LIONS Pierre Louis

We consider a class of closed loop stochastic optimal control problems in finite time horizon, in which the cost is an expectation conditional on the event that the process has not exited a given bounded domain. An important difficulty is that the probability of the event that conditionates the strategy decays as time grows. The optimality conditions consist of a system of partial differential equations, including a Hamilton-Jacobi-Bellman equation (backward w.r.t. time) and a (forward w.r.t. time) Fokker-Planck equation for the law of the conditioned process. The two equations are supplemented with Dirichlet conditions. Next, we discuss the asymptotic behavior as the time horizon tends to`8. This leads to a new kind of optimal control problem driven by an eigenvalue problem related to a continuity equation with Dirichlet conditions on the boundary. We prove existence for the latter. We also propose numerical methods and supplement the various theoretical aspects with numerical simulations.

This paper presents a novel approach to characterize the dynamics of the limit spectrum of large random matrices. This approach is based upon the notion we call "spectral dominance". In particular, we show that the limit spectral measure can be determined as the derivative of the unique viscosity solution of a partial integro-differential equation. This also allows to make general and "short" proofs for the convergence problem. We treat the cases of Dyson Brownian motions, Wishart processes and present a general class of models for which this characterization holds.

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This is a work in progress. The aim is to propose a plausible mechanism for the short term dynamics of the oil market based on the interaction of economic agents. This is a theoretical research which by no means aim at describing all the aspects of the oil market. In particular, we use the tools and terminology of game theory, but we do not claim that this game actually exists in the real world. In parallel, we are currently studying and calibrating a long term model for the oil industry, which addresses the interactions of a monopolists with a competitive fringe of small producers. It is the object of another paper that will be available soon. The present premiminary version does not contain all the economic arguments and all the connections with our long term model. It mostly addresses the description of the model, the equations and numerical simulations focused on the oil industry short term dynamics. A more complete version will be available soon.

We present results of existence, regularity and uniqueness of solutions of the master equation associated with the mean field planning problem in the finite state space case, in the presence of a common noise. The results hold under monotonicity assumptions, which are used crucially in the different proofs of the paper. We also make a link with the trajectories induced by the solution of the master equation and start a discussion on the case of boundary conditions.

This note is concerned with a modeling question arising from the mean field games theory. We show how to model mean field games involving a major player which has a strategic advantage, while only allowing closed loop markovian strategies for all the players. We illustrate this property through three examples.

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We consider a class of closed loop stochastic optimal control problems in finite time horizon, in which the cost is an expectation conditional on the event that the process has not exited a given bounded domain. An important difficulty is that the probability of the event that conditionates the strategy decays as time grows. The optimality conditions consist of a system of partial differential equations, including a Hamilton-Jacobi-Bellman equation (backward w.r.t. time) and a (forward w.r.t. time) Fokker-Planck equation for the law of the conditioned process. The two equations are supplemented with Dirichlet conditions. Next, we discuss the asymptotic behavior as the time horizon tends to`8. This leads to a new kind of optimal control problem driven by an eigenvalue problem related to a continuity equation with Dirichlet conditions on the boundary. We prove existence for the latter. We also propose numerical methods and supplement the various theoretical aspects with numerical simulations.

We study in this paper three aspects of Mean Field Games. The first one is the case when the dynamics of each player depend on the strategies of the other players. The second one concerns the modeling of " noise " in discrete space models and the formulation of the Master Equation in this case. Finally, we show how Mean Field Games reduce to agent based models when the intertemporal preference rate goes to infinity, i.e. when the anticipation of the players vanishes.

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"We study the existence and the uniqueness of the solution to parabolic type equations with irregular coefficients and/or initial conditions. The coefficients considered in the equation typically belong to Lebesgue or Sobolev spaces, the initial condition may be only Lebesgue integrable, the second order term in the equation may be degenerate. The arguments elaborate on the DiPerna-Lions theory of renormalized solutions to linear transport equations and related equations. The connection between the results on the partial differential equation and the well-posedness of the underlying stochastic/ordinary differential equation is examined. We in particular follow up on two previous articles. These notes, written up jointly by the two authors, lay out the background on the various issues and present the recent results obtained by the second author. They are an expanded version of the lectures delivered at Collège de France during the academic year 2012-13.

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We follow-up on our works devoted to homogenization theory for linear second-order elliptic equations with coefficients that are perturbations of periodic coefficients. We have first considered equations in divergence form in [6, 7, 8]. We have next shown, in our recent work [9], using a slightly different strategy of proof than in our earlier works, that we may also address the equation −aij∂iju = f. The present work is devoted to advection-diffusion equations: −aij∂iju + bj∂ju = f. We prove, under suitable assumptions on the coefficients aij, bj, 1 ≤ i, j ≤ d (typically that they are the sum of a periodic function and some perturbation in L p , for suitable p < +∞), that the equation admits a (unique) invariant measure and that this measure may be used to transform the problem into a problem in divergence form, amenable to the techniques we have previously developed for the latter case.

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We show that the initial value problem for Hamilton-Jacobi equations with multiplicative rough time dependence, typically stochastic, and convex Hamiltonians satisfies finite speed of propagation. We prove that in general the range of dependence is bounded by a multiple of the length of the "skeleton" of the path, that is a piecewise linear path obtained by connecting the successive extrema of the original one. When the driving path is a Brownian motion, we prove that its skeleton has almost surely finite length. We also discuss the optimality of the estimate.

This thesis deals with the study of new medium field game models. We first study optimal stopping and impulse control models in the absence of common noise. We construct for these models a notion of adapted solution for which we prove existence and uniqueness results under natural assumptions. Then, we focus on several properties of mean-field games. We study the limit of these models to pure evolution models when the players' anticipation tends to 0. We show the uniqueness of equilibria for strongly coupled systems (coupled by strategies) under certain assumptions. We then prove some regularity results on a master equation that models a mean field game with common noise in a discrete state space. We then present a generalization of the standard Uzawa algorithm and apply it to the numerical solution of some mean-field game models, in particular optimal stopping or impulse control. Finally, we present a concrete case of mean-field game that comes from problems involving a large number of connected devices in telecommunications.

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We study in this paper three aspects of Mean Field Games. The first one is the case when the dynamics of each player depend on the strategies of the other players. The second one concerns the modeling of " noise " in discrete space models and the formulation of the Master Equation in this case. Finally, we show how Mean Field Games reduce to agent based models when the intertemporal preference rate goes to infinity, i.e. when the anticipation of the players vanishes.

No summary available.

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We show that the initial value problem for Hamilton-Jacobi equations with multiplicative rough time dependence, typically stochastic, and convex Hamiltonians satisfies finite speed of propagation. We prove that in general the range of dependence is bounded by a multiple of the length of the "skeleton" of the path, that is a piecewise linear path obtained by connecting the successive extrema of the original one. When the driving path is a Brownian motion, we prove that its skeleton has almost surely finite length. We also discuss the optimality of the estimate.

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Mean field game theory was introduced in 2006 by Jean-Michel Lasry and Pierre-Louis Lions. It allows the study of game theory in certain configurations where the number of players is too large to hope for a practical resolution. We study the theory of mean-field games on graphs based on the work of Olivier Guéant which we will extend to more general Hilbertian forms. We will also study the links between K-means and mean-field games, which will in principle allow us to propose new algorithms for K-means using numerical resolution techniques specific to mean-field games. Finally, we will study a mean-field game, namely the "meeting start time" problem by extending it to situations where agents can choose between two meetings. We will study analytically and numerically the existence and multiplicity of solutions of this problem.

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A parcimonious long term model is proposed for a mining industry. Knowing the dynamics of the global reserve, the strategy of each production unit consists of an optimal control problem with two controls, first the flux invested into prospection and the building of new extraction facilities, second the production rate. In turn, the dynamics of the global reserve depends on the individual strategies of the producers, so the models leads to an equilibrium, which is described by low dimensional systems of partial differential equations. The dimen-sionality depends on the number of technologies that a mining producer can choose. In some cases, the systems may be reduced to a Hamilton-Jacobi equation which is degenerate at the boundary and whose right hand side may blow up at the boundary. A mathematical analysis is supplied. Then numerical simulations for models with one or two technologies are described. In particular, a numerical calibration of the model in order to fit the historical data is carried out.

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We study the existence and uniqueness of the stochastic viscosity solutions of fully nonlinear, possibly degenerate, second order stochastic pde with quadratic Hamiltonians associated to a Rie-mannian geometry. The results are new and extend the class of equations studied so far by the last two authors.

This paper deals with a stochastic order-driven market model with waiting costs, for order books with heterogenous traders. Offer and demand of liquidity drives price formation and traders anticipate future evolutions of the order book. The natural framework we use is mean field game theory, a class of stochastic differential games with a continuum of anonymous players. Several sources of heterogeneity are considered including the mean size of orders. Thus we are able to consider the coexistence of Institutional Investors and High Frequency Traders (HFT). We provide both analytical solutions and numerical experiments. Implications on classical quantities are explored: order book size, prices, and effective bid/ask spread. According to the model, in markets with Institutional Investors only we show the existence of inefficient liquidity imbalances in equilibrium, with two symmetrical situations corresponding to what we call liquidity calls for liquidity. During these situations the transaction price significantly moves away from the fair price. However this macro phenomenon disappears in markets with both Institutional Investors and HFT, although a more precise study shows that the benefits of the new situation go to HFT only, leaving Institutional Investors even with higher trading costs.

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We present a possible approach to approximate at both the coarse and fine scales the solution to an elliptic equation with oscillatory coefficient when this coefficient consists of a “nice”, say periodic, function that is locally perturbed. The approach is based on a local profile, solution to an equation similar to the corrector equation in classical homogenization. The well-posedness of that equation is explored, in various functional settings depending upon the locality of the perturbation. Some related problems are discussed.

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The purpose of this article is to get mathematicians interested in studying a number of PDEs that naturally arise in macroeconomics. These PDEs come from models designed to study some of the most important questions in economics. At the same time they are highly interesting for mathematicians because their structure is often quite difficult. We present a number of examples of such PDEs, discuss what is known about their properties, and list some open questions for future research.

The purpose of this article is to get mathematicians interested in studying a number of partial differential equations (PDEs) that naturally arise in macroeconomics. These PDEs come from models designed to study some of the most important questions in economics. At the same time, they are highly interesting for mathematicians because their structure is often quite difficult. We present a number of examples of such PDEs, discuss what is known about their properties, and list some open questions for future research.

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We continue the development of the theory of pathwise stochastic entropy solutions for scalar conservation laws in $\R^N$ with quasilinear multiplicative ''rough path'' dependence by considering inhomogeneous fluxes and a single rough path like, for example, a Brownian motion. Following our previous note where we considered spatially independent fluxes, we introduce the notion of pathwise stochastic entropy solutions and prove that it is well posed, that is we establish existence, uniqueness and continuous dependence in the form of a (pathwise) $L^1$-contraction. Our approach is motivated by the theory of stochastic viscosity solutions, which was introduced and developed by two of the authors, to study fully nonlinear first- and second-order stochastic pde with multiplicative noise. This theory relies on special test functions constructed by inverting locally the flow of the stochastic characteristics. For conservation laws this is best implemented at the level of the kinetic formulation which we follow here.

We study the long time average, as the time horizon tends to infinity, of the solution of a mean field game system with a nonlocal coupling. We show an exponential convergence to the solution of the associated stationary ergodic mean field game. Proofs rely on semiconcavity estimates and smoothing properties of the linearized system. The research that led to the present paper was partially supported by a grant of the group GNAMPA of INdA.

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We develop a pathwise theory for scalar conservation laws with quasilinear multiplicative rough path dependence, a special case being stochastic conservation laws with quasilinear stochastic dependence. We introduce the notion of pathwise stochastic entropy solutions, which is closed with the local uniform limits of paths, and prove that it is well posed, i.e., we establish existence, uniqueness and continuous dependence, in the form of pathwise $L^1$-contraction, as well as some explicit estimates. Our approach is motivated by the theory of stochastic viscosity solutions, which was introduced and developed by two of the authors, to study fully nonlinear first- and second-order stochastic pde with multiplicative noise. This theory relies on special test functions constructed by inverting locally the flow of the stochastic characteristics. For conservation laws this is best implemented at the level of the kinetic formulation which we follow here.

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Introduced by J. -M. Lasry and P.-L. Lions, mean-field game theory simplifies interactions between economic agents using an approach inspired by physical theories. Economic applications are presented concerning the labor market, asset management, population distribution problems, and growth theory. The models presented use mean-field game theory in various forms, sometimes static, often dynamic, with discrete or continuous state space and in a deterministic or stochastic environment. Various notions of stability are discussed, including the notion of educative stability, which has inspired numerical methods of resolution. Indeed, we present numerical methods that allow to obtain solutions to both stationary and dynamic problems, while abstracting from the forward/backward structure, which is a priori problematic from a numerical point of view. In the margin of mean-field game theory, the problem of appropriate discount rates to deal with sustainable development problems is addressed. We discuss the notion of ecological rate introduced by R. Guesnerie and provide new non-asymptotic properties, notably continuity.

This thesis is devoted to the study of the asymptotic behavior of solutions of a class of partial differential equations with strongly oscillating coefficients. In a first step, we focus on a family of nonlinear equations, heterogeneous scalar conservation laws, which are involved in various problems in fluid mechanics or nonlinear electromagnetism. We assume that the flow of this equation is spatially periodic, and that the period of the oscillations tends to zero. We then identify the microscopic and macroscopic asymptotic profiles of the solution, and we prove a strong convergence result. In particular, we show that when the initial condition does not follow the microscopic profile dictated by the equation, an initial layer in time is formed during which the solutions adapt to it. In a second step, we consider a linear transport equation, which models the evolution of the density of a set of charged particles in a random and highly oscillating electric potential. We establish the appearance of microscopic oscillations in time and space in the density, in response to the excitation by the electric potential. Explicit formulas are also given for the homogenized transport operator when the dimension of space is equal to one.

This thesis is devoted to ordinary differential equations and associated transport equations for sparsely regular vector fields, and contains four works. The first one deals with the solution of ODEs and transport equations for vector fields in L^2(R^2) with zero divergence, verifying a regularity condition on the field direction. The results are obtained in the framework of the theory developed by R. DiPerna and P. L. Lions for the solution of transport equations with sparse regular coefficients. We lower the necessary regularity conditions in this particular case of dimension two. The second work concerns the Liouville equation, which governs the behavior of a density of N interacting particles, in the framework of weakly regular fields. The results of DiPerna and Lions, already extended to the kinetic case by François Bouchut, are adapted to take into account a singularity at the origin. In the third work, we are interested in the convergence of interacting particle systems to the Vlasov equation. The convergence is obtained through accurate discrete estimates, in the case of 1/|x|^alpha interaction forces, for alpha < 1. This improves the previously known result for C^1 forces. The fourth one uses the same kind of technique for the Euler approximation by vortices. Convergence is proved for any time when the interaction is slightly less singular than for vortices. Uniform bounds on the field and its accretion in the case of real wormholes are also given.

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In this thesis, we study (from a mathematical point of view) some asymptotic problems from fluid mechanics. This is motivated by physical as well as numerical reasons: the complete equations of physics are often very complicated and cannot be solved in their entirety, which leads us to consider simplified models that take into account the different scales on which the system can be studied. These models can be justified from a mathematical point of view thanks to convergence theorems when a small parameter tends towards zero. This poses mathematical difficulties, often due to the change in the type of equations, which often correspond to a physical reality: persistence of oscillations, presence of boundary layers. We study three examples which are respectively the passage from the Navier-Stokes equation to the Euler equation in a domain with an edge, the compressible-incompressible boundary of a viscous fluid and finally the study of rotating fluids at high speed.

We study kinetic equations of the form f/t + v. *#xf = q(f,f), t0, x ,r#n, v , r#n, which describe the evolution of a gas or a plasma in which the particles undergo collisions modeled by the operator q, known as Boltzmann's -: q(f,f) dv#* d b(v v#*, ) (f'f'#* ff#*), - or Landau: q(f,f) = /v#i dv#*a#i#j(v v#*) (f#*f/v#j ff#*/v#*#,#j). The first three parts of this thesis concern the homogeneous (x-independent) equations #tf = q(f,f). Two points are particularly emphasized: grazing collisions and entropy dissipation. In the first part, we study the Cauchy problem associated to the homogeneous Boltzmann and Landau equations (for realistic and possibly singular cross sections), the regularity properties of the solutions, as well as the asymptotics of grazing collisions which allows us to pass from one equation to the other. The second part is devoted to the special case of Maxwellian molecules. Under this assumption, a detailed study is made of the Cauchy problem and of the return to equilibrium, as well as of the links between kinetic theory and information theory. In the third part, we use the previous results to obtain explicit estimates of the rate of return to equilibrium in the general case. In the fourth part, we obtain partial results on the Cauchy problem for the inhomogeneous Landau equation, and some new estimates on the inhomogeneous Boltzmann equation. Finally, in the fifth part, we establish various conservative forms of the Boltzmann operator, with application to the asymptotics of grazing collisions.

The object of this thesis is the mathematical analysis of models from fluid mechanics. The study is mainly focused on the incompressible inhomogeneous Navier-Stokes equations and the compressible isentropic Navier-Stokes equations. The first part is devoted to ordinary differential equations associated with vector fields with irregular coefficients, typically with integrable derivatives. R. J. Di Perna and P. -L. Lions have been pioneers in the study of vector fields with W#1#,#1 regularity and bounded divergence, by showing the existence and uniqueness of a flow X verifying most of the properties of regular vector field flows, valid however for almost any initial point. The object of the first part is to extend this theory to fields with unbounded divergence. The proof is based on the method of normalized solutions for transport equations, introduced by R. J. Di Perna and P. -L. Lions. In the continuity of the previous results, we show on the other hand an existence theorem of stronger solutions corresponding to initial data in W#1#,#m (m > 1) for #t +b. * = 0, the associated vector field b being assumed to have Sobolev regularity W#s#+#1#,#p with sp = n. These results are then applied to a proof of uniqueness of solutions of the incompressible inhomogeneous Navier-Stokes equations in dimension 2. In the second part of this work, we focus on incompressible fluid models. We consider a family of incompressible immiscible fluids indexed by 1, . m in an open r#n (n 2). These fluids are characterized by their density i#1im and their viscosity #i#1##im. The first chapter deals with the global existence of weak solutions for the incompressible Navier-Stokes equations when the domain is unbounded. The regularity of multiphase plane flows is then studied, stating the results in terms of the relative dispersion of viscosities, while taking into account the possible presence of vacuum pockets in the fluid medium. The third chapter is devoted to some remarks on the regularity of the weak solutions of an equation from a simplified model of magnetohydrodynamics, coupling the incompressible Navier-Stokes equations and the Maxwell equations. Finally, we study the Navier-Stokes equations modeling the evolution of an isentropic compressible fluid. The work of P.-L. Lions ensures the global existence in time of weak solutions under certain assumptions on the pressure law. In dimension n = 2 or 3, one can show results of regularity in small time for initial densities canceling. When n = 2, one obtains global results in time, provided that the density remains bounded. We use a logarithmic estimate, demonstrated in the context of the incompressible models mentioned above. In the second chapter, we analyze the regularity of weak solutions in dimension n 2, by showing an a priori estimate which gives information on the regularity in time of the velocity field.

The work presented in this thesis can be grouped into two themes. 1. The ergodic problem for the Hamilton-Jacobi-Bellman (H-J-B) equations. 2. Regularizing effects for a class of Hamilton-Jacobi equations. The ergodic problem concerns the long time average behavior of deterministic or stochastic controlled systems. We study here deterministic systems (including those defined in infinite dimension) and the corresponding H-J-B equations, using the theory of viscosity solutions. We first establish in chapter 2 the ergodicity of infinite dimensional systems under classical finite dimensional assumptions. Then, we focus on necessary or sufficient conditions to ensure the ergodicity of finite dimensional systems. In chapter 3, we prove the existence of an ergodic attractor on which the system is controllable. And in chapter 4, we give a kind of reciprocal, the estimate of the controllability on the ergodic attractor. Controllability also plays an essential role in the regularizing effects of the first order Hamilton-Jacobi equations. We show in chapter 5, three types of regularizing effects: lipschitzian regularization, semi-concave regularization and regularization in c#1#,#1#l#o#c.

We present in this work two mathematical techniques for the study of the Hartree-Fock equations, the Fourier analysis and the wavelet analysis, trying to show their advantages as well as their drawbacks. The Fourier analysis, known for a long time but little used by chemists, leads to an impulse representation and allows to simplify the writing of the equations whose resolution becomes possible for small chemical structures. In particular, the equations related to diatomic molecules, H2 and HeH+, could be solved thanks to an iterative method defined in the impulse space obtained by Fourier transform. The analytical derivation of the first iteration is described in detail, and the improvements, both qualitative and quantitative, brought to the initial wave function by this first iteration are analyzed and commented. Wavelet analysis has never been applied in quantum chemistry. Initially developed in signal processing, it finds here a new field of application. After a brief reminder of the general foundations of wavelet theory, the representation of the Hartree-Fock equations in a mixed position-impulse space is obtained thanks to a continuous wavelet transform. An interpretation of this representation is presented, and an iterative method of resolution is proposed. The improvements brought by a first analytical iteration are also analyzed and commented. Another type of wavelet has also been used: ortho-normal wavelets which lead to a fully numerical treatment of the Hartree-Fock equations. The matrix representation provided by the BCR algorithm (Beylkin, Coifman, Rokhlin) is used in a resolution method based on an iterative process whose convergence as well as the problems related to the discretization of the data are studied more particularly.

The thesis gathers three papers which all establish the existence and uniqueness of viscosity solutions for some partial differential equations coming from optimal control problems or mathematical economics models. Most of these equations break out of the classical framework by the presence of either a constraint on the gradient or a nonlinear integral term. We also focus on the determination of an asymptotic criterion compatible with uniqueness results. Thus the first work shows that there exists a unique minor solution for first order Hamilton-Jacobi equations, when the Hamiltonian is convex and nonlinear. In a second work, we establish that the value function of a singular control problem (investment-consumption model with local substitution) is the unique solution of the associated dynamic programming equation, in an unbounded open, with state constraint at the edge. In the last work, we are interested in a class of integro-differential equations whose nonlinearity comes essentially from the integral term. Again, the origin of this problem is economic since the unique solution is the recursive utility function when the uncertainty is generated by a Brownian motion and a jump process.

This thesis gathers a set of works devoted to the mathematical study of different molecular models used in quantum chemistry in numerical simulations. We are first interested in the thomas-fermi models, in particular the thomas-fermi-dirac-von weizsacker model and the thomas-fermi model with fermi-amaldi correction, then in the hartree-fock models, such as the multideterminant models. For each model, the results we prove concern the compactness of the minimizing sequences, the existence of a minimum, and its qualities (uniqueness, decay to infinity, non-degeneracy of the Lagrange multiplier,. . . ). A numerical application of these theoretical models is also presented. In the appendix, a result of group theory is presented.

This thesis gathers a set of works related to the study of minimization problems which arise in the quantum mechanical modeling of atoms and molecules, in atomic physics, on the one hand, and of nuclei, in nuclear physics, on the other hand. The two parts of this thesis deal successively with these two types of problems. In the first part, we are interested in the existence of an optimal geometry of nuclei, for a given ion or molecule, in the framework of Thomas-Fermi, Hartree and Hartree-Fock models. The second part is devoted to the study of two families of nuclear physics models: Hartree-type models and a Hartree-Fock model with a simplified Skyrme-type potential. In both cases, the questions posed are translated in terms of minimization problems in three-dimensional space, invariant by translation, by means of an (energy) functional depending on one or more functions subject to various normalization constraints. The loss of compactness of the minimizing sequences, linked to the invariance by translation, is analyzed by the concentration-compactness method.

We apply the theory of viscosity solutions due to Michael Grain Crandall and Pierre-Louis Lions, to the numerical solution of two nonlinear partial differential equations. The first part is devoted to the study of the Horn equation, which models the shape-from-shading problem. It concerns the reconstruction of a surface illuminated by light sources, from a single gray level coded photograph. The theory of viscosity solutions provides us with a framework in which the problem is mathematically well posed: we prove the uniqueness of the viscosity solution of the Horn equation verifying appropriate edge conditions. Then, in order to compute a first order approximation of the viscosity solution, we construct a monotone finite difference scheme, obtained by discretization of the dynamic programming principle, due to Bellman. In the second part, we propose a finite difference scheme that approximates the unique viscosity solution of the variational inequality that models the stochastic control problem: portfolio management with transaction costs. The algorithm allows not only the computation of an approximation of the value function but also the search for free boundaries that delimit the region in which no transactions are made.

We show an analogy, induced by an algebraic morphism, between dynamic programming and probabilities. We develop the analysis of a probability theory in a dioid, in which we show remarkable results that correspond to the usual case. For example: the law of large numbers and the central limit theorem in dynamic programming. The entities of dynamic programming are thus, in a certain sense, linear.

This thesis concerns the numerical approximation of the viscosity solutions, as defined by Michael Grain Crandall and Pierre-Louis Lions, of the first order Hamilton-Jacobi equations which are nonlinear partial differential equations, as well as the study of an example from image processing, shape-from-shading, which consists in the reconstruction of a three-dimensional relief from the data of a two-dimensional image, a photograph for example. The first chapter is a brief presentation of the viscosity solutions of the Hamilton-Jacobi equations and of some existence and uniqueness results. The second chapter describes the different methods developed to approximate these solutions, and is based on numerical analysis. The third chapter, more applied, aims at explaining how, concretely, one can write an approximation scheme for viscosity solutions. Finally, the example is studied in a precise way (by taking up the different developments of the first chapters of the thesis): we show how the relief can be interpreted as the viscosity solution of a Hamilton-Jacobi equation. We study the different possible formalizations for the edges of the image in order to reach satisfactory existence and uniqueness results. Then a scheme is constructed and applied to the numerical reconstruction of different images.

The work presented here focuses on two distinct topics: in the first part, we deal with two domain decomposition methods inspired by Schwarz's method, on non-overlapping subdomains, for general elliptic problems. These methods are based on the alternative solution of subproblems on the subdomains, with mixed boundary conditions on the interfaces, of Robin for the first method, while for the second one, we introduce an operator acting on the trace term. We demonstrate the convergence of these two methods applied to continuous problems, and establish that they can be interpreted as Peaceman-Rachford methods. After a brief spectral study, we propose general convergence results in the framework of finite difference approximations. We then compare them with those of numerical experiments. The second method, which we can interpret as a preconditioned version of the first one, is more efficient from the continuous point of view, for which we demonstrate the geometric convergence, and from the discrete point of view, for which we establish that the convergence is independent of the discretization step. In the second part, we study a molecular scale detonation initiation problem, modeled by a one-dimensional quantum system perturbed by a potential shock wave. Our goal is to predict the final energy state of the system. We propose the direct numerical integration by a Runge-Kutta method of the Schrodinger equation verified by the wave function of the system decomposed on the basis of the eigenstates. We validate the method for small values of the exciter potential, and conduct some numerical experiments. From the performance analysis we deduce that this method is not powerful enough to be generalized to three-dimensional models, but can be used as a validation tool.

The works gathered in this thesis are related to the mathematical treatment of two distinct topics in mathematical physics. In part A: minimization problems in hadronic matter physics, we find results concerning variational models used in nuclear physics to describe the strong interaction in the low energy limit. The existence of solutions realizing the energy infimum with a constraint of topological degree for different models of the Skyrme type is discussed, with emphasis on the choice of appropriate functional spaces. We use the concentration-compactness method (because of the compactness defects due to the translation invariance), and different functional analysis techniques. We then study a model with non-local coupling, the Adkins and Nappi model, in the framework of the Skyrme ansatz, using in particular the local behavior of the solutions of the Euler-Lagrange equations. In part B: kinetic equations several models describing rarefied gases or plasmas are discussed. We start by studying a modified Boltzmann equation to take into account the quantum corrections for a fermion gas (existence, uniqueness, conserved quantities, h theorem, convergence to the classical limit to the ordinary Boltzmann equation, asymptotic solutions in large time and stationary solutions). We then study the stationary Maxwellian solutions of the Vlasov-Poisson system representing charged particles subjected to the mean field created by their charge distribution and to an external electrostatic field ensuring their confinement (existence and uniqueness in Marcinkiewicz spaces). The last chapter deals with the convergence to asymptotic states in large time for the Vlasov-Poisson-Boltzmann system and the study of the corresponding stationary solutions in the case where the mass and energy are conserved.

The first part presents in detail the practical realization of a simple inverse pendulum and the feasibility study of a double inverse pendulum using the linear stochastic control theory. The originality of these pendulums is the choice of a low-end hardware, associated with a rather complex, multi-task and real time assembler programming. The power of the motors being limited, the tests on the simple pendulum show that the difficulty lies in the constraint on the control. After a general study of the feedbacks minimizing the norm of the control, on the set of stable commands, the study of the global stochastic system allows to estimate the minimal acceleration necessary for the double pendulum and to conclude that it is not feasible. The second part presents the theoretical and numerical study of a tracking game, more precisely the solution of the Isaacs equation of this differential game, thanks to the notion of viscosity solution. Moreover, it is a game modeling a pursuit in a given domain, i.e. with constraints on the edge of the domain. The value functions of this game verify the dynamic programming, are continuous, and are viscosity solutions of the same Isaacs equation. This equation with boundary conditions has a unique viscosity solution. The monotonic schemes, with differential form, and consistent with the equation, allow to approximate the solutions. The numerical codes then provide the value function of the set and thus the optimal trajectories for any initial condition.

This work studies some mathematical properties of different models from atomic physics on the one hand, and from the electrodynamics of nonlinear continuum on the other hand.