The research work presented in this thesis is related to the development of structural optimisation algorithms based on the boundary element and level set methods for two and three-dimensional linear elastic problems. In the initial implementation, a stress based evolutionary structural optimisation (ESO) approach has been used to add and remove material simultaneously for the solution of two-dimensional optimisation problems. The level set method (LSM) is used to provide an implicit description of the structural geometry, which is also capable of automatically handling topological changes, i.e. holes merging with each other or with the boundary. The classical level set based optimisation methods are dependent on initial designs with pre-existing holes. However, the proposed method automatically introduces internal cavities utilising a stress based hole insertion criteria, and thereby eliminates the use of initial designs with pre-existing holes. A detailed study has also been carried out to investigate the relationship between a stress and topological derivative based hole insertion criteria within a boundary element method (BEM) and LSM framework. The evolving structural geometry (i.e. the zero level set contours) is represented by non-uniform rational b-splines (NURBS), providing a smooth geometry throughout the optimisation process and completely eliminating jagged edges. The BEM and LSM are further combined with a shape sensitivity approach for the solution of minimum compliance problems in two-dimensions. The proposed sensitivity based method is capable of automatically inserting holes during the optimisation process using a topological derivative approach. In order to investigate the associated advantages and disadvantages of the evolutionary and sensitivity based optimisation methods a comparative study has also been carried out. There are two advantages associated with the use of LSM in three-dimensional topology optimisation. Firstly, the LSM may readily be applied to three-dimensional space, and it is shown how this can be linked to a 3D BEM solver. Secondly, the holes appear automatically through the intersection of two surfaces moving towards each other. Therefore, the use of LSM eliminates the need for an additional hole insertion mechanism as both shape and topology optimisation can be performed at the same time. A complete algorithm is proposed and tested for BEM and LSM based topology optimisation in three-dimensions. Optimal geometries compare well against those in the literature for a range of benchmark examples.

Cette thèse essaie de reconstituer l’histoire de la réception du calcul leibnizien dans les milieux savants français (1690-1706). Nous repérons deux lieux : d’abord au sein d’un groupe autour de Malebranche, initié au calcul par Jean Bernoulli, puis à l’Académie des sciences. Dans les deux cas nous mettons en avant les horizons d’attente des acteurs. Alors que cet épisode a été beaucoup étudié en termes de rupture, nous insistons, par une analyse des sources primaires – dont plusieurs inédites – sur le fait que l’appropriation du calcul s’effectue aussi grandement sur le fond de pratiques en usage. Dans la première partie, nous examinons l’héritage mathématique à partir duquel est reçu le calcul de Leibniz par le groupe autour de Malebranche. Cette analyse nous permet de montrer que leur appropriation s’appuie sur des pratiques partagées et non sur un terrain vierge comme on l’a trop souvent supposé. Nos mathématiciens réalisent que l’algorithme différentiel permet de donner une étoffe nouvelle à des notions déjà impliquées dans les méthodes d’invention précédentes. Dans la seconde partie, nous étudions la genèse et la structuration du premier ouvrage de calcul différentiel écrit par l’Hospital et publié en 1696 sous le titre Analyse des infiniment petits pour l’intelligence des courbes. Après cette publication, le calcul devient très présent à l’Académie. Une crise y éclate entre partisans et adversaires du calcul. L’examen de leurs discours, objet de notre troisième partie, permet de préciser les notions telles que celle de différentielle ou de courbe, ainsi que la manière dont il est possible d’interpréter géométriquement les résultats issus des calculs. / This thesis is an attempt to reconstruct the reception history of Leibnizian calculus in French learned milieux (1690-1706). Two areas have been located: first among members of Malebranche’s circle, introduced to calculus by Jean Bernoulli, then the Académie des Sciences. In either case, the purpose is to highlight the horizon of expectation of the participants. Whereas this episode has been widely studied in terms of disruption, it is argued, through an analysis of primary sources –some of which un-edited– that calculus was greatly appropriated against a background of practices in use. The first chapter examines the mathematical heritage from which calculus was received by Malebranche’s circle. This analysis enables me to show that their appropriation rested on shared practices, and was not a virgin land, as has often been supposed. Our mathematicians realized that the differential algoritm fleshed out notions already involved in previous invention methods. The second chapter studies the genesis and construction of the first book of differential calculus written by L’Hospital and published in 1696, entitled Analyse des infiniment petits pour l’intelligence des courbes [Analysis of the infinitely small for the intelligence of curbs]. After this publication, calculus became very present at the Académie. A crisis arose between supporters and detractors of calculus. A close examination of their discourses –the object of my third chapter– helps clarify such notions as those of differential and curb, as well as the way it is possible to geometrically interpret the results from calculus.

This thesis concerns the behaviour of maps with a unique critical point which is either a maximum or a minimum: so-called unimodal maps. Our first main result proves that for C2+η unimodal maps with non-flat critical point we have good control on the behaviour of cross-ratios on small scales. This result, an improvement on a result of Kozlovski in [K2], proves that in many cases the negative Schwarzian condition (which is not even defined if a map is not C3) is unnecessary. This result follows recent work of Shen, van Strien and Vargas. The main tools are standard cross-ratio estimates, the usual principal nest, the Koebe Lemma, the real bounds from [SV] and the 'Yoccoz Lemma'. Our second main result concerns questions of structural stability. Prompted by the final section of Kozlovski's thesis [K1], we prove that in some cases we can characterise those points at which a small local perturbation changes the type of the map. We prove for these cases that this set of 'structurally sensitive points' is precisely the postcritical set. The main tools are the Koebe Lemma, the real bounds of [LS1], and the quasiconformal deformation argument of [K3]. The thesis is arranged in the form of two chapters dealing with each of the main results separately, followed by an appendix to prove an auxiliary result. The chapters may be read independently of each other.

A theoretical analysis is made of combustion instabilities of three combustor configurations. The equations governing aeroacoustics and combustion are derived, arriving at an acoustic analogy in terms of the pressure and total enthalpy. A solution for the acoustic analogy is determined in terms of a Green's function and initial instability results are presented for the pressure Green's function. These predictions are limited by assumptions made about the combustion zone. Finally a `reduced complexity' equation is derived accounting for a generalised combustion zone. The equation is nonlinear and furnishes limit cycle solutions for �finite amplitude burner modes. It is a generalisation to combustion flows of the Fant equation used to investigate the production of voiced speech (G Fant. Acoustic Theory of Speech Production. Mouton, The Hague, 1960). The Fant equation governs the unsteady volume ow past the flame holder which, in turn, determines the acoustics of the entire system. The equation includes a fully determinate part that depends on the geometry of the flame-holder and the thermo-acoustic system, and terms defined by integrals involving thermo-aerodynamic sources, such as the flame and vortex sound sources. Illustrative numerical results are presented for both the linearised equation and the full nonlinear equation. The linearised equation governs the growth rate of the natural acoustic modes, which are excited into instability by unsteady heat release from the flame and damped by large scale vorticity production and radiation losses. The full nonlinear equation, however, governs the 'limit cycle' formation when absorption of sound by vortex shedding at trailing edges equally opposes sound generation by the flame. Limit cycle modes are of particular interest because they cannot be captured in linear predictions and are the primary source of combustor instabilities.

This thesis investigates the revenue management (RM) problem encountered in an airport carpark of finite capacity, where the available parking spaces should be sold optimally in advance in order to maximise the revenues on a given day. Customer demand is stochastic, where random pre-booking times and stay lengths overlap with each other, a setting that generates strong inter-dependence among consecutive days and hence leads to a complex network optimisation problem. Several mathematical models are introduced to approximate the problem; a model based on a discrete-time formulation which is solved using Monte Carlo (MC) simulations and two single-resource models, the first based on a stochastic process and the other on a deterministic one, both developed in continuous-time that lead to a partial differential equation (PDE). The optimisation for the spaces is based on the expected displacement costs which are then used in a bid-price control mechanism to optimise the value of the carpark. Numerical tests are conducted to examine the methods’ performance under the network setting. Taking into account the methods’ efficiency, the computation times and the resulting expected revenues, the stochastic PDE approach is shown to be the preferable method. Since the pricing structure among operators varies, an adjusted model based on the stochastic PDE is derived in order to facilitate the solution applicable in all settings. Further, for large carparks facing high demand levels, an alternative second-order PDE model is proposed. Finally, an attempt to incorporate more information about the network structure and the inter-dependence between consecutive days leads to a weighted PDE scheme. Given a customer staying on day T, a weighting kernel is introduced to evaluate the conditional probability of stay on a neighbouring day. Then a weighted average is applied on the expected marginal values over all neighbouring days. The weighted PDE scheme shows significant improvement in revenue for small-size carparks. The use of the weighted PDE opens the possibility for new ways to approximate network RM problems and thus motivates further research in this direction.

Le travail de la thèse est constitué de deux parties indépendantes traitant toutes les deux du comportement de solutions d’équations différentielles partielles. On s’intéressera dans un premier temps aux fonctions orthogonales positives à certains espaces puis à quelques résultats de type « Ingham ». L’existence ou non de fonctions orthogonales positives à certains espaces de fonctions quasi-périodiques a d’importantes implications, en particulier pour l’étude du comportement oscillatoire des solutions d’équations de membranes vibrantes. On se propose ici de clarifier la situation d’un sous-espace défini par trois périodes et de donner des pistes de réflexion pour le cas de quatre périodes ou plus. On peut utiliser les séries de Fourier non harmoniques pour résoudre certains problèmes de contrôle en utilisant des variantes du théorème d’Ingham. On s’intéressera spécifiquement ici aux problèmes que pose la version vectorielle de ce théorème. / The existence or non-existence of positive orthogonal functions for subspaces of almost periodical function has important applications in studying the oscillatory behavior of vibrations. Cazenave, Haraux and Komornik have obtained many theorems of this type. The purpose of this work is to answer an open question formulated in the 1980’s, and to completely clarify the situation for subspaces defined by three periods. We also give some results for subspaces defined by more periods than three periods. We also prove some vectorial result for Ingham type theorems.

Cette thèse est consacrée principalement à l’étude de quelques problèmes elliptiques singuliers dans un domaine Ωɛ*, périodiquement perforé par des trous de taille ɛ. On montre l’existence et l’unicité d’une solution, pour tout ɛ fixé, ainsi que des résultats d’homogénéisation et correcteurs pour le problème singulier suivant :{█(-div (A (x/ɛ,uɛ)∇uɛ)=fζ(uɛ) dans Ωɛ*@uɛ=0 sur Γɛ0@@(A (x/ɛ,uɛ)∇uɛ)υ+ɛγρ (x/ɛ) h(uɛ)= ɛg (x/ɛ) sur Γɛ1@)┤Où l’on prescrit des conditions de Dirichlet homogènes sur la frontière extérieure Γɛ0 et des conditions de Robin non linéaires sur la frontière des trous Γɛ1. Le champ matriciel quasi linéaire A est elliptique, borné, périodique dans la primière variable et de Carathéodory. Le terme singulier non linéaire est le produit d’une fonction continue ζ (singulier en zéro) et de f, dont la sommabilité dépend de la croissance de ζ près de sa singularité. Le terme de bord non linéaire h est une fonction croissante de classe C1, ρ et g sont des fonctions périodiques non négatives avec sommabilité convenables. Pour étudier le comportement asymptotique du problème quand ɛ -> 0, on applique la méthode de l’éclatement périodique due à D. Cioranescu-A. Damlamian-G. Griso (cf. D. Cioranescu-A. Damlamian-P. Donato-G. Griso-R. Zaki pour les domaines perforés). Enfin, on montre l’existence et l’unicité de la solution faible pour la même équation, dans un domaine à deux composants Ω = Ω1 υ Ω2 υ Γ, étant Γ l’interface entre le composant connecté Ω1 et les inclusions Ω2. Plus précisément on considère{█(-div (A(x, u)∇u)+ λu=fζ(u) dans Ω\Γ,@u=0 sur δΩ@(A(x, u1)∇u1)υ1= (A(x, u2)∇u2)υ1 sur Γ,@(A(x, u1)∇u1)υ1= -h(u1-u2) sur Γ@)┤Où λ est un réel non négatif et h représente le coefficient de proportionnalité entre le flux de chaleur et le saut de la solution, et il est supposé être borné et non négatif sur Γ. / This thesis is mainly devoted to the study of some singular elliptic problems posed in perforated domains. Denoting by Ωɛ* e domain perforated by ɛ-periodic holes of ɛ-size, we prove existence and uniqueness of the solution , for fixed ɛ, as well as homogenization and correctors results for the following singular problem :{█(-div (A (x/ɛ,uɛ)∇uɛ)=fζ(uɛ) dans Ωɛ*@uɛ=0 sur Γɛ0@@(A (x/ɛ,uɛ)∇uɛ)υ+ɛγρ (x/ɛ) h(uɛ)= ɛg (x/ɛ) sur Γɛ1@)┤Where homogeneous Dirichlet and nonlinear Robin conditions are prescribed on the exterior boundary Γɛ0 and on the boundary of the holles Γɛ1, respectively. The quasilinear matrix field A is elliptic, bounded, periodic in the first variable and Carathéodory. The nonlinear singular lower order ter mis the product of a continuous function ζ (singular in zero) and f whose summability depends on the growth of ζ near its singularity. The nonlinear boundary term h is a C1 increasing function, ρ and g are periodic nonnegative functions with prescribed summabilities. To investigate the asymptotic behaviour of the problem, as ɛ -> 0, we apply the Periodic Unfolding Method by D. Cioranescu-A. Damlamian-G. Griso, adapted to perforated domains by D. Cioranescu-A. Damlamian-P. Donato-G. Griso-R. Zaki. Finally, we show existence and uniqueness of a weak solution of the same equation in a two-component domain Ω = Ω1 υ Ω2 υ Γ, being Γ the interface between the connected component Ω1 and the inclusions Ω2. More precisely we consider{█(-div (A(x, u)∇u)+ λu=fζ(u) dans Ω\Γ,@u=0 sur δΩ@(A(x, u1)∇u1)υ1= (A(x, u2)∇u2)υ1 sur Γ,@(A(x, u1)∇u1)υ1= -h(u1-u2) sur Γ@)┤Where ν1 is the unit external vector to Ω1 and λ a nonnegative real number. Here h represents the proportionality coefficient between the continuous heat flux and the jump of the solution and it is assumed to be bounded and nonnegative on Γ.

L'analyse du comportement d'un solide ou de plusieurs solides à l'intérieur d'un fluide est un problème de longue date, que l'on peut voir décrit dans de nombreux manuels classiques d'hydrodynamique. Son étude d'un point de vue mathématique a suscité une attention croissante, en particulier au cours des 15 dernières années. Ce projet de recherche vise à mettre l'accent sur plusieurs aspects de cette analyse mathématique, en particulier sur le contrôle et les problèmes asymptotiques. Un modèle simple d'évolution fluide-solide est celui d'un seul corps rigide entouré d'un fluide incompressible parfait. Le fluide est modelé par les équations d'Euler, tandis que le solide évolue selon la loi de Newton et est influencé par la pression du fluide sur la limite. L'objectif de cette thèse de doctorat consisterait en diverses études dans cette branche et, en particulier, étudierait les questions de contrôlabilité de ce système, ainsi que des modèles de limite pour les solides minces qui convergent vers une courbe. Nous souhaitons également étudier le système de contrôle Navier-Stokes / solid d'une manière similaire au problème de contrôlabilité du système Euler / solid. Une autre direction pour ce projet de doctorat est d'obtenir une limite lorsque le solide se concentre dans une courbe. Est-il possible d'obtenir un modèle simplifié d'un objet mince évoluant dans un fluide parfait, de la même manière que des modèles simplifiés ont été obtenus pour des objets qui sont petits dans toutes les directions? Cela pourrait ouvrir la voie à des recherches futures sur la dérivation des flux de cristaux liquides comme limite du système décrivant l'interaction entre le fluide et un filet de tubes solides lorsque le diamètre des tubes converge à zéro. / The analysis of the behavior of a solid or several solids inside a fluid is a long-standing problem, that one can see described in many classical textbooks of hydrodynamics. Its study from a mathematical viewpoint has attracted a growing attention, in particular in the last 15 years. This research project aims at focusing on several aspect of this mathematical analysis, in particular on control and asymptotic issues. A simple model of fluid-solid evolution is that of a single rigid body surrounded by a perfect incompressible fluid. The fluid is modeled by the Euler equations, while the solid evolves according to Newton’s law, and is influenced by the fluid’s pressure on the boundary. The goal of this PhD thesis would consist in various studies in this branch, and in particular would investigate questions of controllability of this system, as well as limit models for thin solids converging to a curve. We would also like to study the Navier-Stokes/solid control system in a similar manner to the previously discussed controllability problem for the Euler/solid system. Another direction for this PhD project is to obtain a limit when the solid concentrates into a curve. Is it possible to obtain a simplified model of a thin object evolving in a perfect fluid, in the same way as simplified models were obtained for objects that are small in all directions? This could open the way to future investigations on derivation of liquid crystal flows as the limit of the system describing the interaction between the fluid and a net of solid tubes when the diameter of the tubes is converging to zero.

Ce travail est consacré à l'étude du comportement en temps grand d'équations aux dérivées partielles de type parabolique. Plus particulièrement, on s'intéresse à des équations non linéaires de type diffusion, qui interviennent dans de nombreux modèles issus de la physique (par exemple l'équation des milieux poreux) ou de la biologie (par exemple le modèle de Patlak-Keller-Segel pour la chimiotaxie). Dans les chapitres I et II on s'intéresse à une amélioration de l'inégalité de Sobolev à travers son inégalité duale, l'inégalité de Hardy-Littlewood-Sobolev, dans le cadre du laplacien ordinaire et du laplacien fractionnaire, respectivement. Le chapitre III est un passage en revue de l'inégalité d'Onofri, qui joue le rôle de l'inégalité de Sobolev pour la dimension deux. De nouveaux résultats sont apportés, dont certains sont étendus aux variétés riemanniennes au chapitre IV. Enfin, le chapitre V traite des états stationnaires de deux modèles paraboliques, utilisés pour l'étude du déplacement de foules et la modélisation en biologie (chimiotaxie). / This work is dedicated to the study of the large time behaviour of some parabolic type partial differential equations. More specifically, we look into non linear diffusion equations that appear in a number of models arising in physics (e.g. the porous medium equation) or biology (e.g. the Patlak-Keller-Segel model for chemotaxis)Chapters I and II deal with an improved Sobolev inequality by means of its dual, the Hardy-Littlewood-Sobolev inequality, in the framework of the standard and fractional Laplacian, respectively. Chapter III is a review of the Onofri inequality,which acts as the Sobolev inequality for dimension two. New results are provided, and some of them are extended to Riemannian manifolds in Chapter IV. Finally, Chapter V deals with the stationary states of two parabolic models, used for thestudy of crowd motion and modeling in biologie (chemotaxis).

Les opérateurs de Berezin--Toeplitz permettent de quantifier des fonctions, ou des symboles, sur des variétés kähleriennes compactes, et sont définies à partir du noyau de Bergman (ou de Szeg\H{o}). Nous étudions le spectre des opérateurs de Toeplitz dans un régime asymptotique qui correspond à une limite semiclassique. Cette étude est motivée par le comportement magnétique atypique observé dans certains cristaux à basse température. Nous étudions la concentration des fonctions propres des opérateurs de Toeplitz, dans des cas où les effets sous-principaux (du même ordre que le paramètre semiclassique) permet de différencier entre plusieurs configurations classiques, un effet connu en physique sous le nom de sélection quantique Nous exhibons un critère général pour la sélection quantique et nous donnons des développements asymptotiques précis de fonctions propres dans le cas Morse et Morse--Bott, ainsi que dans un cas dégénéré. Nous développons également un nouveau cadre pour le traitement du noyau de Bergman et des opérateurs de Toeplitz en régularité analytique. Nous démontrons que le noyau de Bergman admet un développement asymptotique, avec erreur exponentiellement petite, sur des variétés analytiques réelles. Nous obtenons aussi une précision exponentiellement fine dans les compositions et le spectre d'opérateurs à symbole analytique, et la décroissance exponentielle des fonctions propres. / Berezin-Toeplitz operators allow to quantize functions, or symbols, on compact Kähler manifolds, and are defined using the Bergman (or Szeg\H{o}) kernel. We study the spectrum of Toeplitz operators in an asymptotic regime which corresponds to a semiclassical limit. This study is motivated by the atypic magnetic behaviour observed in certain crystals at low temperature. We study the concentration of eigenfunctions of Toeplitz operators in cases where subprincipal effects (of same order as the semiclassical parameter) discriminate between different classical configurations, an effect known in physics as quantum selection . We show a general criterion for quantum selection and we give detailed eigenfunction expansions in the Morse and Morse-Bott case, as well as in a degenerate case. We also develop a new framework in order to treat Bergman kernels and Toeplitz operators with real-analytic regularity. We prove that the Bergman kernel admits an expansion with exponentially small error on real-analytic manifolds. We also obtain exponential accuracy in compositions and spectra of operators with analytic symbols, as well as exponential decay of eigenfunctions.