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Continuation dans les problèmes de contact pour des plaques en flexionPozzolini, Cédric 15 January 2009 (has links) (PDF)
Avec les codes de calculs généralistes de la mécanique il est possible de suivre numériquement l'évolution de structures soumises a un chargement variable, avec des conditions aux bords classiques. Un des outils pour ces méthodes numériques (dites de continuation) est le théorème des fonctions implicites C1. Mais dans le cas des problèmes de contact avec ou sans frottement cet outil ne s'applique plus, car la solution n'est en général plus dérivable par rapport aux paramètres du problème. La difficulté a été surmontée pour les opérateurs semi-lineaires d'ordre 2 (cas d'une membrane élastique en grandes déformations), mais pas encore pour les plaques. Pour cela, nous avons généralise au bilaplacien le Théorème de stabilité de Schaeffer valable pour le laplacien. Ce qui fournit la dérivée de la frontière libre par rapport aux forces extérieures de classe C^infini, si la frontière libre est C^infini. Nous savons qu'il existe une dérivée par rapport aux forces de classe L2 de la solution pour le problème d'obstacle d'une poutre et d'une plaque élastique, avec des hypothèses sur la zone de contact assurant la polyédricité. Nous explorons l'analyse de sensibilité du problème de l'obstacle pour une poutre et une plaque linéaire, par des méthodes nouvelles d'analyse par perturbation au second ordre. Enfin nous expliquons comment ces résultats pourraient servir a comprendre la stabilité et la sensibilité des plaques de von Karman.
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Infinite-dimensional Hamiltonian systems with continuous spectra : perturbation theory, normal forms, and Landau dampingHagstrom, George Isaac 28 October 2011 (has links)
Various properties of linear infinite-dimensional Hamiltonian systems are studied. The structural stability of the Vlasov-Poisson equation linearized around a homogeneous stable equilibrium [mathematical symbol] is investigated in a Banach space setting. It is found that when perturbations of [mathematical symbols] are allowed to live in the space [mathematical symbols], every equilibrium is structurally unstable. When perturbations are restricted to area preserving rearrangements of [mathematical symbol], structural stability exists if and only if there is negative signature in the continuous spectrum. This analogizes Krein's theorem for linear finite-dimensional Hamiltonian systems. The techniques used to prove this theorem are applied to other aspects of the linearized Vlasov-Poisson equation, in particular the energy of discrete modes which are embedded within the continuous spectrum. In the second part, an integral transformation that exactly diagonalizes the Caldeira-Leggett model is presented. The resulting form of the Hamiltonian, derived using canonical transformations, is shown to be identical to that of the linearized Vlasov-Poisson equation. The damping mechanism in the Caldeira-Leggett model is identified with the Landau damping of a plasma. The correspondence between the two systems suggests the presence of an echo effect in the Caldeira-Leggett model. Generalizations of the Caldeira-Leggett model with negative energy are studied and interpreted in the context of Krein's theorem. / text
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Existência e multiplicidade de soluções para uma classe de equações de Schrödinger com expoente supercríticoMoreira Neto, Sandra Imaculada 30 June 2014 (has links)
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Previous issue date: 2014-06-30 / Neste trabalho, estabelecemos a existência e multiplicidade de soluções para uma classe de equações de Schrodinger quase lineares com não linearidades subcrítica ou supercrítica. A fim de utilizarmos métodos variacionais, aplicamos uma mudança de variável para reduzirmos as equações quase lineares a equações semilineares, cujos funcionais associados estão bem definidos em um espaço de Banach reflexivo, e em alguns casos, eles estão bem definidos em espaços de Sobolev clássicos. Nosso principal foco e tratar não linearidades supercríticas, e nossa principal dificuldade e a perda das imersães de Sobolev tanto contínuas quanto compactas. Para contornar isso, no primeiro problema, inspirados por [4], impomos condições de integrabilidade que relacionam as não linearidades, as quais podem mudar de sinal e necessitamos também, nesse caso, de provar a existência do primeiro autovalor para o operador Lu = Au A(u2)u, usando para isso os métodos de bifurcação e sub e supersolução. No outro problema, nos baseamos num argumento de truncamento, introduzido por del Pino e Felmer em [27], assim o problema fica reduzido a um problema subcrítico. E seguimos com a prova dos resultados usando métodos variacionais combinados com a iteração de Moser. Estabelecemos também a existência de solução para um problema ressonante, cuja prova faremos usando uma variação do Teorema de Operadores Monítonos, encontrado em [29].
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Multiplicidade de soluções para problemas elípticos singulares envolvendo crescimento críticoXavier de Souza, Manassés 31 January 2010 (has links)
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Previous issue date: 2010 / Conselho Nacional de Desenvolvimento Científico e Tecnológico / Usando métodos variacionais e o método de sub e super soluções, neste trabalho estudamos
existência e multiplicidade de soluções para algumas classes de problemas elípticos singulares
envolvendo crescimento crítico do tipo Trudinger-Moser.
Tratamos também de uma generalização para desigualdade de Trudinger-Moser e a
existência de uma função extremal. A prova deste resultado é baseada na análise de blow-up
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A kernel function approach to exact solutions of Calogero-Moser-Sutherland type modelsAtai, Farrokh January 2016 (has links)
This Doctoral thesis gives an introduction to the concept of kernel functionsand their signicance in the theory of special functions. Of particularinterest is the use of kernel function methods for constructing exact solutionsof Schrodinger type equations, in one spatial dimension, with interactions governedby elliptic functions. The method is applicable to a large class of exactlysolvable systems of Calogero-Moser-Sutherland type, as well as integrable generalizationsthereof. It is known that the Schrodinger operators with ellipticpotentials have special limiting cases with exact eigenfunctions given by orthogonalpolynomials. These special cases are discussed in greater detail inorder to explain the kernel function methods with particular focus on the Jacobipolynomials and Jack polynomials. / <p>QC 20161003</p>
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Contribution à l'étude de la réduction formelle des systèmes différentiels méromorphes linéairesAbbas, Hassane 01 September 1993 (has links) (PDF)
Cette thèse est consacrée au calcul des solutions formelles d'un système différentiel linéaire méromorphe dans un voisinage de l'origine de c de la forme y(z)=a(z)y(z). Il est bien connu qu'une matrice fondamentale de solutions s'écrit formellement sous forme h(z)=f(z)g(z), ou f(z) est une série formelle en racine de z et g(z) est une matrice de fonctions élémentaires qui constituent des exponentiels polynomiaux en racine de z#1, puissance complexe de z##1, et puissance entière positive de log z. H. L. Turrittin et w. Wasow ont propose une methode algorithmique pour calculer h(z). Cette methode coute chére en calcul. Devant ce fait, nous proposons une nouvelle approche algorithmique pour trouver h(z). Cette approche a l'avantage d'utiliser des transformations simples et moins couteuses en calcul. De plus, notre approche permet de calculer le plus grand degré des polynômes exponentiels qui se trouvent dans la matrice g(z). En pratique, les systèmes a deux dimensions sont importants. Dans ce cas, nous proposons une methode programmable, inspirée de l'approche générale précédente pour calculer les solutions au voisinage d'une singularité
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Unidimensional and Evolution Methods for Optimal TransportationBonnotte, Nicolas 16 December 2013 (has links) (PDF)
In dimension one, optimal transportation is rather straightforward. The easiness with which a solution can be obtained in that setting has recently been used to tackle more general situations, each time thanks to the same method. First, disintegrate your problem to go back to the unidimensional case, and apply the available 1D methods to get a first result; then, improve it gradually using some evolution process.This dissertation explores that direction more thoroughly. Looking back at two problems only partially solved this way, I show how this viewpoint in fact allows to go even further.The first of these two problems concerns the computation of Yann Brenier's optimal map. Guillaume Carlier, Alfred Galichon, and Filippo Santambrogio found a new way to obtain it, thanks to an differential equation for which an initial condition is given by the Knothe--Rosenblatt rearrangement. (The latter is precisely defined by a series of unidimensional transformations.) However, they only dealt with discrete target measures; I~generalize their approach to a continuous setting. By differentiation, the Monge--Ampère equation readily gives a PDE satisfied by the Kantorovich potential; but to get a proper initial condition, it is necessary to use the Nash--Moser version of the implicit function theorem.The basics of optimal transport are recalled in the first chapter, and the Nash--Moser theory is exposed in chapter 2. My results are presented in chapter 3, and numerical experiments in chapter 4.The last chapter deals with the IDT algorithm, devised by François Pitié, Anil C. Kokaram, and Rozenn Dahyot. It builds a transport map that seems close enough to the optimal map for most applications. A complete mathematical understanding of the procedure is, however, still lacking. An interpretation as a gradient flow in the space of probability measures is proposed, with the sliced Wasserstein distance as the functional. I also prove the equivalence between the sliced and usual Wasserstein distances.
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Existence results for some elliptic equations involving the fractional Laplacian operator and critical growthAraújo, Yane Lísley Ramos 18 December 2015 (has links)
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Previous issue date: 2015-12-18 / In this work we prove some results of existence and multiplicity of solutions for equations
of the type
( ) u + V (x)u = f(x; u) in RN;
where 0 < < 1, N 2 , ( ) denotes the fractional Laplacian, V : RN ! R is a
continuous function that satisfy suitable conditions and f : RN R ! R is a continuous
function that may have critical growth in the sense of the Trudinger-Moser inequality
or in the sense of the critical Sobolev exponent. In order to obtain our results we
use variational methods combined with a version of the Concentration-Compactness
Principle due to Lions. / Neste trabalho provamos alguns resultados de existência e multiplicidade de soluções
para equações do tipo
( ) u + V (x)u = f(x; u) em RN;
onde 0 < < 1, N 2 , ( ) denota o Laplaciano fracionário, V : RN ! R é uma
função contínua que satisfaz adequadas condições e f : RN R ! R é uma função cont
ínua que pode ter crescimento crítico no sentido da desigualdade de Trudinger-Moser
ou no sentido do expoente crítico de Sobolev. A m de obter nossos resultados usamos
métodos variacionais combinados com uma versão do Princípio de Concentração-
Compacidade devido à Lions.
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Investigation of soliton equations with integral operators and their dynamicsVikars Hall, Ruben, Svennerstedt, Carl January 2023 (has links)
We present Lax pairs and functions called Lax functions corresponding to Calogero- Moser-Sutherland (CMS) systems. We present the Benjamin-Ono (BO) equation and a pole ansatz to the BO equation, constructed from a specific type of Lax function called a special Lax function corresponding to Rational and Trigonometric CMS systems. We present a generalization of the BO equation called the non-chiral Intermediate wave (ncILW) equation and show that a family of solutions to the ncILW equation can be constructed from the special Lax function corresponding to the hyperbolic CMS system. We present the Szegö equation on the circle and the real line. We obtain a family of solutions to the Szegö equation on the real line using a pole ansatz. Using numerical methods, we display solution plots to the BO equation and Szegö equation.
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Uma desigualdade do tipo Trudinger-Moser em espaços de Sobolev com peso e aplicaçõesAlbuquerque, Francisco Sibério Bezerra 14 April 2014 (has links)
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Previous issue date: 2014-04-14 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This work addresses a class of Trudinger-Moser type inequalities in weighted Sobolev
spaces in R2. As an application of these inequalities and by using variational methods,
we establish sufficient conditions for the existence, multiplicity and nonexistence of
solutions for some classes of nonlinear Schrödinger elliptic equations (and systems
of equations) with unbounded, singular or decaying radial potentials and involving
nonlinearities with exponential critical growth of Trudinger-Moser type. / Este trabalho aborda uma classe de desigualdades do tipo Trudinger-Moser em espaços
de Sobolev com peso em R2. Como aplicação destas desigualdades e usando métodos
variacionais, estabeleceremos condições suficientes para a existência, multiplicidade e
não-existência de soluções para algumas classes de equações (e sistemas de equações)
de Schrödinger elípticas não-lineares com potenciais radiais ilimitados, singulares na
origem ou decaindo a zero no infinito e envolvendo não-linearidades com crescimento
crítico exponencial do tipo Trudinger-Moser.
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