Dissertations / Theses on the topic 'Homogeneous Neumann boundary conditions'

This Thesis proposes the use of the Dirichlet-to-Neumann (DtN) operator to improve the accuracy and the efficiency of the numerical solution of an electromagnetic scattering problem, described in terms of a differential formulation. From a general perspective, the DtN operator provides the “connection” (the mapping) between the Dirichlet and the Neumann data onto a proper closed surface. This allows truncation of the computational domain when treating a scattering problem in an unbounded media. Moreover, the DtN operator provides an exact boundary condition, in contrast to other methods such as Perfectly Matching Layer (PML) or Absorbing Boundary Conditions (ABC). In addition, when the surface where the DtN is introduced has a canonical shape, as in the present contribution, the DtN operator can be computed analytically. This thesis is focused on a 2D geometry under TM illumination. The numerical model combines a differential formulation with the DtN operator defined onto a canonical surface where it can be computed analytically. Test cases demonstrate the accuracy and the computational advantage of the proposed technique.

The work of this thesis is devoted to the development of an original and general numerical method for solving the elliptic equation in an arbitrary domain (described by a level-set function) with general boundary conditions (Dirichlet, Neumann, Robin, ...) using Cartesian grids. It can be then considered an immersed boundary method, and the scheme we use is based on a finite-difference ghost-cell technique. The entire problem is solved by an effective multigrid solver, whose components have been suitably constructed in order to be applied to the scheme. The method is extended to the more challenging case of discontinuous coefficients, and the multigrid is suitable modified in order to attain the optimal convergence factor of the whole iteration procedure. The development of the multigrid is an important feature of this thesis, since multigrid solvers for discontinuous coefficients maintaining the optimal convergence factor without depending on the jump in the coefficient and on the problem size is recently studied in literature. The method is second order accurate in the solution and its gradient. A convergence proof for the first order scheme is provided, while second order is confirmed by several numerical tests.

This dissertation investigates the development of numerical algorithms for coupling computational fluid dynamics (CFD) and computational solid dynamics (CSD) solvers, and the use of these solvers for simulating fluid-solid interaction (FSI) problems involving large deformation, shock waves, and multiphase flow. The dissertation consists of two parts. The first part investigates the use of Robin interface conditions to resolve the well-known numerical added-mass instability, which affects partitioned coupling procedures for solving problems with incompressible flow and strong added-mass effect. First, a one-parameter Robin interface condition is developed by linearly combining the conventional Dirichlet and Neumann interface conditions. Next, a numerical algorithm is developed to implement the Robin interface condition in an embedded boundary method for coupling a parallel, projection-based incompressible viscous flow solver with a nonlinear finite element solid solver. Both an analytical study and a numerical study reveal that the new algorithm can clearly outperform conventional Dirichlet-Neumann procedures in terms of both stability and accuracy, when the parameter value is carefully selected. Moreover, the studies also indicate that the optimal parameter value depends on the materials and geometry of the problem. Therefore, to efficiently solve FSI problems involving non-uniform structures, a generalized Robin interface condition is presented, in which the constant parameter is replaced by a spatially varying function that depends on the local material and geometric properties of the structure. Numerical experiments using two benchmark problems show that the spatially varying Robin interface condition can clearly improve numerical accuracy compared to the constant- parameter version with the same computational cost. The second part of this dissertation focuses on simulating complex FSI problems featuring shock waves, multiphase flow (e.g., bubbles), and shock-induced material damage and fracture. A recently developed three-dimensional computational framework is employed, which couples a multiphase, compressible CFD solver and a nonlinear finite element CSD solver using an embedded boundary method and a partitioned procedure. In particular, the CFD solver applies a level-set method to capture the evolution of the bubble surface, and the CSD solver utilizes a continuum damage mechanics model and an element erosion method to simulate the dynamic fracture of the material. Two computational studies are presented. The first one investigates the dynamic response and failure of a brittle material exposed to a prescribed shock wave. The predictive capability of the computational framework is first demonstrated by simulating a series of laboratory experiments in the context of shock wave lithotripsy. Then, a parametric study is conducted to elucidate the significant effects of the shock wave's profile on material damage. In the second study, the computational framework is applied to simulate shock-induced bubble collapse near various solid and soft materials. The reciprocal effect of the material's properties (e.g., acoustic impedance, Young's modulus) on bubble dynamics is discussed in detail.<br>Doctor of Philosophy<br>Numerical simulations that couple computational fluid dynamics (CFD) solvers and computational solid dynamics (CSD) solvers have been widely used in the solution of nonlinear fluid-solid interaction (FSI) problems underlying many engineering applications. This is primarily because they are based on partitioned solutions of fluid and solid subsystems, which facilitates the use of existing numerical methods and computational codes developed for each subsystem. The first part of this dissertation focuses on developing advanced numerical algorithms for coupling the two subsystems. The aim is to resolve a major numerical instability issue that occurs when solving problems involving incompressible, heavy fluids and thin, lightweight structures. Specifically, this work first presents a new coupling algorithm based on a one-parameter Robin interface condition. An embedded boundary method is developed to enforce the Robin interface condition, which can be advantageous in solving problems involving complex geometry and large deformation. The new coupling algorithm has been shown to significantly improve numerical stability when the constant parameter is carefully selected. Next, the constant parameter is generalized into a spatially varying function whose local value is determined by the local material and geometric properties of the structure. Numerical studies show that when solving FSI problems involving non-uniform structures, using this spatially varying Robin interface condition can outperform the constant-parameter version in both stability and accuracy under the same computational cost. In the second part of this dissertation, a recently developed three-dimensional multiphase CFD - CSD coupled solver is extended to simulate complex FSI problems featuring shock wave, bubbles, and material damage and fracture. The aim is to understand the material’s response to loading induced by a shock wave and the collapse of nearby bubbles, which is important for advancing the beneficial use of shock wave and bubble collapse for material modification. Two computational studies are presented. The first one investigates the dynamic response and failure of a brittle material exposed to a prescribed shock wave. The causal relationship between shock loading and material failure, and the effects of the shock wave’s profile on material damage are discussed. The second study investigates the shock-induced bubble collapse near various solid and soft materials. The two-way interaction between bubble dynamics and materials response, and the reciprocal effects of the material’s properties are discussed in detail.

The analytical solution of a given inhomogeneous boundary value problem of a linear, elliptic, inhomogeneous partial differential equation and a set of inhomogeneous mixed Dirichlet- and Neumann-type boundary conditions is derived in the present paper. In the context of elasticity theory, the problem arises for a non-conservative symmetric ansatz and an extended constitutive law shown earlier. For convenient user application, the scalar function expressed in cylindrical coordinates is primarily obtained for the general case before being expatiated on a special case of linear boundary conditions.

Motivée par deux problématiques liées aux plasmas de tokamak, cette thèse propose deux méthodes d'approximation numérique pour deux problèmes mathématiques s'y rattachant. D'une part, pour l'étude du transport turbulent de particules, une méthode numérique basée sur les schémas de subdivision est présentée pour la simulation de trajectoires de particules dans un champ de vitesse fortement variable. D'autre part, dans le cadre de la modélisation de l'interaction plasma-paroi, une méthode de pénalisation est proposée pour la prise en compte de conditions aux limites de type Neumann ou Robin. Analysées sur des problèmes modèles de complexité croissante, ces méthodes sont enfin appliquées dans des situations plus réalistes d'intérêt pratique dans l'étude du plasma de bord<br>Motivated by two issues related to tokamak plasmas, this thesis proposes two numerical approximation methods for two mathematical problems associated with them. On the one hand, in order to study the turbulent transport of particles, a numerical method based on subdivision schemes is presented for the simulation of particle trajectories in a strongly varying velocity field. On the other hand, in the context of modeling the plasma-wall interaction, a penalization method is proposed to take into account Neumann or Robin boundary conditions. Analyzed on model problems of increasing complexity, these methods are finally applied in more realistic situations of practical interest in the study of the edge plasma

Neste trabalho definimos o operador não local, raiz quadrada do Laplaciano com condição de fronteira de Neumann, através do método de extensão harmônica. O estudo foi feito com o auxílio das séries de Fourier em domínios limitados, como sendo o intervalo, o quadrado e a bola. Posteriormente, aplicamos nosso estudo, à problemas elípticos não lineares envolvendo o operador não local raiz quadrada do Laplaciano com condição de fronteira de Neumann.<br>In this work we define the non-local operator, square root of the Laplacian with Neumann boundary condition, using the method of harmonic extension. The study was done with the aid of Fourier series in bounded domains, as the interval, the square and the ball. Subsequently, we apply our study, the nonlinear elliptic problems involving non-local operator square root of the Laplacian with Neumann boundary condition.

The monograph is devoted to the study of nonlinear first order systems in the plane where the principal term is the gradient of a positive and positively 2-homogeneous Hamiltonian (or the convex combination of two of such gradients). After some preliminaries about positively 2-homogeneous autonomous systems, some results of existence and multiplicity of T-periodic solutions are presented in case of bounded or sublinear nonlinear perturbations. Our attention is mainly focused on the occurrence of resonance phenomena, and the corresponding results rely essentially on conditions of Landesman-Lazer or Ahmad-Lazer-Paul type. The techniques used are predominantly topological, exploiting the theory of coincidence degree and the use of the Poincaré-Birkhoff fixed point theorem. At the end, other boundary conditions, including the Sturm-Liouville ones, are taken into account, giving the corresponding existence and multiplicity results in a nonresonant situation via the shooting method and topological arguments.

Die Least-Squares Finite-Elemente-Methoden (LSFEMn) basieren auf der Minimierung des Least-Squares-Funktionals, das aus quadrierten Normen der Residuen eines Systems von partiellen Differentialgleichungen erster Ordnung besteht. Dieses Funktional liefert einen a posteriori Fehlerschätzer und ermöglicht die adaptive Verfeinerung des zugrundeliegenden Netzes. Aus zwei Gründen versagen die gängigen Methoden zum Beweis optimaler Konvergenzraten, wie sie in Carstensen, Feischl, Page und Praetorius (Comp. Math. Appl., 67(6), 2014) zusammengefasst werden. Erstens scheinen fehlende Vorfaktoren proportional zur Netzweite den Beweis einer schrittweisen Reduktion der Least-Squares-Schätzerterme zu verhindern. Zweitens kontrolliert das Least-Squares-Funktional den Fehler der Fluss- beziehungsweise Spannungsvariablen in der H(div)-Norm, wodurch ein Datenapproximationsfehler der rechten Seite f auftritt. Diese Schwierigkeiten führten zu einem zweifachen Paradigmenwechsel in der Konvergenzanalyse adaptiver LSFEMn in Carstensen und Park (SIAM J. Numer. Anal., 53(1), 2015) für das 2D-Poisson-Modellproblem mit Diskretisierung niedrigster Ordnung und homogenen Dirichlet-Randdaten. Ein neuartiger expliziter residuenbasierter Fehlerschätzer ermöglicht den Beweis der Reduktionseigenschaft. Durch separiertes Markieren im adaptiven Algorithmus wird zudem der Datenapproximationsfehler reduziert. Die vorliegende Arbeit verallgemeinert diese Techniken auf die drei linearen Modellprobleme das Poisson-Problem, die Stokes-Gleichungen und das lineare Elastizitätsproblem. Die Axiome der Adaptivität mit separiertem Markieren nach Carstensen und Rabus (SIAM J. Numer. Anal., 55(6), 2017) werden in drei Raumdimensionen nachgewiesen. Die Analysis umfasst Diskretisierungen mit beliebigem Polynomgrad sowie inhomogene Dirichlet- und Neumann-Randbedingungen. Abschließend bestätigen numerische Experimente mit dem h-adaptiven Algorithmus die theoretisch bewiesenen optimalen Konvergenzraten.<br>The least-squares finite element methods (LSFEMs) base on the minimisation of the least-squares functional consisting of the squared norms of the residuals of first-order systems of partial differential equations. This functional provides a reliable and efficient built-in a posteriori error estimator and allows for adaptive mesh-refinement. The established convergence analysis with rates for adaptive algorithms, as summarised in the axiomatic framework by Carstensen, Feischl, Page, and Praetorius (Comp. Math. Appl., 67(6), 2014), fails for two reasons. First, the least-squares estimator lacks prefactors in terms of the mesh-size, what seemingly prevents a reduction under mesh-refinement. Second, the first-order divergence LSFEMs measure the flux or stress errors in the H(div) norm and, thus, involve a data resolution error of the right-hand side f. These difficulties led to a twofold paradigm shift in the convergence analysis with rates for adaptive LSFEMs in Carstensen and Park (SIAM J. Numer. Anal., 53(1), 2015) for the lowest-order discretisation of the 2D Poisson model problem with homogeneous Dirichlet boundary conditions. Accordingly, some novel explicit residual-based a posteriori error estimator accomplishes the reduction property. Furthermore, a separate marking strategy in the adaptive algorithm ensures the sufficient data resolution. This thesis presents the generalisation of these techniques to three linear model problems, namely, the Poisson problem, the Stokes equations, and the linear elasticity problem. It verifies the axioms of adaptivity with separate marking by Carstensen and Rabus (SIAM J. Numer. Anal., 55(6), 2017) in three spatial dimensions. The analysis covers discretisations with arbitrary polynomial degree and inhomogeneous Dirichlet and Neumann boundary conditions. Numerical experiments confirm the theoretically proven optimal convergence rates of the h-adaptive algorithm.

Cette thèse se décompose en trois grandes parties, qui traitent des EDP quasi linéaires paraboliques sur une jonction, des diffusions stochastiques sur une jonction, et du contrôle optimal également sur une jonction, avec contrôle au point de jonction. Nous commençons au premier Chapitre par introduire une nouvelle classe d'EDP non dégénérée et quasi linéaire, satisfaisant une condition de Neumann (ou de Kirchoff) non linéaire et non dynamique au point de jonction. Nous prouvons l'existence d'une solution classique, ainsi que son unicité. L'une des motivations portant sur l'étude de ce type d'EDP, est de faire le lien avec la théorie du contrôle optimale sur les jonctions, et de caractériser la fonction valeur de ce type de problème à l'aide des équations d'Hamilton Jacobi Bellman. Ainsi, au Chapitre suivant, nous formulons une preuve donnant l'existence d'une diffusion sur une jonction. Ce processus admet un temps local, dont l'existence et la variation quadratique dépendent essentiellement de l'hypothèse d'ellipticité des termes du second ordre au point de jonction. Nous formulerons une formule d'Itô pour ce processus. Ainsi, grâce aux résultats de ces deux Chapitres, nous formulerons dont le dernier Chapitre un problème de contrôle stochastique sur les jonctions, avec contrôle au point de jonction. L'espace des contrôles est celui des mesures de Probabilités résolvant un problème martingale. Nous prouvons la compacité de l'espace des contrôles admissibles, ainsi que le principe de la programmation dynamique<br>This thesis consists of three parts which deal with quasi linear parabolic PDE on a junction, stochastic diffusion on a junction and stochastic control on a junction with control at the junction point. We begin in the first Chapter by introducing and studying a new class of non degenerate quasi linear parabolic PDE on a junction, satisfying a Neumann (or Kirchoff) non linear and non dynamical condition at the junction point. We prove the existence and the uniqueness of a classical solution. The main motivation of studying this new mathematical object is the analysis of stochastic control problems with control at the junction point, and the characterization of the value function of the problem in terms of Hamilton Jacobi Bellman equations. For this end, in the second Chapter we give a proof of the existence of a diffusion on a junction. The process is characterized by its local time at the junction point, whose quadratic approximation is centrally related to the ellipticty assumption of the second order terms around the junction point.We then provide an It's formula for this process. Thanks to the previous results, in the last Chapter we study a problem of stochastic control on a junction, with control at the junction point. The set of controls is the set of the probability measures (admissible rules) satisfying a martingale problem. We prove the compactness of the admissible rules and the dynamic programming principle

碩士<br>國立中興大學<br>應用數學系<br>85<br>We exploit symmetries in certain semilinear elliptic eigenvalue problems withNeumann boundary conditions for the continuation of solution curves. We showthat symmetry makes the problem decomposable into small ones, and thediscretization matrix obtained via central differences associated to theLaplacian is similar to a symmetric one. Furthermore, the discrete problemspreserve some basic properties on eigenvalues of the continuous problems.Thus the continuation-Lanczos algorithm can be adapted to trace the solutioncurves of the reduced problems. Sample numerical results are reported.

碩士<br>國立臺灣科技大學<br>營建工程系<br>103<br>This article is based on Beam Theory and is shown to establish the concept of functionally graded beam transforms into equivalent homogeneous beam. Moreover, extending the concept of relationship between functionally graded material and equivalent homogeneous material to FGM plate, FGM coated plate and FGM undercoated plate to complete this paper.Using beam moment-curvature equation derived the relationship between functionally graded beam and equivalent homogeneous beam, and solved beam buckling solution by Euler’s Column Formula. Using concept of equivalent homogeneous beam for FGM plate material coefficient transforms into equivalent homogeneous plate. The Young’s modulus of the beam are assumed to be graded continuously in the direction z of thickness. The variation of the Young’s modulus follows a power-law, sigmoid function and exponential function distribution in terms of the volume fractions of functionally graded material. It’s assumed that the Poisson’s ratio of the beam remain constant. Two kinds of boundary conditions are considered. One is to subjected simple-support at the opposite of x-axis and another is cantilever-support support at the opposite of x-axis.To check the accuracy of equivalent homogeneous plate analytical buckling solution, the numerical buckling solutions by MARC program of finite element method are obtained and compare with the theoretical buckling solution of literature. The result shows that the error of buckling load of finite element method is reasonable in FGM plates and equivalent homogeneous plates compare with theoretical value, it means FGM material can be replaced by equivalent homogeneous materials. Young’s modulus distribution curve of S-FGM is the best to resist the critical buckling load, second is E-FGM, the last is P-FGM. Reduce material index can improve the stiffness of member in the same material distribution curve. Add conditions of constraint can make critical buckling load increase under different boundary conditions. FGM coated plate is the best type to resist the critical buckling load. It’s the most effective plate type to relief the stress concentration problem in whole kinds of plates.

碩士<br>國立臺灣科技大學<br>營建工程系<br>104<br>The main purpose of this article is utilizing Equivalent Homogeneous Beam Theory to analyze the buckling behavior of FGM rectangular plates with uniform elastic support under axial load. Using Equivalent Homogeneous Beam Theory to establish the relationship between material coefficient C11 to analyze the equivalent homogeneous results of FGM plates. The research is concentrated on three kinds of material distribution type which are power-law function (P-FGM), sigmoid function (S-FGM) and exponential function (E-FGM). Two kinds of boundary conditions are considered. One is simply-support at the opposite of x-axis, the others is simply-support at the x-axis with fixed end at the opposite of y axis. Moreover, extending the concept of equivalent homogeneous to FGM coated plates and FGM undercoated plates and calculated by finite element method using MARC program. To prove the accuracy of solution, compare the results with FGM plates and FGM theory value which was researched by Yu-Ting Cheng[32]. Discuss the behavior of buckling load under different kinds of parameters such as material index, stiffness of elastic support…etc. The result shows that buckling load will change seriously under same material index and Young’s modulus, equivalent homogeneous plates behave same as functionally graded material plates, increasing stiffness of elastic support could improve the buckling load of all structure and S-FGM coated plates is the best material to resist the buckling load under three kinds of material distribution and plate type.

碩士<br>國立新竹教育大學<br>應用數學系碩士班<br>98<br>Based on the master stability function (MSF) [7] for local synchronization in coupled chaotic systems, the stability of chaotic synchronization is actually controlled by the second largest eigenvalue of the coupling matrix of coupled chaotic systems. In addition, it is demonstrated that the wavelet transform method, which is proposed by Wei et al.[11], can greatly increase the applicable ranges of coupling strengths for local synchronization of coupled chaotic systems. In this research, there are two-fold. First, the concept of the wavelet transform method by using median to improve the best choice of wavelet parameters is proposed. Second, we give an application to the individual chaotic system diusively coupled with Neumann boundary conditions.

Les façons d'aborder l'étude du spectre du laplacien sont multiples. Ce mémoire se concentre sur les partitions spectrales optimales de domaines planaires. Plus précisément, lorsque nous imposons des conditions aux limites de Dirichlet, nous cherchons à trouver la ou les partitions qui réalisent l'infimum (sur l'ensemble des partitions à un certain nombre de composantes) du maximum de la première valeur propre du laplacien sur tous ses sous-domaines. Dans les dernières années, cette question a été activement étudiée par B. Helffer, T. Hoffmann-Ostenhof, S. Terracini et leurs collaborateurs, qui ont obtenu plusieurs résultats analytiques et numériques importants. Dans ce mémoire, nous proposons un problème analogue, mais pour des conditions aux limites de Neumann cette fois. Dans ce contexte, nous nous intéressons aux partitions spectrales maximales plutôt que minimales. Nous cherchons alors à vérifier le maximum sur toutes les $k$-partitions possibles du minimum de la première valeur propre non nulle de chacune des composantes. Cette question s'avère plus difficile que sa semblable dans la mesure où plusieurs propriétés des valeurs propres de Dirichlet, telles que la monotonicité par rapport au domaine, ne tiennent plus. Néanmoins, quelques résultats sont obtenus pour des 2-partitions de domaines symétriques et des partitions spécifiques sont trouvées analytiquement pour des domaines rectangulaires. En outre, des propriétés générales des partitions spectrales optimales et des problèmes ouverts sont abordés.<br>There exist many ways to study the spectrum of the Laplace operator. This master thesis focuses on optimal spectral partitions of planar domains. More specifically, when imposing Dirichlet boundary conditions, we try to find partitions that achieve the infimum (over all the partitions of a given number of components) of the maximum of the first eigenvalue of the Laplacian in all the subdomains. This question has been actively studied in recent years by B. Helffer, T. Hoffmann-Ostenhof, S. Terracini and their collaborators, who obtained a number of important analytic and numerical results. In the present thesis we propose a similar problem, but for the Neumann boundary conditions. In this case, we are looking for spectral maximal, rather than minimal, partitions. More precisely, we attempt to find the maximum over all possible $k$-partitions of the minimum of the first non-zero Neumann eigenvalue of each component. This question appears to be more difficult than the one for the Dirichlet conditions, since many properties of Dirichlet eigenvalues, such as domain monotonicity, no longer hold in the Neumann case. Nevertheless, some results are obtained for 2-partitions of symmetric domains, and specific partitions are found analytically for rectangular domains. In addition, some general properties of optimal spectral partitions and open problems are also discussed.

La présente thèse porte sur différentes questions émanant de la géométrie spectrale. Ce domaine des mathématiques fondamentales a pour objet d'établir des liens entre la géométrie et le spectre d'une variété riemannienne. Le spectre d'une variété compacte fermée M munie d'une métrique riemannienne $g$ associée à l'opérateur de Laplace-Beltrami est une suite de nombres non négatifs croissante qui tend vers l’infini. La racine carrée de ces derniers représente une fréquence de vibration de la variété. Cette thèse présente quatre articles touchant divers aspects de la géométrie spectrale. Le premier article, présenté au Chapitre 1 et intitulé « Superlevel sets and nodal extrema of Laplace eigenfunctions », porte sur la géométrie nodale d'opérateurs elliptiques. L’objectif de mes travaux a été de généraliser un résultat de L. Polterovich et de M. Sodin qui établit une borne sur la distribution des extrema nodaux sur une surface riemannienne pour une assez vaste classe de fonctions, incluant, entre autres, les fonctions propres associées à l'opérateur de Laplace-Beltrami. La preuve fournie par ces auteurs n'étant valable que pour les surfaces riemanniennes, je prouve dans ce chapitre une approche indépendante pour les fonctions propres de l’opérateur de Laplace-Beltrami dans le cas des variétés riemanniennes de dimension arbitraire. Les deuxième et troisième articles traitent d'un autre opérateur elliptique, le p-laplacien. Sa particularité réside dans le fait qu'il est non linéaire. Au Chapitre 2, l'article « Principal frequency of the p-laplacian and the inradius of Euclidean domains » se penche sur l'étude de bornes inférieures sur la première valeur propre du problème de Dirichlet du p-laplacien en termes du rayon inscrit d’un domaine euclidien. Plus particulièrement, je prouve que, si p est supérieur à la dimension du domaine, il est possible d'établir une borne inférieure sans aucune hypothèse sur la topologie de ce dernier. L'étude de telles bornes a fait l'objet de nombreux articles par des chercheurs connus, tels que W. K. Haymann, E. Lieb, R. Banuelos et T. Carroll, principalement pour le cas de l'opérateur de Laplace. L'adaptation de ce type de bornes au cas du p-laplacien est abordée dans mon troisième article, « Bounds on the Principal Frequency of the p-Laplacian », présenté au Chapitre 3 de cet ouvrage. Mon quatrième article, « Wolf-Keller theorem for Neumann Eigenvalues », est le fruit d'une collaboration avec Guillaume Roy-Fortin. Le thème central de ce travail gravite autour de l'optimisation de formes dans le contexte du problème aux valeurs limites de Neumann. Le résultat principal de cet article est que les valeurs propres de Neumann ne sont pas toujours maximisées par l'union disjointe de disques arbitraires pour les domaines planaires d'aire fixée. Le tout est présenté au Chapitre 4 de cette thèse.<br>The main topic of the present thesis is spectral geometry. This area of mathematics is concerned with establishing links between the geometry of a Riemannian manifold and its spectrum. The spectrum of a closed Riemannian manifold M equipped with a Riemannian metric g associated with the Laplace-Beltrami operator is a sequence of non-negative numbers tending to infinity. The square root of any number of this sequence represents a frequency of vibration of the manifold. This thesis consists of four articles all related to various aspects of spectral geometry. The first paper, “Superlevel sets and nodal extrema of Laplace eigenfunction”, is presented in Chapter 1. Nodal geometry of various elliptic operators, such as the Laplace-Beltrami operator, is studied. The goal of this paper is to generalize a result due to L. Polterovich and M. Sodin that gives a bound on the distribution of nodal extrema on a Riemann surface for a large class of functions, including eigenfunctions of the Laplace-Beltrami operator. The proof given by L. Polterovich and M. Sodin is only valid for Riemann surfaces. Therefore, I present a different approach to the problem that works for eigenfunctions of the Laplace-Beltrami operator on Riemannian manifolds of arbitrary dimension. The second and the third papers of this thesis are focused on a different elliptic operator, namely the p-Laplacian. This operator has the particularity of being non-linear. The article “Principal frequency of the p-Laplacian and the inradius of Euclidean domains” is presented in Chapter 2. It discusses lower bounds on the first eigenvalue of the Dirichlet eigenvalue problem for the p-Laplace operator in terms of the inner radius of the domain. In particular, I show that if p is greater than the dimension, then it is possible to prove such lower bound without any hypothesis on the topology of the domain. Such bounds have previously been studied by well-known mathematicians, such as W. K. Haymann, E. Lieb, R. Banuelos, and T. Carroll. Their papers are mostly oriented toward the case of the usual Laplace operator. The generalization of such lower bounds for the p-Laplacian is done in my third paper, “Bounds on the Principal Frequency of the p-Laplacian”. It is presented in Chapter 3. My fourth paper, “Wolf-Keller theorem of Neumann Eigenvalues”, is a joint work with Guillaume Roy-Fortin. This paper is concerned with the shape optimization problem in the case of the Laplace operator with Neumann boundary conditions. The main result of our paper is that eigenvalues of the Neumann boundary problem are not always maximized by disks among planar domains of given area. This joint work is presented in Chapter 4.

Title: Bifurcation of Ordinary Differential Equations from Points of Fučík Spectrum Author: Vendula Exnerová Department: Department of Mathematical Analysis Supervisor: doc. RNDr. Jana Stará, CSc., Department of Mathematical Analysis MFF UK, Prague Abstract: The main subject of the thesis is the Fučík spectrum of a system of two differential equations of the second order with mixed boundary conditions. In the first part of the thesis there are described Fučík spectra of problems of a differential equation with Dirichlet, mixed and Neumann boundary conditions. The other part deals with systems of two differential equations. It attends to basic properties of systems and their nontrivial solutions, to a possibility of a reduction of number of parameters and to a dependance of a problem with mixed boundary condition on one with Dirichlet boundary conditions. The thesis takes up the results of E. Massa and B. Ruff about the Dirichlet problem and improves some of their proofs. In the end the Fučík spectrum of a problem with mixed boundary conditions is described as the union of countably many continuously differentiable surfaces and there is proven that this spectrum is closed.