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The boundary element method (BEM) has become a powerful method for the numerical solution of boundary-value problems (BVPs), due to its ability (at least for problems with constant coefficients) of reducing a BVP for a linear partial differential equation (PDE) defined in a domain to an integral equation defined on the boundary, leading to a simplified discretisation process with boundary elements only. On the other hand, the coefficients in the mathematical model of a physical problem typically correspond to the material parameters of the problem. In many physical problems, the governing equation is likely to involve variable coefficients. The application of the BEM to these equations is hampered by the difficulty of finding a fundamental solution. The first part of this thesis will focus on the derivation of the boundary integral equation (BIE) for the Laplace equation, and numerical results are presented for some examples using constant elements. Then, the formulations of the boundary-domain integral or integro-differential equation (BDIE or BDIDE) for heat conduction problems with variable coefficients are presented using a parametrix (Levi function), which is usually available. The second part of this thesis deals with the extension of the BDIE and BDIDE formulations to the treatment of the two-dimensional Helmholtz equation with variable coefficients. Four possible cases are investigated, first of all when both material parameters and wave number are constant, in which case the zero-order Bessel function of the second kind is used as fundamental solution. Moreover, when the material parameters are variable (with constant or variable wave number), a parametrix is adopted to reduce the Helmholtz equation to a BDIE or a BDIDE. Finally, when material parameters are constant (with variable wave number), the standard fundamental solution for the Laplace equation is used in the formulation. In the third part, the radial integration method (RIM) is introduced and discussed in detail. Modifications are introduced to the RIM, particularly the fact that the radial integral is calculated by using a pure boundary-only integral which relaxes the “star-shaped” requirement of the RIM. Then, the RIM is used to convert the domain integrals appearing in both BDIE and BDIDE for heat conduction and Helmholtz equations to equivalent boundary integrals. For domain integrals consisting of known functions the transformation is straightforward, while for domain integrals that include unknown variables the transformation is accomplished with the use of augmented radial basis functions (RBFs). The most attractive feature of the method is that the transformations are very simple and have similar forms for both 2D and 3D problems. Finally, the application of the RIM is discussed for the diffusion equation, in which the parabolic PDE is initially reformulated as a BDIE or a BDIDE and the RIM is used to convert the resulting domain integrals to equivalent boundary integrals. Three cases have been investigated, for homogenous, non-homogeneous and variable coefficient diffusion problems.

A numerical implementation of the direct Boundary-Domain Integral Equation (BDIE)/ Boundary-Domain Integro-Differential Equations (BDIDEs) and Localized Boundary-Domain Integral Equation (LBDIE)/Localized Boundary-Domain Integro-Differential Equations (LBDIDEs) related to the Neumann and Dirichlet boundary value problem for a scalar elliptic PDE with variable coefficient is discussed in this thesis. The BDIE and LBDIE related to Neumann problem are reduced to a uniquely solvable one by adding an appropriate perturbation operator. The mesh-based discretisation of the BDIE/BDIDEs and LBDIE/LBDIDEs with quadrilateral domain elements leads to systems of linear algebraic equations (discretised BDIE/BDIDEs/LBDIE/BDIDEs). Then the systems obtained from BDIE/BDIDE (discretised BDIE/BDIDE) are solved by the LU decomposition method and Neumann iterations. Convergence of the iterative method is analyzed in relation with the eigen-values of the corresponding discrete BDIE/BDIDE operators obtained numerically. The systems obtained from LBDIE/LBDIDE (discretised LBDIE/LBDIDE) are solved by the LU decomposition method as the Neumann iteration method diverges.

Les plantes croissent de manière continue tout au long de leur vie. Elles sont notamment capablesde produire de nouveaux axes de croissance, ce qui nécessite la mise place d’une zone frontière, induitepar l’expression des facteurs de transcription CUP-SHAPED COTYLEDON 1-3 (CUC). Au cours de mathèse, j’ai utilisé les dents formées à la marge des feuilles chez Arabidopsis thaliana comme un modèlepour mieux comprendre le rôle du réseau régulateur centré sur les gènes CUC au cours de lamorphogenèse.La première partie de mon travail a consisté en l’étude des processus en aval de CUC2, le principalrégulateur de la formation des dents. Grâce à l’utilisation d’un système d’expression inductible pour CUC2combiné à des analyses morphométriques et à la quantification de gènes rapporteurs, j’ai montré queCUC2 agit comme un déclencheur primaire et quantitatif de la formation des dents. Plusieurs relaisagissent en aval de CUC2, à des moments et dans des domaines différents, et ensemble permettent à ladent de continuer de croitre.Dans une seconde partie de mon travail, j’ai identifié et caractérisé des régulateurs en amont desgènes CUC. En suivant une approche candidat, j’ai montré que le microARN miR164 et le complexepolycombe PRC2 interagissent et contrôlent finement l’expression de CUC2. De plus, j’ai réalisé un criblesimple hybride en levure suivi d’expériences de validation in planta pour identifier de nouveauxrégulateurs de l’expression des gènes CUC/MIR164. Enfin, j’ai initié une validation fonctionnelle pourcertains de ces nouveaux candidats et montré qu’il s’agit de régulateurs généraux de l’architecture de lapartie aérienne. En décryptant les mécanismes en amont et en aval des gènes CUC, ce travail a permis demettre en évidence de nouveaux aspects de la mise en place des zones frontières et de la manière dont elles régulent l’architecture des plantes. / Throughout their lives, plants are able to produce new axes by differential growth. The formationof such new growth axes depends on the establishment of a boundary domain, which requires the CUPSHAPEDCOTYLEDON 1-3 (CUC) transcription factors. In this work, I used the small outgrowthsformed at the margin of Arabidopsis thaliana leaves as a model to decipher the CUC-centered networkregulating morphogenesis.In the first part of my work, I focused on the events downstream of CUC2, the master regulator ofleaf margin morphogenesis. Using conditional CUC2 expression combined with morphometric analysesand quantification of reporter genes, I showed that CUC2 functions as a primary and quantitative triggerfor morphogenesis. This trigger then acts through multiple relays, which actions spatially and temporallydiffer, and together allow sustained differential growth.In the second part of this work, I identified and characterized upstream regulators of the CUCgenes. In a candidate-based approach, I showed that miR164 and the polycomb complex PRC2 interact totightly control CUC2 expression. Next, I uncovered new potential transcriptional regulators of theCUC/MIR164 genes through a yeast one-hybrid screen followed by an in planta assay. Finally, I initiateda functional study for some of these candidates, which showed that they are general regulators of shootarchitecture. By revealing upstream and downstream components of the CUC-centered network, this workprovides new insights into how boundaries are regulated and how they shape plants.

In this thesis, we study homogenization of optimal control problems in various oscillatory domains. Specifically, we consider four types of domains given in Figure 1 below. Figure 1: Oscillating Domains The thesis is organized into six chapters. Chapter 1 provides an introduction to our work and the rest of the thesis. The main contributions of the thesis are contained in Chapters 2-5. Chapter 6 presents the conclusions of the thesis and possible further directions. A brief description of our work (Chapters 2-5) follows: Chapter 2: Asymptotic behaviour of a fourth order boundary optimal control problem with Dirichlet boundary data posed on an oscillating domain as in Figure 1(A) is analyzed. We use the unfolding operator to study the asymptotic behavior of this problem. Chapter 3: Homogenization of a time dependent interior optimal control problem on a branched structure domain as in Figure 1(B) is studied. Here we pose control on the oscillating interior part of the domain. The analysis is carried out by appropriately defined unfolding operators suitable for this domain. The optimal control is characterized using various unfolding operators defined at each branch of every level. Chapter 4: A new unfolding operator is developed for a general oscillating domain as in Figure 1(C). Homogenization of a non-linear elliptic problem is studied using this new un-folding operator. Using this idea, homogenization of an optimal control problem on a circular oscillating domain as in Figure 1(D) is analyzed. Chapter 5: Homogenization of a non-linear optimal control problem posed on a smooth oscillating domain as in Figure 1(C) is studied using the unfolding operator.