CoordenaÃÃo de AperfeÃoamento de Pessoal de NÃvel Superior / Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / In this work we establish local regularity estimates for at solutions to non-convex fully nonlinear elliptic equations and we study cavitation type equations modeled within coef- icients bounded and measurable. / Neste trabalho estabelecemos estimativas de regularidade local para soluÃÃes "flat" de equaÃÃes elÃpticas totalmente nÃo-lineares nÃo-convexas e estudamos equations do tipo cavidade com coeficientes meramente mensurÃveis.

[pt] Esquemas semi-Lagrangeanos usados para a aproximação do princípio da programação dinâmica são baseados em uma discretização temporal reconstruída no espaço de estado. O uso de uma malha estruturada torna essa abordagem inviável para problemas de alta dimensão devido à maldição da dimensionalidade. Nesta tese, apresentamos uma nova abordagem para problemas de controle ótimo de horizonte infinito onde a função valor é calculada usando Funções de Base Radial (RBFs) pelo método de aproximação de mínimos quadrados móveis de Shepard em malhas irregulares. Propomos um novo método para gerar uma malha irregular guiada pela dinâmica e uma rotina de otimizada para selecionar o parâmetro responsável pelo formato nas RBFs. Esta malha ajudará a localizar o problema e aproximar o princípio da programação dinâmica em alta dimensão. As estimativas de erro para a função valor também são fornecidas. Testes numéricos para problemas de alta dimensão mostrarão a eficácia do método proposto. Além do controle ótimo de EDPs clássicas mostramos como o método também pode ser aplicado ao controle de equações não-locais. Também fornecemos um exemplo analisando a convergência numérica de uma equação não-local controlada para o modelo contínuo. / [en] Semi-Lagrangian schemes for the approximation of the dynamic programming principle are based on a time discretization projected on a state-space grid. The use of a structured grid makes this approach not feasible for highdimensional problems due to the curse of dimensionality. In this thesis, we present a new approach for infinite horizon optimal control problems where the value function is computed using Radial Basis Functions (RBF) by the Shepard s moving least squares approximation method on scattered grids. We propose a new method to generate a scattered mesh driven by the dynamics and an optimal routine to select the shape parameter in the RBF. This mesh will help to localize the problem and approximate the dynamic programming principle in high dimension. Error estimates for the value function are also provided. Numerical tests for high dimensional problems will show the effectiveness of the proposed method. In addition to the optimal control of classical PDEs, we show how the method can also be applied to the control of nonlocal equations. We also provide an example analyzing the numerical convergence of a nonlocal controlled equation towards the continuous model.

Depuis la publication de l'ouvrage de Choquet (1955), la théorie d'espérance non linéaire a attiré avec grand intérêt des chercheurs pour ses applications potentielles dans les problèmes d'incertitude, les mesures de risque et le super-hedging en finance. Shige Peng a construit une sorte d'espérance entièrement non linéaire dynamiquement cohérente par l'approche des EDP. Un cas important d'espérance non linéaire cohérente en temps est la G-espérance, dans laquelle le processus canonique correspondant (B_{t})_{t≥0} est appelé G-mouvement brownien et joue un rôle analogue au processus de Wiener classique. L'objectif de cette thèse est d'étudier, dans le cadre de la G-espérance, certaines équations différentielles stochastiques rétrogrades (G-EDSR) à croissance quadratique avec applications aux problèmes de maximisation d'utilité robuste avec incertitude sur les modèles, certaines équations différentielles stochastiques (G-EDS) réfléchies et équations différentielles stochastiques rétrogrades réfléchies avec générateurs lipschitziens. On considère d'abord des G-EDSRs à croissance quadratique. Dans le Chapitre 2 nous fournissons un resultat d'existence et unicité pour des G-EDSRs à croissance quadratique. D'une part, nous établissons des estimations a priori en appliquant le théorème de type Girsanov, d'où l'on en déduit l'unicité. D'autre part, pour prouver l'existence de solutions, nous avons d'abord construit des solutions pour des G-EDSRs discretes en résolvant des EDPs non-linéaires correspondantes, puis des solutions pour les G-EDSRs quadratiques générales dans les espaces de Banach. Dans le Chapitre 3 nous appliquons les G-EDSRs quadratiques aux problèmes de maximisation d'utilité robuste. Nous donnons une caratérisation de la fonction valeur et une stratégie optimale pour les fonctions d'utilité exponentielle, puissance et logarithmique. Dans le Chapitre 4, nous traitons des G-EDSs réfléchies multidimensionnelles. Nous examinons d'abord la méthode de pénalisation pour résoudre des problèmes de Skorokhod déterministes dans des domaines non convexes et établissons des estimations pour des fonctions α-Hölder continues. A l'aide de ces résultats obtenus pour des problèmes déterministes, nous définissons le G-mouvement Brownien réfléchi et prouvons son existence et son unicité dans un espace de Banach. Ensuite, nous prouvons l'existence et l'unicité de solution pour les G-EDSRs multidimensionnelles réfléchies via un argument de point fixe. Dans le Chapitre 5, nous étudions l'existence et l'unicité pour les équations différentielles stochastiques rétrogrades réfléchies dirigées par un G-mouvement brownien lorsque la barrière S est un processus de G-Itô. / Since the publication of Choquet's (1955) book, the theory of nonlinear expectation has attracted great interest from researchers for its potential applications in uncertainty problems, risk measures and super-hedging in finance. Shige Peng has constructed a kind of fully nonlinear expectation dynamically coherent by the PDE approach. An important case of time-consistent nonlinear expectation is G-expectation, in which the corresponding canonical process (B_{t})_{t≥0} is called G-Brownian motion and plays a similar role to the classical Wiener process. The objective of this thesis is to study, in the framework of the G-expectation, some backward stochastic differential equations (G-BSDE) under a quadratic growth condition on their coefficients with applications to robust utility maximization problems with uncertainty on models, Reflected stochastic differential equations (reflected G-SDE) and reflected backward stochastic differential equations with Lipschitz coefficients (reflected G-BSDE). We first consider G-BSDE with quadratic growth. In Chapter 2 we provide a result of existence and uniqueness for quadratic G-BSDEs. On the one hand, we establish a priori estimates by applying the Girsanov-type theorem, from which we deduce the uniqueness. On the other hand, to prove the existence of solutions, we first constructed solutions for discrete G-BSDEs by solving corresponding nonlinear PDEs, then solutions for the general quadratic G-BSDEs in the spaces of Banach. In Chapter 3 we apply quadratic G-BSDE to robust utility maximization problems. We give a characterization of the value function and an optimal strategy for exponential, power and logarithmic utility functions. In Chapter 4, we discuss multidimensional reflected G-SDE. We first examine the penalization method to solve deterministic Skorokhod problems in non-convex domains and establish estimates for continuous α-Hölder functions. Using these results for deterministic problems, we define the reflected G-Brownian motion and prove its existence and its uniqueness in a Banach space. Then we prove the existence and uniqueness of the solution for the multidimensional reflected G-SDE via a fixed point argument. In Chapter 5, we study the existence and uniqueness of the reflected backward stochastic differential equations driven by a G-Brownian motion when the obstacle S is a G-Itô process.

Cette thèse est dédiée à l'étude mathématique de quelques modèles de trafic routier congestionné. La notion essentielle est l'équilibre de Wardrop. Elle poursuit des travaux de Carlier et Santambrogio avec des coauteurs. Baillon et Carlier ont étudié le cas de grilles cartésiennes dans $\RR^2$ de plus en plus denses, dans le cadre de la théorie de $\Gamma$-convergence. Trouver l'équilibre de Wardrop revient à résoudre des problèmes de minimisation convexe. Dans le chapitre 2, nous regardons ce qui se passe dans le cas de réseaux généraux, de plus en plus denses, dans $\RR^d$. Des difficultés nouvelles surgissent par rapport au cas initial de réseaux cartésiens et pour les contourner, nous introduisons la notion de courbes généralisées. Des hypothèses structurelles sur ces suites de réseaux discrets sont nécessaires pour s'assurer de la convergence. Cela fait alors apparaître des fonctions qui sont des sortes de distances de Finsler et qui rendent compte de l'anisotropie du réseau. Nous obtenons ainsi des résultats similaires à ceux du cas cartésien. Dans le chapitre 3, nous étudions le modèle continu et en particulier, les problèmes limites. Nous trouvons alors des conditions d'optimalité à travers une formulation duale qui peut être interprétée en termes d'équilibres continus de Wardrop. Cependant, nous travaillons avec des courbes généralisées et nous ne pouvons pas appliquer directement le théorème de Prokhorov, comme cela a été le cas dans \cite{baillon2012discrete, carlier2008optimal}. Pour pouvoir néanmoins l'utiliser, nous considérons une version relaxée du problème limite, avec des mesures d'Young. Dans le chapitre 4, nous nous concentrons sur le cas de long terme, c'est-à-dire, nous fixons uniquement les distributions d'offre et de demande. Comme montré dans \cite{brasco2013congested}, le problème de l'équilibre de Wardrop est équivalent à un problème à la Beckmann et il se réduit à résoudre une EDP elliptique, anisotropique et dégénérée. Nous utilisons la méthode de résolution numérique de Lagrangien augmenté présentée dans \cite{benamou2013augmented} pour proposer des exemples de simulation. Enfin, le chapitre 5 a pour objet l'étude de problèmes de Monge avec comme coût une distance de Finsler. Cela se reformule en des problèmes de flux minimal et une discrétisation de ces problèmes mène à un problème de point-selle. Nous le résolvons alors numériquement, encore grâce à un algorithme de Lagrangien augmenté. / This thesis is devoted to the mathematical analysis of some models of congested road traffic. The essential notion is the Wardrop equilibrium. It continues Carlier and Santambrogio's works with coauthors. With Baillon they studied the case of two-dimensional cartesian networks that become very dense in the framework of $\Gamma$-convergence theory. Finding Wardrop equilibria is equivalent to solve convex minimisation problems.In Chapter 2 we look at what happens in the case of general networks, increasingly dense. New difficulties appear with respect to the original case of cartesian networks. To deal with these difficulties we introduce the concept of generalized curves. Structural assumptions on these sequences of discrete networks are necessary to obtain convergence. Sorts of Finsler distance are used and keep track of anisotropy of the network. We then have similar results to those in the cartesian case.In Chapter 3 we study the continuous model and in particular the limit problems. Then we find optimality conditions through a duale formulation that can be interpreted in terms of continuous Wardrop equilibria. However we work with generalized curves and we cannot directly apply Prokhorov's theorem, as in \cite{baillon2012discrete, carlier2008optimal}. To use it we consider a relaxed version of the limit problem with Young's measures. In Chapter 4 we focus on the long-term case, that is, we fix only the distributions of supply and demand. As shown in \cite{brasco2013congested} the problem of Wardrop equilibria can be reformulated in a problem à la Beckmann and reduced to solve an elliptic anisotropic and degenerated PDE. We use the augmented Lagrangian scheme presented in \cite{benamou2013augmented} to show a few numerical simulation examples. Finally Chapter 5 is devoted to studying Monge problems with as cost a Finsler distance. It leads to minimal flow problems. Discretization of these problems is equivalent to a saddle-point problem. We then solve it numerically again by an augmented Lagrangian algorithm.

[pt] Esta tese tem o intuito do estudo da existência, não existência e estimativas a priori de soluções não negativas de alguns tipos de problemas elípticos degenerados coercivos e não coercivos com um termo adicional dependendo do gradiente. Dentre outras coisas, obtemos condições integrais generalizadas tipo Keller-Osserman para a existência e não existência de soluções. Também mostramos que condições adicionais e diferentes são necessárias quando p é maior ou igual à 2 ou p é menor ou igual à 2, devido ao caráter degenerado do operador. As estimativas a priori são obtidas para super-soluções e soluções de EDPs elípticas superlineares o sistemas de tais tipos de equações em forma divergente com diferentes operadores e não linearidades. Além do mais, obtemos extensões até a fronteira de algumas desigualdades de Harnack fracas e lemas quantitativos de Hopf para operadores elípticos como o p-Laplaciano. / [en] This thesis concerns the study of existence, nonexistence and a priori estimates of nonnegative solutions of some types of degenerate coercive and non coercive elliptic problems involving an additional term which depends on the gradient. Among other things, we obtain generalized integral conditions of Keller-Osserman type for the existence and nonexistence of solutions. Also, we show that different conditions are needed when p is higher or equal to 2 or p is less than or equal to 2, due to the degeneracy of the operator. The uniform a priori estimates are obtained for supersolutions and solutions of superlinear elliptic PDE or systems of such PDE in divergence form that can contain different operators and nonlinearities. We also give full boundary extensions to some half Harnack inequalities and quantitative Hopf lemmas, for degenerate elliptic operators like the p-Laplacian.

Since the US National Security Agency’s former contractor Edward Snowden exposed the Agency’s mass surveillance, the EU has been making a series of attempts toward a more safeguarded and stricter path concerning its data privacy protection. On 8 April 2014, the Court of Justice of the European Union (the CJEU) invalidated the EU Data Retention Directive 2006/24/EC on the basis of incompatibility with the Charter of Fundamental Rights of the European Union (the Charter). After this judgment, the CJEU examined the legality of the Safe Harbour Agreement, which had been the main legal basis for transfers of personal data from the EU to the US under Decision 2000/520/EC. Subsequently, on 6 October 2015, in the case of Schrems v Data Protection Commissioner, the CJEU declared the Safe Harbour Decision invalid. The ground for the Court’s judgment was the fact that the Decision enabled interference, by US public authorities, with the fundamental rights to privacy and personal data protection under Article 7 and 8 of the Charter, when processing the personal data of EU citizens. According to the judgment, this interference has been beyond what is strictly necessary and proportionate to the protection of national security and the persons concerned were not offered any administrative or judicial means of redress enabling the data relating to them to be accessed, rectified or erased. The Court’s analysis of the Safe Harbour was borne out of the EU Commission’s own previous assessments. Consequently, since the transfers of personal data between the EU and the US can no longer be carried out through the Safe Harbour, the EU legislature is left with the task to create a safer option, which will guarantee that the fundamental rights to privacy and protection of personal data of the EU citizens will be respected. However, although the EU is the party dictating the terms for these transatlantic transfers of personal data, the current provisions of the US law are able to provide for derogations from every possible renewed agreement unless they become compatible with the EU data privacy law. Moreover, as much business is at stake and prominent US companies are involved in this battle, the pressure toward the US is not only coming from the EU, but some American companies are also taking the fight for EU citizens’ right to privacy and protection of their personal data.