Σκοπός της παρούσας εργασίας είναι η ανάλυση προβλημάτων συνοριακών τιμών. Αρχικά αναφέρονται στοιχεία γραμμικής ανάλυσης και συγκεκριμένα εισάγεται η έννοια ενός τελεστή και τα είδη τελεστών που υπάρχουν, καθώς και η σημασία τους στη Φυσική. Επίσης, δίνεται ο ορισμός της διαφορικής εξίσωσης (Σ.Δ.Ε), ο ορισμός ενός προβλήματος αρχικών τιμών και ο ορισμός ενός προβλήματος συνοριακών τιμών. Έπειτα, αναλύεται η θεωρία Sturm-Liouville και περιγράφονται παραδείγματα συνοριακών τιμών τα οποία επιλύονται με αυτή. Ακόμη, μελετώνται οι συναρτήσεις Green και δίνονται παραδείγματα εφαρμογών τους. Στη συνέχεια εξάγεται η κυματική εξίσωση με τη βοήθεια του μοντέλου της ταλαντούμενης χορδής και επιλύεται με τη μέθοδο του χωρισμού των μεταβλητών για διάφορους τύπους αρχικών και συνοριακών τιμών. Κατόπιν, περιγράφονται μέθοδοι για την επίλυση προβλημάτων συνοριακών τιμών που συνδέονται με την εξίσωση της θερμότητας και μετά αναφέρονται εφαρμογές που προκύπτουν από την επίλυση προβλημάτων διάδοσης θερμότητας. Τέλος αναφέρεται η θεωρία Fredholm και η έννοια της κατανομής και δίνονται παραδείγματα λύσεων των διαφορικών εξισώσεων με την έννοια των κατανομών. Η θεωρία Fredholm είναι ιδιαίτερα σημαντική σε προβλήματα διαφορικών εξισώσεων που είναι μη ομογενή. / In the present project, the initial boundary value problems are analyzed. Firstly, elements of linear analysis are introduced. Particularly the concept of an operator and its types are introduced as well as the importance in the physics sector. Also, the definition of a differential equation and the initial boundary value problems are presented. Additionally, the theory of Sturm-Liouville and its example are described. Moreover, Green function and their applications are introduced. Furthermore, the wave equation was elicited with the basis of vibrating spring model and solved with the method of separating variables. Also with this method and by using Fourier series the heat equation was solved. Finally the theory of Fredholm and the concept of distribution are described. The theory of Fredholm is important in problems of not homogeneous differential equation problems.

Submitted by Sandra Maria Neri Santiago (sandra.neri@ufpe.br) on 2016-03-07T19:42:45Z No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) DISSERTAÇÃO versão finalt.pdf: 2479188 bytes, checksum: 40682d874a9a13182595ec8c40992750 (MD5) / Made available in DSpace on 2016-03-07T19:42:45Z (GMT). No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) DISSERTAÇÃO versão finalt.pdf: 2479188 bytes, checksum: 40682d874a9a13182595ec8c40992750 (MD5) Previous issue date: 2013 / Neste trabalho resumimos o estudo do empacotamento de fios em uma região bidimensional planar. Abordamos o problema de um ponto de vista teórico, usando técnicas de campo conforme, e propriedades de escala do modelo, no regime de empacotamento-rígido, são derivadas, de sorte que os expoentes críticos para a energia elástica e para o número de laços da conformação são obtidos. Os resultados apresentam razoável concordância com dados advindos de experimentos e simulações. Também esboçamos uma analogia entre esse sistema e gravitação em duas dimensões, via gravitação de Liouville. / In this work we summarize the study of the packaging of wire in a planar two-dimensional region. We approach the problem from a theoretical point of view, using techniques of conformal field, and scaling properties of the model, in the tight-packing configuration, are derived, so that the critical exponents for the elastic energy and the number of loops of the conformation are obtained. The results show reasonable agreement with data coming from experiments and simulations. We also outline an analogy between this system and gravitation in two dimensions, via Liouville gravity.

Considere a variedade Tn x S1 com coordenadas (t;x) e considere uma 1-forma diferencial fechada e real a(t) em Tn. Neste trabalho consideramos o operador Lpa = dt +a(t) Λ ∂x de D\'p em D\'p+1, onde D\'p é o espaço das p-correntes da forma u = ∑ Ι I Ι = puI (t, x)dtI. O operador acima define um complexo de cocadeia formado pelos espaços vetoriais D\'p e pelos homomorfismos lineares Lpa : D\'p → D\'p+1. Definiremos o que significa resolubilidade global no complexo acima e caracterizaremos para quais 1-formas a o complexo é globalmente resolúvel. Faremos o mesmo com respeito a hipoeliticidade global no primeiro nível do complexo. / Consider the manifold Tn x S1 with coordinates (t;x) and let a(t) be a real and closed differential 1-form on Tn. In this work we consider the operator Lpsub>a = dt +a(t) Λ ∂x de D\'p from D\'p to D\'p+1, where D\'p is the space of all p-currents u = ∑ Ι I Ι = puI (t, x)dtI . The above operator defines a cochain complex consisting of the vector spaces D\'p and of the linear maps Lpa : D\'p → D\'p+1. We define what global solvability means for the above complex and characterize for which 1-forms a the complex is globally solvable. We will do the same with respect to global hypoellipticity on the first level of the complex.

Cette thèse de doctorat porte sur l’étude de deux objets probabilistes, les mesures de chaos multiplicatif gaussien (GMC) et la théorie conforme des champs de Liouville (LCFT). Le GMC fut introduit par Kahane en 1985 et il s’agit aujourd’hui d’un objet extrêmement important en théorie des probabilités et en physique mathématique. Très récemment le GMC a été utilisé pour définir les fonctions de corrélation de la LCFT, une théorie qui est apparue pour la première fois en 1981 dans le célèbre article de Polyakov, “Quantum geometry of bosonic strings”. Grâce à ce lien établi entre GMC et LCFT, nous pouvons traduire les techniques de la théorie conforme des champs dans un langage probabiliste pour effectuer des calculs exacts sur les mesures de GMC. Ceci est précisément ce que nous développerons pour le GMC sur le cercle unité. Nous écrirons les équations BPZ qui fournissent des relations non triviales sur le GMC. Le résultat final est la densité de probabilité pour la masse totale de la mesure de GMC sur cercle unité ce qui résout une conjecture établie par Fyodorov et Bouchaud en 2008. Par ailleurs, il s'avère que des techniques similaires permettent également de traiter un autre cas, celui du GMC sur le segment unité, et nous obtiendrons de même des formules qui avaient été conjecturées indépendamment par Ostrovsky et par Fyodorov, Le Doussal, et Rosso en 2009. La dernière partie de cette thèse consiste en la construction de la LCFT sur un domaine possédant la topologie d’une couronne. Nous suivrons les méthodes introduites par David- Kupiainen-Rhodes-Vargas même si de nouvelles techniques seront requises car la couronne possède deux bords et un espace des modules non trivial. Nous donnerons également des preuves plus concises de certains résultats connus. / Throughout this PhD thesis we will study two probabilistic objects, Gaussian multiplicative chaos (GMC) measures and Liouville conformal field theory (LCFT). GMC measures were first introduced by Kahane in 1985 and have grown into an extremely important field of probability theory and mathematical physics. Very recently GMC has been used to give a probabilistic definition of the correlation functions of LCFT, a theory that first appeared in Polyakov’s 1981 seminal work, “Quantum geometry of bosonic strings”. Once the connection between GMC and LCFT is established, one can hope to translate the techniques of conformal field theory in a probabilistic framework to perform exact computations on the GMC measures. This is precisely what we develop for GMC on the unit circle. We write down the BPZ equations which lead to non-trivial relations on the GMC. Our final result is an exact probability density for the total mass of the GMC measure on the unit circle. This proves a conjecture of Fyodorov and Bouchaud stated in 2008. Furthermore, it turns out that the same techniques also work on a more difficult model, the GMC on the unit interval, and thus we also prove conjectures put forward independently by Ostrovsky and by Fyodorov, Le Doussal, and Rosso in 2009. The last part of this thesis deals with the construction of LCFT on a domain with the topology of an annulus. We follow the techniques introduced by David-Kupiainen- Rhodes-Vargas although novel ingredients are required as the annulus possesses two boundaries and a non-trivial moduli space. We also provide more direct proofs of known results.

Dans cette thèse, nous étudions la contrôlabilité à zéro de quelques systèmes paraboliques continus et semi-discrétisés. Nous considérons tout d'abord des systèmes en cascade d'équations paraboliques de la forme ∂t −(∂xγ∂x +q). La variable spatiale évolue dans un intervalle réel borné et ce système est semi-discrétisé en espace par un schéma aux différences finies. En appliquant la méthode des moments, nous démontrons des résultats de contrôlabilité à zéro et de φ(h) contrôlabilité à zéro, suivant les hypothèses formulées sur le maillage et les fonctions γ et q. Puis nous étendons ces résultats lorsque la variable d'espace évolue dans un domaine cylindrique, la zone de contrôle se situant dans une partie d'une section au bord du cylindre. Ce domaine cylindrique se décompose en un produit de deux espaces. Sur le premier, de dimension 1, nous appliquons les résultats décrits précédemment. Sur le second, nous appliquons la méthode de Lebeau-Robbiano. Cette approche permet à la fois de montrer que le problème discrétisé est φ(h) contrôlable à zéro et de retrouver un résultat de contrôlabilité à zéro sur le système continu. Dans une autre partie, nous nous intéressons au temps minimal de contrôle à zéro de l'équation de Grushin posée sur un domaine rectangulaire dont le domaine de contrôle est une bande verticale. L'étude se ramène à une infinité dénombrable, indexée par le paramètre de Fourier $n$, de problèmes de contrôle à zéro d'équations paraboliques, traitée, ici encore, à l'aide de la méthode des moments. / In this thesis, we study the null controllability of some continous and semi discretized parabolic systems. We first consider cascade systems of parabolic equations of the form ∂t −(∂xγ∂x +q). The space variable belongs to a real and bounded interval and this system is semi-discretized in space by a finite differences scheme. Applying the so called moments method, we prove null controllability and φ(h) null controllability results, depending on the hypotheses on the mesh and on functions γ and q. Then, we extend this results when the space variable belongs to a cylindrical domain which control zone is in a section at the border of the cylinder. This cylindrical domain is decomposed into a product of two spaces. On the first, of dimension 1, we apply the results described previously. On the second, we use the Lebeau-Robbiano's procedure. In this framework, we prove φ(h) null controllability results on the discretized domain as well as null controllability results on the continous problem. In another section, we investigate the computation of minimal time of null controllability of Grushin's equation defined on a rectangular domain which control region is a vertical strip. This problem of control amounts to study a countably infinite family, indexed by the Fourier parameter $n$, of null control problems of parabolic equations, tackled, once again, with the moments method.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In this dissertation, we study the Sturm Liouvile's theorems for the zeros of the solutions of linear differential equations of second order. These classical theorems are applied to analysis of the monotonicity of functions involving the zeros of classical orthogonal polynomials. in particular, Gegenbauer polynomials. / Neste trabalho estudamos os Teoremas de Sturm Liouville para zeros de soluções de equações diferenciais lineares de segunda ordem. Estes teoremas clássicos são aplicados para análise do crescimento e decrescimento de certas funções que envolvem os zeros de Polinômios Ortogonais Clássicos, como os Polinômios de Gegenbauer. / Mestre em Matemática

Tez (Yüksek Lisans) - Süleyman Demirel Üniversitesi, Fen Bilimleri Enstitüsü, Matematik Anabilim Dalı, 2007. / Bibliyografya var.

On s’intéresse dans cette thèse à la PGD (Proper Generalized Decomposition), l’une des méthodes de réduction de modèles qui consiste à chercher, a priori, la solution d’une équation aux dérivées partielles sous forme de variables séparées. Ce travail est formé de cinq chapitres dans lesquels on vise à étendre la PGD aux espaces fractionnaires et aux espaces des fonctions à variation bornée, et à donner des interprétations théoriques de cette méthode pour une classe de problèmes elliptiques et paraboliques. Dans le premier chapitre, on fait un bref aperçu sur la littérature puis on présente les notions et outils mathématiques utilisés dans le corps de la thèse. Dans le second chapitre, la convergence des suites des directions alternées (AM) pour une classe de problèmes variationnels elliptiques est étudiée. Sous une condition de non-orthogonalité uniforme entre les itérés et le terme source, on montre que ces suites sont en général bornées et compactes. Alors, si en particulier la suite (AM) converge faiblement alors elle converge fortement et la limite serait la solution du problème de minimisation alternée. Dans le troisième chapitre, on introduit la notion des dérivées fractionnaires au sens de Riemann-Liouville puis on considère un problème variationnel qui est une généralisation d’ordre fractionnaire de l’équation de Poisson. En se basant sur la nature quadratique et la décomposabilité de l’énergie associée, on démontre que la suite PGD progressive converge fortement vers la solution faible de ce problème. Dans le quatrième chapitre, on profite de la structure tensorielle des espaces BV par rapport à la topologie faible étoile pour définir les suites PGD dans ce type d’espaces. La convergence de telle suite reste une question ouverte. Le dernier chapitre est consacré à l’équation de la chaleur d-dimensionnelle, où on discrétise en temps puis à chaque pas de temps on cherche la solution de l’équation elliptique en utilisant la PGD. On montre alors que la fonction affine par morceaux en temps obtenue à partir des solutions construites en utilisant la PGD converge vers la solution faible de l’équation. / In this thesis, we are interested in the PGD (Proper Generalized Decomposition), one of the reduced order models which consists in searching, a priori, the solution of a partial differential equation in a separated form. This work is composed of five chapters in which we aim to extend the PGD to the fractional spaces and the spaces of functions of bounded variation and to give theoretical interpretations of this method for a class of elliptic and parabolic problems. In the first chapter, we give a brief review of the litterature and then we introduce the mathematical notions and tools used in this work. In the second chapter, the convergence of rank-one alternating minimisation AM algorithms for a class of variational linear elliptic equations is studied. We show that rank-one AM sequences are in general bounded in the ambient Hilbert space and are compact if a uniform non-orthogonality condition between iterates and the reaction term is fulfilled. In particular, if a rank-one (AM) sequence is weakly convergent then it converges strongly and the common limit is a solution of the alternating minimization problem. In the third chapter, we introduce the notion of fractional derivatives in the sense of Riemann-Liouville and then we consider a variational problem which is a generalization of fractional order of the Poisson equation. Basing on the quadratic nature and the decomposability of the associated energy, we prove that the progressive PGD sequence converges strongly towards the weak solution of this problem. In the fourth chapter, we benefit from tensorial structure of the spaces BV with respect to the weak-star topology to define the PGD sequences in this type of spaces. The convergence of this sequence remains an open question. The last chapter is devoted to the d-dimensional heat equation, we discretize in time and then at each time step one seeks the solution of the elliptic equation using the PGD. Then, we show that the piecewise linear function in time obtained from the solutions constructed using the PGD converges to the weak solution of the equation.

Made available in DSpace on 2015-05-15T11:46:10Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 440995 bytes, checksum: d194a6a60d04b251160ec2e62f106e77 (MD5) Previous issue date: 2011-06-10 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work, we will do some applications of the Moving Spheres method, a variant of the method of Moving Planes, in order to obtain some Liouville-type theorems and some Harnack-type inequalities in Rn, as well as in the Euclidian half space Rn +. Our study focuses on, mostly, in the article written by Yan Yan Li and Lei Zhan [32], as well as some references of the same article. We concentrate in studying some properties of positive solutions of some semilinear elliptic partial differential equations with critical exponent and giving different proofs, improvements, and extensions of some previously established Liouville-type theorems and Harnack-type inequalities. / Neste trabalho, faremos algumas aplicações do método Moving Spheres, uma variante do método Moving Planes, na obtenção de alguns teoremas tipo Liouville e de algumas desigualdades tipo Harnack em Rn, bem como no semi-espaço euclidiano Rn +. Nosso estudo se concentra, marjoritariamente, no artigo do Yan Yan Li e Lei Zhang [32], bem como algumas referências do mesmo. Nos concentramos em estudar propriedades de soluções positivas de algumas equações diferenciais parciais elípticas semilineares com expoente crítico e dar provas diversificadas, refinamentos e extensões de alguns Teoremas tipo Liouville e desigualdades tipo Harnack já estabelecidos.

CoordenaÃÃo de AperfeÃoamento de Pessoal de NÃvel Superior / Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / FundaÃÃo Cearense de Apoio ao Desenvolvimento Cientifico e TecnolÃgico / No presente trabalho, provaremos resultados de comparaÃÃo para o nÃcleo do calor de subvariedades mÃnimas de variedades Riemannianas com curvatura seccional limitada superiormente pela curvatura de uma variedade modelo. Em seguida, iremos obter resultados sobre a propriedade L1-Liouville de submersÃes Riemannianas com fibras mÃnimas. Por Ãltimo, provaremos desigualdades para o tom fundamental ponderado de subconjuntos transversalmente folheados de variedades Riemannianas ponderadas em termos das curvaturas mÃdias ponderadas das folhas da folheaÃÃo. / In this work we will prove comparison results for the heat kernel of minimal submanifolds in Riemannian manifolds with sectional curvature bounded above by the curvature of a model manifold. Next we will obtain results about the L1-Liouville property of Riemannian submersions with minimal fibers. Finnaly, we will prove inequalities for the weighted fundamental tone of transversally foliated subsets of weighted Riemannian manifolds in terms of the weighted mean curvatures of the leaves of the foliation.