆ à la courbure de Ricci de M, nous assimilons ∆ à un opérateur de Schrödinger à valeurs vectorielles. C’est un argument de dualité, basé sur une formule de commutation algébrique, qui lie l’étude de ∆ à celle de ∆. [...] / This thesis has two main parts. The first one deals with the study of the boundedness on Lp of the Riesz transform d∆-½ , where ∆ denotes the nonnegative Laplace-Beltrami operator. The second one deals with the Sobolev regularity W1,p of the solution of the heat equation. We also establish some results on the Riesz transforms of Schrödinger operators with a potential possibly having a negative part. In this work, we consider a complete non-compact Riemannian manifold (M, g). We assume that M satisfies the volume doubling property (with doubling constant equal to D) as well as a Gaussian upper estimate for its heat kernel associated to the operator ∆. We work with the Hodge-de Rham Laplacian ∆, acting on 1-differential forms of M. With the Bochner formula, linking ∆to the Ricci curvature of M, we see ∆ has a vector-valued Schrödinger operator. It is a duality argument, based on a commutation formula, which links the study of ∆to the one of ∆. [...]

[pt] A estrutura dos mean-filed games foi desenvolvida com o intuito de estudar problemas com um infinito número de jogadores em algum tipo de competição, ao qual pode ser aplicado em diversos problemas. O estudo formalizado desses problemas começou, na comunidade matemática com Lasry and Lions, e mais ou menos na mesma época, porém independentemente, na comunidade de engenharia por P. Caines, Minyi Huang, and Roland Malhamé. Desde então a pesquisa nos mean-field games cresceu exponencialmente, e nesse trabalho apresentaremos regularidade para um caso de mean-field games utilizando tecnicas particulares. Nesse trabalho, estudamos time-dependent mean-field games no caso subquadrático, isto é, mean-field games, o qual é escrito como um sistema de duas equações, uma equação de Hamilton-Jacobi e uma equação do transporte ou uma equação de Fokker-Plank, em que o Hamiltoniano na equação de Hamilton-Jacobi possui um crescimento subquadratico. Começamos em assumir dez suposições, e então sob os mesmos deduzir regularidade Lipschitz para o sistema. / [en] The mean-field games framework was developed to study problems with an infinite number of rational players in competition, which could be applied in many problems. The formalized study of these problems has begun, in the mathematical community by Lasry and Lions, and beside them, but independently close to the same time in the engineering community by P. Caines, Minyi Huang, and Roland Malhamé. Since these seminal contributions, the research in mean-field games has grown exponentially, and in this work we present a regularity to a case of mean-field games using particulars techniques. In this work, we study time-dependent mean-field games in the subquadratic case, that is, mean-field games, which are written as a system of a Hamilton–Jacobi equation and a transport or Fokker–Planck equation, where The Hamiltonian presented on the Hamilton–Jacobi equation has a subquadratic growth. We begin by assuming ten assumptions, and then, under these assumptions derive Lipschitz regularity of the system.