This paper considers a free energy functional and corresponding free boundary problem for multilayered structures which arise from a mixture of a block copolymer and a weak solvent. The free boundary problem is formally derived from the limit of large solvent/polymer segregation and intermediate segregation between monomer species. A change of variables based on Legendre transforms of the effective bulk energy is used to explicitly construct a family of equilibrium solutions. The second variation of the effective free energy of these solutions is shown to be positive. This result is used to show more generally that equilibria are local minimizers of the free energy.

We analyze the geometry and regularity of the largest subsolution of a Free Boundary Problem. We showed that the largest subsolution is a viscosity solution of (1) with Lipschitz and Non-Degenerate properties under a very general free boundary condition. In addition to this, we provide density bounds for the positivity set and its complement near the free boundary. / text

The main contribution of this thesis is to examine binary options within the British payoff mechanism introduced by Peskir and Samee. This includes British cash-or-nothing put, British asset-or-nothing put, British binary call and American barrier binary options. We assume the geometric Brownian motion model and reduce the optimal stopping problems to free-boundary problems under the Markovian nature of the underlying process. With the help of the local time-space formula on curves, we derive a closed form expression for the arbitrage-free price in terms of the rational exercise boundary and show that the rational exercise boundary itself can be characterised as the unique solution to a non-linear integral equation. We begin by investigating the binary options of American-type which are also called `one-touch' binary options. Then we move on to examine the British binary options. Chapter~2 reviews the existing work on all different types of the binary options and sets the background for the British binary options. We price and analyse the American-type (one-touch) binary options using the risk-neutral probability method. In Chapters~3 ~4 and ~5, we present the British binary options where the holder enjoys the early exercise feature of American binary options whereupon his payoff is the `best prediction' of the European binary options payoff under the hypothesis that the true drift equals a contract drift. Based on the observed price movements, if the option holder finds that the true drift of the stock price is unfavourable then he can substitute it with the contract drift and minimise his losses. The key to the British binary option is the protection feature as not only can the option holder exercise at unfavourable stock price to a substantial reimbursement of the original option price (covering the ability to sell in a liquid option market completely endogenously) but also when the stock price movements are favourable he will generally receive high returns. Chapters~3 and~4 focus on the British binary put options and Chapter~5 on call options. We also analyse the financial meaning of the British binary options and show that with the contract drift properly selected the British binary options become very attractive alternatives to the classic European/American options. Chapter~6 extends the binary options into barrier binary options and discusses the application of the optimal structure without a smooth-fit condition in the option pricing. We first review the existing work for the knock-in options and present the main results from the literature. Then we examine the method in \cite{dai2004knock} in the application to the knock-in binary options. For the American knock-out binary options, the smooth-fit property does not hold when we apply the local time-space formula on curves. We transfer the expectation of the local time term into a computational form under the basic properties of Brownian motion. Using standard arguments based on Markov processes, we analyse the properties of the value function.

Most models for the spread of an invasive species into a new environment are based on Fisher's reaction-diffusion equation. They assume that habitat quality is independent of the presence or absence of the invading population. Ecosystem engineers are species that modify their environment to make it (more) suitable for them. A potentially more appropriate modeling approach for such an invasive species is to adapt the well-known Stefan problem of melting ice. Ahead of the front, the habitat is unsuitable for the species (the ice); behind the front, the habitat is suitable (the open water). The engineering action of the population moves the boundary ahead (the melting). This approach leads to a free boundary problem. In this thesis, we mathematically analyze a novel free-boundary model for the spread of ecosystem engineers that was recently derived from an individual random walk model. The Stefan condition for the moving boundary is replaced by a biologically derived two-sided condition that models the movement behavior of individuals at the boundary as well as the process by which the population moves the boundary to expand their territory. We first consider the model with logistic growth and study its well-posedness. We assign a convex functional to this problem so that the evolution system governed by this convex potential is exactly the system of evolution equations describing the above model. We then apply variational and fixed-point methods to deal with this free boundary problem and prove the existence of local in-time solutions. We next study traveling wave solutions of the model with the strong Allee growth function. We use phase plane analysis to find traveling wave solutions of different types and their corresponding existence range of speed for the model with an imposed speed of the moving boundary. We then find the speeds in those ranges at which the corresponding traveling wave follows the speed of the free boundary.

The main contribution of this thesis is to derive the properties and present a closed from solution of the exotic options under some specific types of Levy processes, such as American put options, American call options, British put options, British call options and American knock-out put options under either double exponential jump-diffusion processes or one-sided exponential jump-diffusion processes. Compared to the geometric Brownian motion, exponential jump-diffusion processes can better incorporate the asymmetric leptokurtic features and the volatility smile observed from the market. Pricing the option with early exercise feature is the optimal stopping problem to determine the optimal stopping time to maximize the expected options payoff. Due to the Markovian structure of the underlying process, the optimal stopping problem is related to the free-boundary problem consisting of an integral differential equation and suitable boundary conditions. By the local time-space formula for semi-martingales, the closed form solution for the options value can be derived from the free-boundary problem and we characterize the optimal stopping boundary as the unique solution to a nonlinear integral equation arising from the early exercise premium (EEP) representation. Chapter 2 and Chapter 3 discuss American put options and American call options respectively. When pricing options with early exercise feature under the double exponential jump-diffusion processes, a non-local integral term will be found in the infinitesimal generator of the underlying process. By the local time-space formula for semi-martingales, we show that the value function and the optimal stopping boundary are the unique solution pair to the system of two integral equations. The significant contributions of these two chapters are to prove the uniqueness of the value function and the optimal stopping boundary under less restrictive assumptions compared to previous literatures. In the degenerate case with only one-sided jumps, we find that the results are in line with the geometric Brownian motion models, which extends the analytical tractability of the Black-Scholes analysis to alternative models with jumps. In Chapter 4 and Chapter 5, we examine the British payoff mechanism under one-sided exponential jump-diffusion processes, which is the first analysis of British options for process with jumps. We show that the optimal stopping boundaries of British put options with only negative jumps or British call options with only positive jumps can also be characterized as the unique solution to a nonlinear integral equation arising from the early exercise premium representation. Chapter 6 provides the study of American knock-out put options under negative exponential jump-diffusion processes. The conditional memoryless property of the exponential distribution enables us to obtain an analytical form of the arbitrage-free price for American knock-out put options, which is usually more difficult for many other jump-diffusion models.

In the first part of this dissertation, we produce and study a generalized mathematical model of solid combustion. Our generalized model encompasses two special cases from the literature: a case of negligible heat diffusion in the product, for example, when the burnt product is a foam-like substance; and another case in which diffusivities in the reactant and product are assumed equal. In addition to that, our model pinpoints the dynamics in a range of settings, in which the diffusivity ratio between the burned and unburned materials varies between 0 and 1. The dynamics of temperature distribution and interfacial front propagation in this generalized solid combustion model are studied through both asymptotic and numerical analyses. For asymptotic analysis, we first analyze the linear instability of a basic solution to the generalized model. We then focus on the weakly nonlinear case where a small perturbation of a neutrally stable parameter is taken so that the linearized problem is marginally unstable. Multiple scale expansion method is used to obtain an asymptotic solution for large time by modulating the most linearly unstable mode. On the other hand, we integrate numerically the exact problem by the Crank-Nicolson method. Since the numerical solutions are very sensitive to the derivative interfacial jump condition, we integrate the partial differential equation to obtain an integral-differential equation as an alternative condition. The result system of nonlinear algebraic equations is then solved by the Newton’s method, taking advantage of the sparse structure of the Jacobian matrix. By a comparison of our asymptotic and numerical solutions, we show that our asymptotic solution captures the marginally unstable behaviors of the solution for a range of model parameters. Using the numerical solutions, we also delineate the role of the diffusivity ratio between the burned and unburned materials. We find that for a representative set of this parameter values, the solution is stabilized by increasing the temperature ratio between the temperature of the fresh mixture and the adiabatic temperature of the combustion products. This trend is quite linear when a parameter related to the activation energy is close to the stability threshold. Farther from this threshold, the behavior is more nonlinear as expected. Finally, for small values of the temperature ratio, we find that the solution is stabilized by increasing the diffusivity ratio. This stabilizing effect does not persist as the temperature ratio increases. Competing effects produce a “cross-over” phenomenon when the temperature ratio increases beyond about 0.2. In the second part, we study the existence and decay rate of a transmission problem for the plate vibration equation with a memory condition on one part of the boundary. From the physical point of view, the memory effect described by our integral boundary condition can be caused by the interaction of our domain with another viscoelastic element on one part of the boundary. In fact, the three different boundary conditions in our problem formulation imply that our domain is composed of two different materials with one condition imposed on the interface and two other conditions on the inner and outer boundaries, respectively. These transmission problems are interesting not only from the point of view of PDE general theory, but also due to their application in mechanics. For our mathematical analysis, we first prove the global existence of weak solution by using Faedo-Galerkin’s method and compactness arguments. Then, without imposing zero initial conditions on one part of the boundary, two explicit decay rate results are established under two different assumptions of the resolvent kernels. Both of these decay results allow a wider class of relaxation functions and initial data, and thus generalize some previous results existing in the literature.

Cette thèse présente des travaux concernant des phénomènes d'invasion tumorale, aux échelles tissulaire et cellulaire. La première partie est consacrée à deux modèles mathématiques continus. Le premier est un modèle macroscopique de croissance d'un cancer du sein qui se focalise sur la description du passage du stade in situ au stade invasif. Basé sur des équations d'advection d'espèces cellulaires, il tient compte de la géométrie et de l'éventuelle dégradation des tissus, dans le cas où la tumeur produit des enzymes protéolytiques qui permettent l'invasion. Le second modèle concerne le phénomène d'invadopodia, à l'échelle de la cellule. C'est un problème d'interface mobile qui décrit le changement de morphologie des cellules pré-métastatiques qui leur permet de dégrader les tissus pour migrer dans le reste de l'organisme. Chacun de ces deux modèles tient compte des couplages forts inhérents au phénomènes biologiques en jeu.La seconde partie est consacrée aux méthodes numériques développées pour résoudre ces deux problèmes et surmonter les difficultés liées aux couplages et non linéarités. Elles sont construites sur grille cartésienne uniforme, à partir des différences finies et d'une version stabilisée de la méthode Ghost fluid. Leur particularité est de tirer pleinement parti des propriétés de superconvergence de la solution du problème de Poisson, spécifiquement étudiées afin d'aboutir à la résolution des problèmes de cancer du sein et d'invadopodia à l'ordre un ou deux, en fonction de la précision désirée. Cesméthodes peuvent également être utilisées pour résoudre d'autres problèmes d'interface mobile. / In this thesis, we present a study about phenomena of tumor invasion, at the tissues and cell scales.The first part is devoted to two continuous mathematical models. The first one is a macroscopic model for breast cancer growth, which focuses on the transition between the stage in situ and the invasive phase of growth. This model is based on advection equations for cellular species. The geometry and possible tissue damage are taken into account. Invasion occurs when the tumor cells produce proteolytic enzymes. The second model deals with the phenomenon of invadopodia, at the cell scale.This is a free boundary problem, which describes the change in morphology of pre-metastatic cells,enabling them to degrade the tissues and migrate into the rest of the body. Each of these models reflects the strong coupling of biological phenomena.The second part is devoted to numerical methods specifically developed to solve these problems and overcome coupling and nonlinearities. They are built on uniform Cartesian grids, thanks to the finite difference method, and a stabilized version of the Ghost fluid method. Their peculiarity is to take full advantage of superconvergence properties of the Poisson problem solution. These properties are specifically studied, leading to the first or second order numerical computation of the problems ofbreast cancer and invadopodia, depending on the desired accuracy. These methods can also be used to solve other free boundary problems.

We study a free boundary problem with initial data given on three-dimensional cones. In particular, we study when the free boundary is allowed to pass through the vertex of the cone, which depends on the cone’s slope, determined by a parameter c. With a mix of analytical and computational methods, we show that the free boundary may pass through the vertex of the cone when c ≤ 0.43.

La propagation d'une fissure chargée par un écoulement de fluide visqueux est un phénomène complexe où la compréhension des phénomènes mécaniques mis en jeu en pointe de fissures reste encore partielle. C'est le cas de la zone de décollement entre le solide et le fluide qui apparaît pour un certain choix de débit d'injection, de viscosité du fluide et de ténacité du matériau. Cette thèse propose une modélisation simplifiée de ce problème d'interaction fortement couplé. Dans un premier chapitre, on étudie un modèle simplifié unidimensionnel de film élastique collé sur un substrat rigide et on considère une injection de fluide visqueux entre le film et le substrat. On suppose que la propagation de la fissure est régie par la loi de Griffith. On néglige l'existence du retard possible entre le fluide et le solide et on choisit une loi de comportement non-linéaire pour le fluide visqueux. A partir d'une analyse asymptotique pour une faible viscosité, on établit une solution approchée du problème. On montre que le champ pression est singulier en pointe de fissure et on montre l'influence du débit d'injection sur la cinétique du trajet de fissuration. Dans le deuxième chapitre on propose de prendre en compte l'existence de la zone de décollement en modifiant la formulation du modèle et en le réécrivant sous la forme d'un problème d'optimisation en temps discret où les zones de décollement font partie des inconnues du problème. On valide la formulation proposée sur l'exemple analytique de l'écrasement d'une goutte par une barre rigide. On montre ensuite que cette formulation et l'algorithme lié à son implémentation sont capables de gérer l'évolution de l'écrasement de plusieurs gouttes de forme quelconque en capturant correctement les phase d'étalement des gouttes ainsi que de leur coalescence. On étend ensuite cette formulation au cas de l'écrasement d'une goutte par un film élastique. Dans le dernier chapitre, on examine la validité de l'hypothèse de lubrification utilisée en fracturation hydraulique. A l'aide de la méthode de développement asymptotique, on construit une équation de Reynolds régularisée avec des termes de gradient supérieur tenant compte de la variation spatiale de la hauteur des parois. On compare alors le comportement des champs de pression donnés par les équations de Reynolds classique et régularisée sur des exemples d'écoulement entre des conduits de formes multiples. / The crack evolution under a viscous fluid action is a complex phenomenon where the understanding of the mechanical phenomena near the crack tip is still largely limited. This is the case for the lag between the solid and the fluid front propagation which appears for some configurations of injection rate, fluid viscosity and material toughness. This thesis proposes a simplified model for this strongly coupled interaction problem.The first chapter studies a simplified one-dimensional model of a elastic film bonded to a rigid substrate. We consider a viscous fluid injection between the film and the substrate. The crack propagation is assumed to follow the Griffith's law. The existence of the lag is neglected and a non-linear behavior law is chosen for the viscous fluid. Using an asymptotic analysis, an approximate solution is established for the low viscosity case. It is shown that the pressure field diverges at the crack tip and that the kinetics of the crack is influenced by the injection rate. The second chapter proposes to take into account the existence of the lag by modifying the model formulation and rewriting it as a discrete time optimisation problem where the delamination zones are part of the unknowns of the problem. This formulation is validated for the analytical example of a drop crushed by a rigid bar. It is shown that this formulation and its implementation can manage the evolution of several drops of any shape and correctly captures the drops spreading and coalescence. This formulation is then extended to the case of a drop crushed by an elastic film. In the last chapter, the validity of the lubrication hypothesis is examinated. Using an asymptotic analysis, a regularized Reynolds equation is constructed with higher gradient terms taking into account the spatial variation of the walls height. A comparison between the pressure fields behaviour given by the classical and the regularized Reynolds equation is shown for different conducts.

In financial markets, spread option is a derivative security with two underlying assets and the payoff of the spread option depends on the difference of these assets. We consider American style spread option which allows the owners to exercise it at any time before the maturity. The complexity of pricing American spread option is that the boundary of the corresponding partial differential equation which determines the option price is unknown and the model for the underlying assets is two-dimensional.In this dissertation, we incorporate the stochasticity to the interest rate and assume that it satisfies the Vasicek model or the CIR model. We derive the partial differential equations with terminal and boundary conditions which determine the American spread option with stochastic interest rate and formulate the associated free boundary problem. We convert the free boundary problem to the linear complimentarity conditions for the American spread option, so that we can go around the free boundary and compute the option price numerically. Alternatively, we approximate the option price using methods based on the Monte Carlo simulation, including the regression-based method, the Lonstaff and Schwartz method and the dual method. We make the comparisons among the option prices derived by the partial differential equation method and Monte Carlo methods to show the accuracy of the result.