Дисертації з теми "Liouville systems"

I studied singular Liouville systems on compact surfaces from a variational point of view. I gave sufficient and necessary conditions for the existence of globally minimizing solutions, then I found min-max solutions for some particular systems. Finally, I also gave some non-existence results.

In this thesis we present a numerical algorithm to solve the singular Inverse Sturm-Liouville problems with symmetric potential functions. The singularity, which comes from the unbounded domain of the problem, is treated by considering the limiting case of the associated problem on the symmetric finite interval. In contrast to regular problems which are considered on a finite interval the singular inverse problem has an ill-conditioned structure despite of the limiting treatment. We use the regularization techniques to overcome the ill-posedness difficulty. Moreover, since the problem is nonlinear the iterative solution procedures are needed. Direct computation of the eigenvalues in iterative solution is handled via psoudespectral methods. The numerical examples of the considered problem are given to illustrate the accuracy and convergence behaviour.

In this thesis, a general pseudospectral formulation for a class of Sturm-Liouville eigenvalue problems is consructed. It is shown that almost all, regular or singular, Sturm-Liouville eigenvalue problems in the Schrö
dinger form may be transformed into a more tractable form. This tractable form will be called here a weighted equation of hypergeometric type with a perturbation (WEHTP) since the non-weighted and unperturbed part of it is known as the equation of hypergeometric type (EHT). It is well known that the EHT has polynomial solutions which form a basis for the Hilbert space of square integrable functions. Pseudospectral methods based on this natural expansion basis are constructed to approximate the eigenvalues of WEHTP, and hence the energy eigenvalues of the Schrö
dinger equation. Exemplary computations are performed to support the convergence numerically.

Thesis (Ph. D.)--University of Oregon, 2000.
Typescript. Includes vita and abstract. Includes bibliographical references (leaves 98-99). Also available for download via the World Wide Web; free to University of Oregon users. Address: http://wwwlib.umi.com/cr/uoregon/fullcit?p9963453.

Estudamos a resolubilidade global de uma classe de sistemas involutivos com n campos vetoriais suaves definidos no toro de dimensão n + 1. Obtemos uma caracterização completa para o caso desacoplado desta classe em termos de formas de Liouville e da conexidade de todos os subníveis e superníveis, no espaço de recobrimento minimal, de uma primitiva global da 1-forma associada ao sistema. Além disso, apresentamos uma situação especial na qual o sistema não é globalmente resolúvel e usamos isso para obter alguns resultados em um caso com acoplamento mais forte
We study the global solvability of a class of involutive systems with n smooth vector fields on the torus of dimension n + 1. We obtain a complete characterization for the uncoupled case of this class in terms of Liouville forms and of the connectedness of all sublevel and superlevel sets of the primitive of a certain 1-form in the minimal covering space. Also, we exhibit a special situation where the system is not globally solvable and we use this to obtain some results in a more general case

We present a new spectral problem, a generalization of the D-Kaup-Newell spectral problem, associated with the Lie algebra sl(2,R). Zero curvature equations furnish the soliton hierarchy. The trace identity produces the Hamiltonian structure for the hierarchy. Lastly, a reduction of the spectral problem is shown to have a different soliton hierarchy with a bi-Hamiltonian structure. The first major motivation of this dissertation is to present spectral problems that generate two soliton hierarchies with infinitely many commuting conservation laws and high-order symmetries, i.e., they are Liouville integrable. We use the soliton hierarchies and a non-seimisimple matrix loop Lie algebra in order to construct integrable couplings. An enlarged spectral problem is presented starting from a generalization of the D-Kaup-Newell spectral problem. Then the enlarged zero curvature equations are solved from a series of Lax pairs producing the desired integrable couplings. A reduction is made of the original enlarged spectral problem generating a second integrable coupling system. Next, we discuss how to compute bilinear forms that are symmetric, ad-invariant, and non-degenerate on the given non-semisimple matrix Lie algebra to employ the variational identity. The variational identity is applied to the original integrable couplings of a generalized D-Kaup-Newell soliton hierarchy to furnish its Hamiltonian structures. Then we apply the variational identity to the reduced integrable couplings. The reduced coupling system has a bi-Hamiltonian structure. Both integrable coupling systems retain the properties of infinitely many commuting high-order symmetries and conserved densities of their original subsystems and, again, are Liouville integrable. In order to find solutions to a generalized D-Kaup-Newell integrable coupling system, a theory of Darboux transformations on integrable couplings is formulated. The theory pertains to a spectral problem where the spectral matrix is a polynomial in lambda of any order. An application to a generalized D-Kaup-Newell integrable couplings system is worked out, along with an explicit formula for the associated Bäcklund transformation. Precise one-soliton-like solutions are given for the m-th order generalized D-Kaup-Newell integrable coupling system.

Dans cette thèse, nous nous intéressons à la contrôlabilité interne et à son coût pour une ou plusieurs équations aux dérivées partielles conservatives. ?Dans la première partie, nous introduisons et détaillons deux méthodes permettant d'estimer le coût du contrôle (et par dualité, de la constante d'observabilité) de l'équation des ondes avec potentiel $l^{\infty}$ en dimension un d'espace. La première utilise la propagation des ondes le long des caractéristiques en s'appuyant sur le rôle symétrique de la variable de temps et d'espace. La deuxième méthode repose sur la décomposition spectrale de l'équation des ondes et sur l'utilisation des inégalités d'ingham. L'estimation de la constante d'observabilité se ramène alors à l'étude d'un problème d'optimisation faisant intervenir les vecteurs propres du laplacien-dirichlet avec potentiel. Nous fournissons ensuite des propriétés qualitatives sur le minimiseurs ainsi qu'une estimation du minimum ne dépendant que de la mesure de l'ensemble d'observation. ?Dans la deuxième partie, nous étudions la contrôlabilité de certains systèmes d'équations avec un nombre de contrôles réduits, autrement dit le nombre de contrôles est plus petit que le nombre d'équations. En particulier, nous caractérisons exactement les données initiales qui peuvent être contrôlées pour des systèmes d'équations couplées de type schrödinger et nous énonçons une condition nécessaire et suffisante de type kalman pour des systèmes d'équations des ondes couplées. La preuve repose sur une méthode de contrôle fictif combinée à la résolution algébrique d'un système sous-déterminé et sur certains résultats de régularité
In this work, we focus on the internal controllability and its cost for some linear partial differential equations. In the first part, we introduce and describe two methods to provide precise estimates of the cost of control (and by duality, of the observability constant) for general one dimensional wave equations with potential. The first one is based on a propagation argument along the characteristics relying on the symmetrical roles of the time and space variables. The second one uses a spectral decomposition of the solution of the wave equation and ingham's inequalities. This relates the estimation of the observability constant to the study of an optimal problem involving dirichlet eigenfunctions of laplacian with potential. We provide some qualitative properties of the minimizers, and also precise bounds on the minimum. In the second part, we are concerned with the controllability of some systems of equations by a reduced number of controls (i.e. the number of controls is less that the number of equations). In particular, in the case of coupled systems of schrödinger equations, we exactly characterize the initial conditions that can be controlled and we give a necessary and sufficient condition of kalman type for the controllability of coupled systems of wave equations. The proof relies on the fictitious control method coupled with the proof of an algebraic solvabilityproperty for some related underdetermined system, as well as on some regularity results

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.

Dans cette thèse nous implémentons une approche Sturmienne, qui se sert de fonctions Sturmiennes généralisées (GSFs, en anglais), pour étudier l'ionisation de molécules par collisions de photons ou d'électrons. Comme l'Hamiltonian de la cible est non central, la description de l'ionisation des molécules n'est pas simple. En plus, puisque l'orientation spatiale de la molécule n'est généralement pas déterminée lors des expériences, une question importante à considérer est l'orientation aléatoire de la cible. Dans la littérature, des nombreuses méthodes théoriques ont été proposées pour traiter les molécules ; néanmoins, la plupart sont adaptées pour étudier, principalement, des états liés. Une description précise des états non-liés (continuum) des molécules reste un défi. Ici, nous proposons d'attaquer le problème avec les GSFs qui ont, par construction, un comportement asymptotique approprié au système étudié. Cette propriété permet de faire des calculs d'ionisation de façon plus efficace. Dans une première partie, nous validons l'implémentation de notre approche Sturmienne par l'étude de la photo-ionisation (PI) d'atomes. Différents potentiels effectifs sont utilisés pour décrire l'interaction de l'électron éjecté avec la cible ionisée. Les sections efficaces de PI sont calculées dans les jauges de longueur et de vitesse. Pour l'atome d'hydrogène la comparaison avec la formule analytique, indique qu'une convergence très rapide est obtenue avec un nombre modéré de GSFs. Pour He et Ne, nos résultats montrent, également, un très bon accord avec d'autres résultats théoriques et expérimentaux. Dans le cas des molécules, nous avons abordé l'orientation aléatoire avec deux stratégies : une utilise un potentiel moléculaire modèle (non-central), et l'autre un potentiel moyenné (central). Nous étudions la PI de CH4, NH3 et H2O à partir des orbitales de valence extérieure et intérieure, et aussi de SiH4 et H2S à partir des orbitales extérieures. Les sections efficaces de PI et les paramètres d'asymétrie (obtenus à partir des distributions angulaires) sont comparés avec ceux publiés dans la littérature. Nos résultats sont globalement satisfaisants et reproduisent les caractéristiques principales de ce processus d'ionisation. Dans une deuxième partie de la thèse, nous utilisons l'approche Sturmienne pour étudier l'ionisation de molécules par impact d'électrons. Pour le processus (e,2e), les sections efficaces triplement différentielles (TDCSs) sont examinées dans la première et deuxième approximation de Born, également en traitant de deux façons l'orientation aléatoire des molécules. Nous avons testé la méthode en comparent nos TDCSs pour l'atome d'hydrogène, montrant aussi son efficacité. Enfin, nous l'avons apliqué à l'ionisation de CH4, H2O et NH3, et nous avons comparé les résultats avec des données expérimentales et théoriques disponibles dans la littérature. Dans la plupart des cas, nos TDCSs sont en accord satisfaisant avec ces données, en particulier pour H2O et pour des électrons lents dans le cas de CH4
In this PhD thesis we implement a Sturmian approach, based on generalized Sturmian functions (GSFs), to study the ionization of molecules by collision with photons or electrons. Since the target Hamiltonian is highly non-central, describing molecular ionization is far from easy. Besides, as the spatial orientation of the molecule in most experimental measurements is not resolved, an important issue to take into account is its random orientation. In the literature, many theoretical methods have been proposed to deal with molecules, but many of them are adapted to study mainly bound states. An accurate description of the unbound (continuum) states of molecules remains a challenge. Here we propose to tackle these problems using GSFs, which are characterized to have, by construction, the correct asymptotic behavior of the studied system. This property allows one to perform ionization calculations more efficiently. We start and validate our Sturmian approach implementation by studying photoionization (PI) of H, He and Ne atoms. Different model potentials were used in order to describe the interaction of the ejected electron with the parental ion. We calculated the corresponding PI cross sections in both length and velocity gauges. For H atom, the comparison with the analytical formula shows that a rapid convergence can be achieved using a moderate number of GSFs. For He and Ne we have also an excellent agreement with other theoretical calculations and with experimental data. For molecular targets, we considered two different strategies to deal with their random orientation: one makes use of a molecular model potential (non-central), while the other uses an angular averaged version of the same potential (central). We study PI for CH4, NH3, and H2O, from the outer and inner valence orbitals, and for SiH4 and H2S from the outer orbitals. The calculated PI cross sections and also the asymmetry parameters (obtained from the corresponding angular distributions) are compared with available theoretical and experimental data. For most cases, we observed an overall fairly good agreement with reference values, grasping the main features of the ionization process. In a second part of the thesis, we apply the Sturmian approach to study ionization of molecules by electron collisions. In the so-called (e,2e) processes, fully differential cross sections are investigated within both the first- or the second-Born approximations. Again, we show how to include in the description the random orientation of the molecule. We start with H atom, as a test system: the comparison of the calculated triple differential cross sections (TDCSs) with analytical results illustrates, similarly to the PI case, the efficiency of our GSF method. It is then applied to ionization of CH4, H2O and NH3, and comparisons are made with the few theoretical and experimental data available in the literature. For most cases, our TDCSs can reproduce such data, particularly for H2O and for slow ejected electrons in CH4

During the past three decades, the subject of fractional calculus (that is, calculus of integrals and derivatives of arbitrary order) has gained considerable popularity and importance, mainly due to its demonstrated applications in numerous diverse and widespread fields in science and engineering. For example, fractional calculus has been successfully applied to problems in system biology, physics, chemistry and biochemistry, hydrology, medicine, and finance. In many cases these new fractional-order models are more adequate than the previously used integer-order models, because fractional derivatives and integrals enable the description of the memory and hereditary properties inherent in various materials and processes that are governed by anomalous diffusion. Hence, there is a growing need to find the solution behaviour of these fractional differential equations. However, the analytic solutions of most fractional differential equations generally cannot be obtained. As a consequence, approximate and numerical techniques are playing an important role in identifying the solution behaviour of such fractional equations and exploring their applications. The main objective of this thesis is to develop new effective numerical methods and supporting analysis, based on the finite difference and finite element methods, for solving time, space and time-space fractional dynamical systems involving fractional derivatives in one and two spatial dimensions. A series of five published papers and one manuscript in preparation will be presented on the solution of the space fractional diffusion equation, space fractional advectiondispersion equation, time and space fractional diffusion equation, time and space fractional Fokker-Planck equation with a linear or non-linear source term, and fractional cable equation involving two time fractional derivatives, respectively. One important contribution of this thesis is the demonstration of how to choose different approximation techniques for different fractional derivatives. Special attention has been paid to the Riesz space fractional derivative, due to its important application in the field of groundwater flow, system biology and finance. We present three numerical methods to approximate the Riesz space fractional derivative, namely the L1/ L2-approximation method, the standard/shifted Gr¨unwald method, and the matrix transform method (MTM). The first two methods are based on the finite difference method, while the MTM allows discretisation in space using either the finite difference or finite element methods. Furthermore, we prove the equivalence of the Riesz fractional derivative and the fractional Laplacian operator under homogeneous Dirichlet boundary conditions – a result that had not previously been established. This result justifies the aforementioned use of the MTM to approximate the Riesz fractional derivative. After spatial discretisation, the time-space fractional partial differential equation is transformed into a system of fractional-in-time differential equations. We then investigate numerical methods to handle time fractional derivatives, be they Caputo type or Riemann-Liouville type. This leads to new methods utilising either finite difference strategies or the Laplace transform method for advancing the solution in time. The stability and convergence of our proposed numerical methods are also investigated. Numerical experiments are carried out in support of our theoretical analysis. We also emphasise that the numerical methods we develop are applicable for many other types of fractional partial differential equations.

_I The theory of impulsive di®
erential equations has become an important area of research in recent years. Linear equations, meanwhile, are fundamental in most branches of applied mathematics, science, and technology. The theory of higher order linear impulsive equations, however, has not been studied as much as the cor- responding theory of ordinary di®
erential equations. In this work, higher order linear impulsive equations at ¯
xed moments of impulses together with certain boundary conditions are investigated by making use of a Green'
s formula, constructed for piecewise di®
erentiable functions. Existence and uniqueness of solutions of such boundary value problems are also addressed. Properties of Green'
s functions for higher order impulsive boundary value prob- lems are introduced, showing a striking di®
erence when compared to classical bound- ary value problems of ordinary di®
erential equations. Necessarily, instead of an or- dinary Green'
s function there corresponds a sequence of Green'
s functions due to impulses. Finally, as a by-product of boundary value problems, eigenvalue problems for higher order linear impulsive di®
erential equations are studied. The conditions for the existence of eigenvalues of linear impulsive operators are presented. Basic properties of eigensolutions of self-adjoint operators are also investigated. In particular, a necessary and su±
cient condition for the self-adjointness of Sturm-Liouville opera- tors is given. The corresponding integral equations for boundary value and eigenvalue problems are also demonstrated in the present work.

Dans cette thèse, on s’intéresse à la contrôlabilité d’équations paraboliques et à son coût. On étudie dans un premier temps, la contrôlabilité à zéro d’un système de deux équations paraboliques couplés en dimension 1 d’espace, pour lesquelles l’opérateur spacial est de type Sturm-Liouville, avec des conditions au bord de type Dirichlet et un contrôle distribué. Par la méthode des moments on montre l’existence d’un temps minimal de contrôlabilité à zéro sous des conditions géométrique sur le couplage des deux équations. Dans un autre travail, rédigé en collaboration avec González-Burgos, on s’intéresse aux familles biorthogonales aux exponentielles complexes (dont la partie réelle est dominante), sous une condition de gap faible. On montre des estimées inférieures et supérieures de ces familles. Celles ci, sont utilisées pour obtenir le coût du contrôle du système parabolique étudié dans la première partie. Enfin, dans la dernière partie, on utilise les résultats précédents. Ils nous permettent de montrer la contrôlabilité à zéro d’un système parabolique en dimension d’espace supérieure à 1. De plus, on montre la contrôlabilité locale à zéro d’un système parabolique non linéaire de type réaction-diffusion avec un contrôle distribué
In this work we investigate the null controllability of parabolic equations and its cost. We start by studying the null controllability of the one dimensional 2 2 coupled parabolic equations, for which the associated spatial operator is of type Sturm-Liouville, with Dirichlet boundary conditions and internal control. Using the moments method we show the existence of a minimal control time connected to some geometrical conditions on the coupling terms. In an other work, with the collaboration of González-Burgos, we analyze the properties of biorthogonal families to complex exponentials (with dominant real part) under weak gap condition. We prove precise upper and lower bounds for these families. Then, we present an application of these estimates to study the control cost of the parabolic system of the first part. Finally, by using the control cost estimate, we study the null controllability properties of parabolic system, on cylindrical domain with boundary control and local null controllability properties of non linear reaction diffusion system with distributed control

This paper concentrates on the description of fractional order LTI SISO systems using generalized Laguerre functions. There are properties of generalized Laguerre functions described in the paper, and an orthogonal base of these functions is shown. Next the concept of fractional derivatives is explained. The last part of this paper deals with the representation of fractional order LTI SISO systems using generalized Laguerre functions. Several examples were solved to demonstrate the benefits of using these functions for the representation of LTI SISO systems.

Cette thèse porte principalement sur l'étude des solutions de certaines équations aux dérivées partielles elliptiques via l'indice de Morse, y compris des solutions stables, i.e. quand l'indice de Morse est égal à zéro. Elle comporte deux parties indépendantes.Dans la première partie, sous des hypothèses sur-linéaires et sous-critiques sur f, on établit d'abord une estimation explicite de la norme L [infini] des solutions de -Δu = f(u) avec u = 0 sur le bord, via leurs indices de Morse. On propose une approche plus transparente et plus souple que le travail de Yang [1998], ce qui nous permet de traiter des non linéarités très proches de la croissance critique. Les résultats obtenus nous ont motivé de travailler sur des équations polyharmoniques (-Δ)ku = f(x; u) avec notamment k = 2 et 3. Avec des hypothèses semblables à Yang [1998] sur f et des conditions au bord convenables, on obtient pour la première fois des estimations explicites de solution des équations polyhamoniques, via l'indice de Morse. Dans la seconde partie, on considère un système de Lane-Emden-Δu = ρ(x)vp; -Δv = ρ(x)u θ ; u; v > 0; dans RN; avec 1 < p< θ et un poids radial ρ strictement positif. Nous montrons la non-existence de solution stable en petites dimensions N. Nos résultats améliorent les travaux précédents de Cowan & Fazly [2012]; Fazly [2012]; Hu [2015], et fournissent notamment des résultats du type Liouville pour solution stable, en petites dimensions N, valables pour tout 1 < ρ min(4 3 ; θ)
The main concern of this thesis deals with the study of solutions of several elliptic partial differential equations via the Morse index, including the stable solutions, i.e. when the Morse index is zero. The thesis has two independent parts. In the first part, under suplinear and subcritical assumptions on f, we establish firstly some explicit estimation for the L1 norms of solutions to -Δu = f(u) avec u = 0 on the boundary, via its Morse index. We propose an approach more transparent and easier than the work of Yang [1998], which allow us to treat some nonlinearities very close to the critical growth. These results motivated us to consider the polyharmonic equations (-Δ)ku = f(x; u) with especially k = 2 and 3. With the hypothesis on f similar to Yang [1998] and appropriate boundary conditions, we obtain for the _rst time some explicit estimations of solution via its Morse index, for the polyharmonic equations.In the second part, we consider a Lane-Emden system -Δu = ρ(x)vp; -Δv = ρ(x)u_; u; v > 0; in RN; with 1 < p< θ and a radial positive weight ρ. We prove the non-existence of stable solution in small dimension case. Our results improve the previous works Cowan & Fazly [2012]; Fazly [2012]; Hu [2015], especially we prove some general Liouville type results for stable solutions in small dimension which hold true for any 1 < ρ min(4 3 ; θ)

Cette thèse est divisée en deux parties. Dans la première partie, on considère le système de réaction-diffusion-advection (Pε), qui est un modèle d'haptotaxie, mécanisme lié à la dissémination de tumeurs cancéreuses. Le résultat principal concerne la convergence de la solution du systeme (Pε) vers la solution d'un problème à frontière libre (P0) qui est bien défini. Dans la seconde partie, on considère une classe générale d'équations elliptiques du type Hénon:−∆u = |x|^{α} f(u) dans Ω ⊂ R^N avec α > -2. On examine deux cas classiques : f(u) = e^u, |u|^{p−1} u et deux autres cas : f(u) = u^{p}_{+} puis f(u) nonlinéarité générale. En étudiant les solutions stables en dehors d'un ensemble compact (en particulier, solutions stables et solutions avec indice de Morse fini) avec différentes méthodes, on obtient des résultats de classification.

This thesis is devoted to the study of several problems arising in the field of nonlinear analysis. The work is divided in two parts: the first one concerns existence of oscillating solutions, in a suitable sense, for some nonlinear ODEs and PDEs, while the second one regards the study of qualitative properties, such as monotonicity and symmetry, for solutions to some elliptic problems in unbounded domains. Although the topics faced in this work can appear far away one from the other, the techniques employed in different chapters share several common features. In the firts part, the variational structure of the considered problems plays an essential role, and in particular we obtain existence of oscillating solutions by means of non-standard versions of the Nehari's method and of the Seifert's broken geodesics argument. In the second part, classical tools of geometric analysis, such as the moving planes method and the application of Liouville-type theorems, are used to prove 1-dimensional symmetry of solutions in different situations.

Dans cette thèse, nous nous intéressons à l'étude de trois équations aux dérivées partielles et d'évolution non-locales en espace et en temps. Les solutions de ces trois solutions peuvent exploser en temps fini. Dans une première partie de cette thèse, nous considérons l'équation de la chaleur nonlinéaire avec une puissance fractionnaire du laplacien, et obtenons notamment que, dans le cas d'exposant sur-critique, le comportement asymptotique de la solution lorsque $t\rightarrow+\infty$ est déterminé par le terme de diffusion anormale. D'autre part, dans le cas d'exposant sous-critique, l'effet du terme non-linéaire domine. Dans une deuxième partie, nous étudions une équation parabolique avec le laplacien fractionnaire et un terme non-linéaire et non-local en temps. On montre que la solution est globale dans le cas sur-critique pour toute donnée initiale ayant une mesure assez petite, tandis que dans le cas sous-critique, on montre que la solution explose en temps fini $T_{\max}>0$ pour toute condition initiale positive et non-triviale. Dans ce dernier cas, on cherche le comportement de la norme $L^1$ de la solution en précisant le taux d'explosion lorsque $t$ s'approche du temps d'explosion $T_{\max}.$ Nous cherchons encore les conditions nécessaires à l'existence locale et globale de la solution. Une toisième partie est consacré à une généralisation de la deuxième partie au cas de systèmes $2\times 2$ avec le laplacien ordinaire. On étudie l'existence locale de la solution ainsi qu'un résultat sur l'explosion de la solution avec les mêmes propriétés étudiées dans le troisième chapitre. Dans la dernière partie, nous étudions une équation hyperbolique dans $\mathbb{R}^N,$ pour tout $N\geq2,$ avec un terme non-linéaire non-local en temps. Nous obtenons un résultat d'existence locale de la solution sous des conditions restrictives sur les données initiales, la dimension de l'espace et les exposants du terme non-linéaire. De plus on obtient, sous certaines conditions sur les exposants, que la solution explose en temps fini, pour toute condition initiale ayant de moyenne strictement positive.

Lau Ching Yan Ada = 耗散系統中的廣義Sturm-Liouville理論 / 劉正欣.
Thesis (M.Phil.)--Chinese University of Hong Kong, 2004.
Includes bibliographical references (leaves 156-157).
Text in English; abstracts in English and Chinese.
Lau Ching Yan Ada = Hao san xi tong zhong de guang yi Sturm-Liouville li lun / Liu Zhengxin.
Abstract --- p.i
Acknowledgement --- p.iii
Chapter 1 --- Introduction --- p.1
Chapter 1.1 --- Vibrational motion in physics --- p.1
Chapter 1.2 --- Normal modes of vibration --- p.2
Chapter 1.3 --- Boundary conditions --- p.4
Chapter 1.4 --- The wave equation --- p.6
Chapter 1.4.1 --- Mechanical waves --- p.7
Chapter 1.4.2 --- Electromagnetic waves --- p.9
Chapter 1.5 --- General form of the wave equation --- p.10
Chapter 1.5.1 --- V(x) as a restoring force --- p.11
Chapter 1.5.2 --- V(x) in gravitational waves --- p.13
Chapter 1.5.3 --- V(x) by transformation --- p.16
Chapter 2 --- Sturm-Liouville systems --- p.18
Chapter 2.1 --- Introduction --- p.18
Chapter 2.2 --- Differential operators --- p.19
Chapter 2.2.1 --- Introduction --- p.19
Chapter 2.2.2 --- Adjoint operators --- p.20
Chapter 2.2.3 --- Self-adjoint operators --- p.21
Chapter 2.2.4 --- More examples --- p.24
Chapter 2.3 --- Sturm-Liouville boundary-value problems --- p.27
Chapter 2.4 --- Sturm-Liouville theory --- p.28
Chapter 2.4.1 --- Real eigenvalues --- p.29
Chapter 2.4.2 --- Orthogonal eigenfunctions --- p.30
Chapter 2.4.3 --- Completeness of eigenfunctions --- p.31
Chapter 2.4.4 --- Interlacing zeros of the eigenfunctions --- p.33
Chapter 2.5 --- Applications of Sturm-Liouville theory --- p.35
Chapter 2.5.1 --- Vibrations of a string --- p.36
Chapter 2.5.2 --- The hydrogen atom --- p.40
Chapter 3 --- Wave equation with damping --- p.46
Chapter 3.1 --- Statement of problem --- p.46
Chapter 3.1.1 --- The equation --- p.46
Chapter 3.1.2 --- The operator --- p.48
Chapter 3.1.3 --- Non-self-adjointness --- p.49
Chapter 3.2 --- Eigenfunctions and Eigenvalues --- p.51
Chapter 3.3 --- The completeness problem --- p.53
Chapter 4 --- Green's function solution --- p.55
Chapter 4.1 --- Introduction --- p.55
Chapter 4.2 --- Green's function solution --- p.56
Chapter 4.3 --- Fourier transform --- p.58
Chapter 4.4 --- Inverse Fourier transform --- p.61
Chapter 5 --- Proof of completeness --- p.66
Chapter 5.1 --- WKB approximation --- p.66
Chapter 5.2 --- "An upper bound for \G(x,y,w)e~iwt\ " --- p.68
Chapter 5.3 --- Proof of completeness --- p.72
Chapter 5.3.1 --- The limit when R→∞ --- p.72
Chapter 5.3.2 --- Eigenfunction expansion --- p.76
Chapter 6 --- The bilinear map --- p.80
Chapter 6.1 --- Introduction --- p.80
Chapter 6.2 --- Evaluation of J1(wj) --- p.82
Chapter 6.3 --- Self-adjointness of H --- p.84
Chapter 6.4 --- Properties of the map --- p.87
Chapter 7 --- Applications --- p.89
Chapter 7.1 --- Eigenfunction expansion --- p.89
Chapter 7.2 --- Perturbation theory --- p.94
Chapter 7.2.1 --- First and second-order corrections --- p.95
Chapter 7.2.2 --- Example --- p.97
Chapter 7.2.3 --- Example (Constant r) --- p.102
Chapter 8 --- Critical points --- p.104
Chapter 8.1 --- Introduction --- p.104
Chapter 8.2 --- Conservative cases (Γ = 0) --- p.105
Chapter 8.3 --- Non-conservative cases (Constant r) --- p.107
Chapter 8.4 --- Critical points away from imaginary axis --- p.108
Chapter 9 --- Jordan block and applications --- p.114
Chapter 9.1 --- Jordan basis --- p.114
Chapter 9.2 --- An analytical example --- p.117
Chapter 9.2.1 --- Solving for the extra basis function --- p.117
Chapter 9.2.2 --- Freedom of choice --- p.118
Chapter 9.2.3 --- Interpolating function --- p.120
Chapter 9.3 --- A numerical example --- p.122
Chapter 9.3.1 --- "Solving for f2,1 " --- p.124
Chapter 9.3.2 --- Interpolating function --- p.126
Chapter 9.4 --- Jordan basis expansion --- p.127
Chapter 9.5 --- Perturbation theory near critical points --- p.131
Appendices --- p.142
Chapter A --- WKB approximation --- p.142
Chapter B --- Green's function (Discontinuous V(x)) --- p.145
Chapter B.l --- Finite discontinuouity in V(x) --- p.145
Chapter B.1.1 --- Green's function --- p.145
Chapter B.1.2 --- "Behaviour of the extra phases Φ, Φ " --- p.147
Chapter B.2 --- Delta function in --- p.148
Chapter B.2.1 --- Green's function --- p.148
Chapter B.2.2 --- "Behaviour of the extra phases Φ, Φ " --- p.150
Chapter C --- Dual basis --- p.151
Chapter C.1 --- Matrix representation --- p.152
Chapter C.2 --- Relation with bilinear map --- p.153
Chapter C.3 --- Construction of dual basis --- p.154
Bibliography --- p.156

碩士
國立中山大學
應用數學研究所
82
We consider the regular Sturm-Liouville equation on [0,1] (p(x) y')'+(.lambda. w(x)-q(x))y=0 together with separated boundary conditions. In 1993, Ashbaugh and Benguria [2] gave various optimal bounds of eigenvalue ratios for the Sturm- Liouville system with Dirichlet boundary conditions. when q .gdsim. 0. For the general regular Sturm-Liouville system, we prove the various estimates of eigenvalue ratios under the same assumption. For the Neumann boundary conditions, the upper bound is a sharp estimate. The modified Prufer substitution and the Comparison Theorem are the key techniques in the proof. A trigonometric inequality given in [1] is also found to be useful. In this thesis, we give an alternatively proof with elementary methods. We also give an elementary proof of an trigonometric inequality were given in [1]. The sign of .lambda. of the index 1 need to be discussed too. Using the modified Prufer substitution and the Comparison Theorem, we prove that .lambda.1 .relbo. 0 for most of the for most of the separated boundary conditions. In conclusion, this thesis can be viewed as an application of the methods of the modified Prufer substitution and the Comparison Theorem to eigenvalue problems.

碩士
國立臺灣大學
數學研究所
90
We generalize the results for a scalar equation by Modica to the case of a system of equations. It is shown that if a bounded entire solution U(x) of a system of semi-linear equations satisfies the gradient bound |\nabla U|^{2}\leq 2F(U) for all x\in \Bbb R^{n}, then the Liouville theorem holds. Also we show that the inequality above holds if $F(U)$ satisfies some suitable assumptions.

Στην παρούσα διατριβή μελετήσαμε την ολοκληρωσιμότητα του ανισοτροπικού προβλήματος Størmer (ASP) και του ισοσκελούς προβλημάτος τριών σωμάτων (IP), με εφαρμογή της θεωρίας Morales-Ramis-Simó. Τα αποτελέσματα της μελέτης δημοσιεύθηκαν στο περιοδικό Physica D: Nonlinear Phenomena. Ένα σύστημα Hamilton SH, Ν βαθμών ελευθερίας, είναι ολοκληρώσιμο (κατά Liouville) όταν επιδέχεται Ν συναρτησιακώς ανεξάρτητα και σε ενέλιξη πρώτα ολοκληρώματα. Οι J.J. Morales-Ruiz, J.P. Ramis και C. Simó απέδειξαν ότι αν ένα SH είναι ολοκληρώσιμο, τότε η ταυτοτική συνιστώσα G0k της διαφορικής ομάδας Galois των εξισώσεων μεταβολών VE¬k τάξης k , που αντιστοιχούν σε μια ολοκληρωτική καμπύλη του SH, είναι αβελιανή. Το ASP μπορεί να θεωρηθεί ότι είναι ένα σύστημα Hamilton δυο βαθμών ελευθερίας που περιέχει τις παραμέτρους pφ και ν2>0, το οποίο περιγράφει την κίνηση ενός φορτισμένου σωματιδίου υπό την επίδραση του μαγνητικού πεδίου ενός διπόλου. Οι Α. Almeida, T. Stuchi είχαν αποδείξει ότι το ASP είναι μη-ολοκληρώσιμο για pφ≠0 και ν2>0, ενω για pφ=0 είχαν αποδείξει τη μη-ολοκληρωσιμότητα των περιπτώσεων που αντιστοιχούν στις τιμές ν2≠5/12, 2/3. Η δική μας διερεύνηση απέδειξε ότι το ASP με pφ=0 (ASP0) είναι, επίσης, μη-ολοκληρώσιμο για ν2=5/12, 2/3. Αρχικά, με χρήση της μεθόδου Yoshida, αναλύσαμε τις G01 των VE¬1, που αντιστοιχούν σε δύο ολοκληρωτικές καμπύλες του ASP0, καταλήγοντας ότι οι G01 είναι μη-αβελιανές για ν2≠2/3. Στη συνέχεια, ορίσαμε τις VE3 κατά μήκος μιας τρίτης ολοκληρωτικής καμπύλης του ASP0 και δείξαμε ότι η αντίστοιχη G03 είναι μη-αβελιανή για ν2=2/3. Σύμφωνα με τη θεωρία Morales-Ramis-Simó, τα προαναφερόμενα αποδεικνύουν τη μη-ολοκληρωσιμότητα του ASΡ για pφ=0 και ν2>0. Το ΙΡ είναι μια υποπερίπτωση του προβλήματος τριών σωμάτων και μπορεί να μελετηθεί ως ένα σύστημα Hamilton δύο βαθμών ελευθερίας με παραμέτρους pφ και m, m3>0. Η προγενέστερη ανάλυση του ΙΡ υπεδείκνυε τη μη-ολοκληρωσιμότητα του συστήματος, όμως είχε πραγματοποιηθεί με χρήση αριθμητικών μεθόδων. Βρίσκοντας από μια ολοκληρωτική καμπύλη για κάθε μια απο τις περιπτώσεις pφ=0, pφ≠0, ορίσαμε τις αντίστοιχες VE1 και αποδείξαμε τη μη-ολοκληρωσιμότητα του ΙΡ. Για pφ=0 χρησιμοποιήσαμε τη μέθοδο Yoshida για να μελετήσουμε την G01, ενώ για pφ≠0 εφαρμόσαμε τον αλγόριθμο Kovacic και ερευνητικά αποτελέσματα των D. Boucher, J.A. Weil για να διερευνήσουμε την αντίστοιχη G01. Οι G01 και στις δυο προαναφερόμενες περιπτώσεις είναι μη-αβελιανές, οπότε το ΙΡ είναι μη-ολοκληρώσιμο, σύμφωνα με τη θεωρία Morales-Ramis-Simó.
In the present dissertation we studied the integrability of the anisotropic Stormer problem (ASP) and the isosceles three-body problem (IP), applying the Morales-Ramis-Simo theory. The results of our study were published by the journal Physica D: Nonlinear Phenomena. A Hamiltonian system SH, of N degrees of freedom, is integrable (in the Liouville sense) if it admits an involutive set of N functionally independent first integrals. J.J. Morales-Ruiz, J.P. Ramis and C. Simó proved that if an SH is integrable, then the identity component G0k of the differential Galois group of the variational equations VE¬k of order k that correspond to an integral curve of the SH, is abelian. The ASP can be considered as a Hamiltonian system of two degrees of freedom that contains the parameters pφ and ν2>0, which describes the motion of a charged particle under the influence of the magnetic field of a dipole. Α. Almeida, T. Stuchi had proved that the ASP is non-integrable for pφ≠0 and ν2>0, while for pφ=0 they had proved the non-integrability of the cases that correspond to ν2≠5/12, 2/3. Our study proved that the ASP with pφ=0 (ASP0) is, also, non-integrable for ν2=5/12, 2/3. Initially, using the Yoshida method, we analysed the G01 of the VE¬1, that correspond to two integrals curves of the ASP0, concluding that they are non-abelian for ν2≠2/3. Then, we defined the VE3 along a third integral curve of the ASP0 and indicated that the corresponding G03 is non-abelian for ν2=2/3. According to the Morales-Ramis-Simó theory, the aforementioned considerations prove the non-integrability of the ASP for pφ=0 and ν2>0. The IP is a special case of the three-body problem and it can be treated as a Hamiltonian system of two degrees of freedom that embodies the parameters pφ and m, m3>0. Previous analysis of the IP suggested the non-integrability of the system, but it was performed with the use of numerical methods. Finding an integral curve for each of the cases pφ=0, pφ≠0, we defined the corresponding VE1 and proved the non-integrability of the IP. For pφ=0 we used the Yoshida method to examine G01 , while for pφ≠0 we applied the Kovacic algorithm and some results of D. Boucher, J.A. Weil to investigate the corresponding G01 . In both of the aforementioned cases the G01 were non-abelian, yielding IP non-integrable, according to the Morales-Ramis-Simó theory.