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

We study the Ambarzumyan problem for Sturm-Liouville operator defined on graph. The classical Ambarzumyan Theorem states that if the Neumann eigenvalues of the Sturm-Liouville operator defined on the interval [0,£k] are exactly {n^2: n ∈ N ⋃ {0} }, then the potential q=0. In 2005, Pivovarchik proved two similar theorems with uniform lengths a for the Sturm-Liouville operator defined on a 3-star graphs. Then Wu considered the Ambarzumyan problem for graphs of nonuniform length in his thesis. In this thesis, we shall study the Ambarzumyan problem on more complicated trees, namely, 4-star graphs and caterpillar graphs with edges of different lengths. We manage to solve the Ambarzumyan problem for both Neumann eigenvalues and Dirichlet eigenvalues. In particular, the whole spectrum can be partitioned into several parts. Each part forms the solution to one Ambarzumyan problem. For example, for a 4-star graphs with edge lengths a, a, 2a, 2a form the solution to 3 different Ambarzumyan problems for the Neumann eigenvalues.

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.

Baigiamajame darbe išnagrinėta antros eilės paprastoji diferencialinė lygtis. Tam panaudojant faktorizacijos metodą. Taip pat sukonstruota sprendinių struktūra. Gautus sprendinius apibendrina suformuotos teoremos. / In this work, we study the ordinary differential equation of the second – order. Using the factorization method. We constructed the structure of solutions. At the result summarize the theorems.

This thesis provides a study of various boundary problems for one and two dimensional random walks. We first consider a one-dimensional random walk that starts at integer-valued height k > 0, with a lower boundary being the x-axis, and on each step moving downward with probability q being greater than or equal to the probability of going upward p. We derive the variance and the standard deviation of the number of steps T needed for the height to reach 0 from k, by first deriving the moment generating function of T. We then study two types of two-dimensional random walks with four boundaries. A Type I walk starts at integer-valued coordinates (h; k), where0 < h < m and 0 < k < n. On each step, the process moves one unit either up, down, left, or right with positive probabilities pu, pd, pl, pr, respectively, where pu + pd + pl + pr = 1. The process stops when it hits a boundary. A Type II walk is similar to a Type I walk except that on each step, the walk moves diagonally, either left and upward, left and downward, right and downward, or right and upward with positive probabilities plu, pld, prd, pru, respectively. We mainly answer two questions on these two types of two-dimensional random walks: (1) What is the probability of hitting one boundary before the others from an initial starting point? (2) What is the average number of steps needed to hit a boundary? To do so, we introduce a Markov Chains method and a System of Equations method. We then apply the obtained results to a boundary problem involving two independent one-dimensional random walks and answer various questions that arise. Finally, we develop a conjecture to calculate the probability of a two-sided downward-drifting Type II walk with even-valued starting coordinates hitting the x-axis before the y-axis, and we test the result with Mathematica simulations

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.