Made available in DSpace on 2015-04-22T22:07:21Z (GMT). No. of bitstreams: 1 adalberto.pdf: 2643580 bytes, checksum: e1a556373c92d8f9719b0691629848d1 (MD5) Previous issue date: 2013-02-08 / Fundação de Amparo à Pesquisa do Estado do Amazonas / The objective of this work is to present a proposal for a theoretical analysis of the theory of superconductivity together with an analysis of the Ginzburg-Landau equations in this context, in which the superconducting state is characterized by an order parameter, given by constructing a wave function Ψ (r, t) to describe the quantum behavior of particles and to show the knowledge of concept maps as a didactics tool. We will present the theoretical aspects of the phenomenon of superconductivity and its applications, and examples of conceptual maps including some models containing concepts of superconductivity. The specific objective is to use the maps as a conceptual study of physics theory in the academic, they are methodological tools to help in understanding the concepts with the interpretations, through hierarchical diagrams, shown in a conceptual framework. The research methods adopted are the development of the Ginzburg-Landau equations, the research that includes students enrolled in undergraduate courses in Physics, as individual basis and for last the implementation of a short course, with the participation of undergraduate and graduate students in physics and related areas, distributed in groups or individually to analyze the results. The survey instrument adopted for the last two methods, in order to obtain the scores for the students performance, will be a simple questionnaire, using pencil, black ballpoint pen and A4 paper, containing eleven questions in the first method and in the second method (short course) it will be ten conceptual questions (open or closed) about the concepts related to the topics provided by the instructor and finally it will be presented the analyzes of the results. / O objetivo deste trabalho é apresentar uma proposta de análise teórica da teoria da supercondutividade conjuntamente com uma análise das equações de Ginzburg-Landau neste contexto, em que um estado do supercondutor é caracterizado por um parâmetro de ordem, dado pela construção de uma função de onda Ψ(r,t) para descrever o comportamento quântico das partículas e mostrar o conhecimento de mapas conceituais como ferramenta didática. Serão apresentados os aspectos teóricos do fenômeno da supercondutividade e suas aplicações, e exemplos de mapas conceituais incluindo alguns modelos contendo conceitos da Supercondutividade. O objetivo específico é o de utilizar os mapas conceituais como um estudo da teoria Física no âmbito acadêmico, porque são instrumentos metodológicos para ajudar na compreensão dos conceitos com as interpretações, através de diagramas hierárquicos, mostrados em uma estrutura conceitual. Os métodos da pesquisa adotados são os de desenvolvimento das equações de Ginzburg-Landau, os da investigação que contarão com discentes matriculados nos cursos de graduação em Física, de forma individual e por ultimo a aplicação de um minicurso, com a participação de graduandos e graduados em Física e áreas afins, distribuídos em grupos ou individual para análise dos resultados. O instrumento de pesquisa adotado para estes dois últimos métodos, fins de obter os escores referentes ao desempenho dos discentes, será um questionário simples, utilizando lápis, caneta esferográfica preta e papel A4 contendo, no primeiro método onze questões e no segundo método (minicurso) dez questões conceituais (abertas ou fechadas) sobre os conceitos relacionados aos temas fornecidos pelo instrutor e finalmente, serão apresentados as análises dos resultados.

Cette thèse est dédiée à l'analyse mathématique de quelques problèmes variationnels motivés par le modèle de Ginzburg-Landau en théorie de la supraconductivité. Dans la première partie on étudie l'existence de solutions pour les équations de Ginzburg-Landau sans champ magnétique et avec données au bord de type semi-rigides. Ces données consistent à prescrire le module de la fonction sur le bord du domaine ainsi que son degré topologique. C'est un cas particulier de problèmes à bord libre, ou la donnée complète de la fonction sur le bord est une inconnue du problème. L'existence de solutions à ce problème n'est pas assurée. En effet la méthode directe du calcul des variations ne peut pas s'appliquer car le degré sur le bord n'est pas continu pour la convergence faible dans l'espace de Sobolev adapté. On dit que c'est un problème sans compacité. En étudiant le phénomène de "bubbling" qui apparaît dans l'étude de tels problèmes on donne des résultats d'existence et de non existence de solutions. Dans le Chapitre 1 on étudie des conditions qui permettent d'affirmer que la différence entre deux niveaux d'énergie est strictement optimale. Pour cela on adapte une technique due à Brezis-Coron. Ceci nous permet de redémontrer un résultat (précédemment obtenu par Berlaynd Rybalko et Dos Santos) d'existence de solutions stables pour les équations de Ginzburg-Landau dans des domaines multiplement connexes. Dans le Chapitre 2 on considère les applications harmoniques a valeurs dans $R^2$ avec des conditions au bord de type degrés prescrits sur un anneau. On fait un lien entre ce problème et la théorie des surfaces minimales dans $R^3$ grâce à la différentielle quadratique de Hopf. Ceci nous conduit à l'étude des surfaces minimales bordées par deux cercles dans des plans parallèles. On prouve l'existence de telles surfaces qui ne sont pas des catenoides grâce a un résultat de bifurcation. On utilise alors les résultats obtenus pour déduire des théorèmes d'existence et de non existence de minimiseurs de l'énergie de Ginzburg-Landau à degrés prescrits dans un anneau. Dans ce troisième Chapitre on obtient des résultats pour une valeur du paramètre " grand. Le Chapitre 4 a pour objet l'étude des problèmes a degrés prescrits en dimension n3. On y montre la non existence des minimiseurs de la n-énergie de Ginzburg-Landau a degrés prescrits dans un domaine simplement connexe. On étudie ensuite des points critiques de type min-max pour une énergie perturbée. La deuxième partie est consacrée a l'analyse asymptotique des solutions des équations deGinzburg-Landau lorsque " tend vers zero. Sandier et Serfaty ont étudié le comportement asymptotique des mesures de vorticité associées aux équations. Ils ont notamment trouvé des conditions critiques sur les mesures limites dans le cas des équations avec et sans champ magnétique. Nous nous intéressons alors à ces conditions critiques dans le cas sans champ magnétique. Le problème de la régularité locale des mesures limites se ramène ainsi a l'étude de la régularité des fonctions stationnaires harmoniques dont le Laplacien est une mesure. Nous montrons que localement de telles mesures sont supportées par une union de lignes appartenant à l'ensemble des zéros d'une fonction harmonique / This thesis is devoted to the mathematical analysis of some variational problems. These problem sare motivated by the Ginzburg-Landau model related to the super conductivity. In the first part we study existence of solutions of the Ginzburg-Landau equations without magnetic eld but with semi-sti boundary conditions. These conditions are obtained by prescribing the modulus of the function on the boundary of the domain along with its topological degree. This is a particular case of free boundary problems, where the function on the boundary is an unknown of the problem. Existence of solutions of that problem does not necessary hold. Indeed we can not apply the direct method of the calculus of variations since the degree on the boundaryis not continuous with respect to the weak convergence in an appropriated Sobolev space. This is problem with loss of compactness. By studying the bublling" phenomenon which come upin such problems we obtain some existence and non existence results .In Chapter 1 we study conditions under which the dierence between two energy levels is strictly optimal. In order to do that we adapt a technique due to Brezis-Coron. This allow us to recover known existence results (previously obtained by Berlyand and Rybalko and DosSantos) for stable solutions of the Ginzburg-Landau equations in multiply connected domains. In Chapter 2 we are interested in harmonic maps with values in $R^2$ with prescribed degree boundary condition in an annulus. We make a link between this problem and the minimal surface theory in $R^3$ thanks to the so-called Hopf quadratic differential. This leads us to study immersed minimal surfaces bounded by two circles in parallel planes. We prove the existence of such surfaces die rent from catenoids by using a bifurcation argument. We then apply the results obtained to deduce existence and non existence results for minimizers of the Ginzburg-Landau energy with prescribed degrees. This is done in Chapter 3 where the results are obtained for large ".Chapter 4 is devoted to prescribed degree problems in dimension n3 . We prove the non existence of minimizers of the Ginzburg-Landau energy in simply connected domains. We then study min-max critical points of a perturbed energy. The second part is devoted to the asymptotic analysis of solutions of the Ginzburg-Landau equations when "goes to zero. Sandier and Serfaty studied the asymptotic behavior of the vorticity measures associated to these equations. They derived critical conditions on the limiting measures both with and without magnetic Field. We are interested by these conditions when there is no magnetic Field. The problem of the local regularity of the limiting measures is then equivalent to the study of regularity of stationary harmonic functions whose Laplacianis a measure. We show that locally such measures are concentrated on a union of lines which belong to the zero set of an harmonic function