Dans un article publié en 2003, M.Gromov propose une reformulation de la théorie des petites simplifications en géométrie des groupes. Dans cette version, un graphe fini définit une présentation finie de groupe; les générateurs du groupe sont les étiquettes du graphe; les relateurs sont les mots associés aux cycles; les morceaux, mots "courts " qui permettent les petites simplifications dans un groupe, sont des mots qui étiquettent deux chemins distincts du graphe. Cette thèse prend pour point de départ une brève description de cette théorie publiée par Y. Ollivier en 2006. Le concept de groupe de présentation finie à "petites simplifications", développé par R. Lyndon, M. Greendlinger et autres dans les années 60 et 70, est précurseur des groupes hyperboliques de M.Gromov à la fin des années 80, pour lesquels les propriétés combinatoires de la présentation entraînent des propriétés algébriques du groupe. Dans notre travail, nous fondons de manière rigoureuse la théorie des petites simplifications du point de vue des graphes, et développons le concept de base de "mégatuiles", utilisé implicitement par Y. Ollivier dans son article. Nous étendons ses résultats aux cas non-hyperboliques et non-métriques (par exemple$C(4)-T(4)$). Ce point de vue permet une nouvelle preuve, plus naturelle, de la résolubilité des problèmes du mot et de conjugaison pour les présentations des groupes des entrelacs alternés premiers. Nous prolongeons également les résultats d'un théorème de M. Greendlinger au cas non-métrique, répondant ainsi à une question d'I. Kapovich. / In a paper published in 2003, M.Gromov proposes a rewording of the small cancellation theory in geometric group theory. In this version, a finite graph defines a finitely presented group; generators of the group are the labels of the graph; relators are the words associated with cycles; pieces, "short" words which allow small cancellations in a group, are words which label two distinct paths in the graph.Our thesis relies on a brief description of this theory published in2006 by Y.Ollivier. The concept of finitely presented "small cancellation" group, developed by R.Lyndon, M.Greendlinger and others in the 60's and 70's, is a precursor of Gromovword-hyperbolic groups in the late of the 80's, for which combinatorial properties of the presentation imply algebraic properties of the group. In our work, we build a rigorous small cancellation theory in terms of graphs, and develop the basic concept of "megatiles", implicitly used by Y. Ollivier in his article. We extend his results to non-hyperbolic and non-metric cases (eg. $C(4)-T(4)$). This point of view allows a new proof, more natural, of thesolvability of word and conjugacy problems for presentations of prime alternating link groups. We also extend the results of a M.Greendlinger theorem to thenon-metric case, in response to a question of I. Kapovich.

The first part of this thesis is devoted to symmetrizations. Symmetrizations are tranformations of functions that preserve many properties of functions and enhance their symmetry. In the calculus of variation they are a simple and powerful tool to prove that minimizers of functionals are symmetric functions. In this work, the approximation of symmetrizations by simpler symmetrizations is investigated: The existence of a universal approximating sequence is proved, sufficient conditions for deterministic and random sequences to be approximating are given. These approximation methods are then used to prove some symmetry properties of critical points obtained by minimax methods: For example if there is a solution obtained by the mountain pass theorem, then there is a symmetric solution with the same energy. This part ends with a study of the properties of anisotropic symmetrizations i.e. symmetrizations performed with respect to noneuclidean norms. The second part is devoted to L^1 estimates.  In general, the second derivative of the solution of the Poisson equation with L^1 data fails to be in L^1. Recently it was proved that if the data is a L^1 divergence-free vector-field, then even if in general it is false that the second derivative of the solution is in L^1, all the consequences thereof by Sobolev embeddings hold. Elementary proofs of such results, as well as a generalization with a second order operator replacing the divergence, are given. / La première partie de cette thèse est consacrée aux symétrisations. Les symétrisations sont des transformations de fonctions qui préservent de nombreuses propriétés des fonctions et qui améliorent leur symétrie. Elles sont un outil simple et puissant pour montrer dans le calcul des variations que les minimiseurs de certaines fonctionnelles sont des fonctions symétriques. Dans ce travail, nous étudions l'approximation des symétrisations par des symétrisations plus simples. Nous prouvons l'existence d'une suite approximante universelle et nous donnons des conditions suffisantes pour que des suites déterministes et aléatoires soient approximantes. Nous utilisons ensuite ces méthodes d'approximation pour prouver des propriétés de symétrie de points critiques obtenus par des méthodes de minimax. Par exemple, s'il y a une solution obtenue par le théorème du col, alors il y a une solution symétrique de même énergie. Nous achevons cette partie par une étude des symétrisations anisotropes (symétrisations par rapport à des normes non euclidiennes). La seconde partie est consacrée aux estimations L^1. En général, les dérivées secondes de la solution de l'équation de Poisson avec des données L^1 ne sont pas dans L^1. Recemment, on a prouvé que si les données sont un champ de vecteurs L^1 à divergence nulle, même si en général les dérivées secondes ne sont toujours pas dans L^1, toutes les conséquences qui en suivraient par les injections de Sobolev sont vraies. Nous donnons des preuves élémentaires de ces résultats, avec une extension où la divergence est remplacée par un opérateur différentiel du second ordre.

We investigate stability for certain geometric and functional inequalities and address the regularity of the free boundary for a problem arising in optimal transport theory. More specifically, stability estimates are obtained for the relative isoperimetric inequality inside convex cones and the Gaussian log-Sobolev inequality for a two parameter family of functions. Thereafter, away from a ``small" singular set, local C^{1,\alpha} regularity of the free boundary is achieved in the optimal partial transport problem. Furthermore, a technique is developed and implemented for estimating the Hausdorff dimension of the singular set. We conclude with a corresponding regularity theory on Riemannian manifolds. / text

Orientador: Sueli Irene Rodrigues Costa / Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-17T21:17:14Z (GMT). No. of bitstreams: 1 Limberger_Roberto_M.pdf: 2247337 bytes, checksum: b1a97b81860de8b0d8d254e3a4f702d0 (MD5) Previous issue date: 2011 / Resumo: Neste trabalho são apresentadas abordagens do problema isoperimétrico que podem ser utilizadas no ensino médio ou ensino universitário. Estas incluem: i) aspectos históricos, ii) deduções formais do problema (dentre as curvas de perímetro fixo, a circunferência é a que engloba a maior área) utilizando apenas geometria euclidiana ou via cálculo diferencial, iii) contextualização em problemas de otimização a serem abordados também utilizando recursos computacionais e iv) descrição detalhada de material audiovisual produzido para o ensino médio, com a participação do autor, para um projeto com suporte MEC - UNICAMP / Abstract: This dissertation presents approaches to the isoperimetric problem that can be used in high school or university education. These include: i)historical aspects, ii) formal deductions of the problem (among the curves of fixed perimeter, the circle encompasses most area) using only Euclidean geometry or calculus iii) contextualization in optimization problems to be also addressed using computational resources iv) detailed description of audiovisual material produced for the high school, with the participation of the author / Mestrado / Matematica / Mestre em Matemática

The goal of this diploma thesis is to gather and describe ways the optimisation problems had been solved before the calculus was invented. Some of the methods identified through history are generalised and these which could be utilised at secondary or primary school education are described more in depth. Thesis descri- bes the method, where educator follows historical development of the problem in classroom and what benefits it might bring, as well. There is a section dedicated to evaluation of experiment which goal was to investigate understanding of pupils to certain concepts in the field of optimisation, mainly concerning isoperimetric problem. It was also focused on the pupils fe- edback to different approaches of optimisation problem solving. 1

The present thesis is devoted to the study of various properties of Banach func- tion spaces, with a particular emphasis on applications in the theory of Sobolev spaces and in harmonic analysis. The thesis consists of four papers. In the first one we investigate higher-order embeddings of Sobolev-type spaces built upon rearrangement-invariant Banach function spaces. In particular, we show that optimal higher-order Sobolev embeddings follow from isoperimetric inequal- ities. In the second paper we focus on the question when the above-mentioned Sobolev-type space is a Banach algebra with respect to a pointwise multiplica- tion of functions. An embedding of the Sobolev space into the space of essentially bounded functions is proved to be the answer to this question in several standard as well as nonstandard situations. The third paper is devoted to the problem of validity of the Lebesgue differentiation theorem in the context of rearrangement- invariant Banach function spaces. We provide a necessary and sufficient condition for the validity of this theorem given in terms of concavity of certain functional depending on the norm in question and we find also alternative characterizations expressed in terms of properties of a maximal operator related to the norm. The object of the final paper is the boundedness of the...

Made available in DSpace on 2016-06-02T20:27:39Z (GMT). No. of bitstreams: 1 3744.pdf: 1323892 bytes, checksum: 71a7a7180d61db167b8cbec4db2bbe8b (MD5) Previous issue date: 2011-05-30 / Universidade Federal de Sao Carlos / We address the question of existence of stationary stable solutions to a class of reaction-diffusion equations with spatial dependence in 2 and 3-dimensional bounded domains. The approach consists of proving the existence of local minimizer of the corres-ponding energy functional. For existence, it was enough to give sufficient conditions on the diffusion coefficient and on the reaction term to ensure the existence of isolated mi¬nima of the Γlimit functional of the energy functional family. In the second part we take the techniques developed in the first part to minimize functional in 2 and 3-dimensional rectangles, with and without constraint, solving in a more general form this problem, which was originaly proposed in 1989 by Robert Kohn and Peter Sternberg. / Na primeira parte deste trabalho, abordamos a existência de soluções estacioná-rias estáveis para uma classe de equações de reação-difusão com dependência espacial em domínios limitados 2 e 3-dimensionais. Esta abordagem foi feita via existência de míni¬mos locais dos funcionais de energia correspondentes. Para tal, foi suficiente encontrar condições no coeficiente de difusão e no termo de reação que garantam existência de míni¬mos isolados do funcional Γlimite da família de funcionais de energia. Na segunda parte, aproveitamos as técnicas desenvolvidas na primeira parte para minimizar funcionais em retângulos e paralelepípedos, com e sem vínculo, resolvendo de forma bem mais geral este problema, originalmente proposto em 1989 por Robert Kohn e Peter Sternberg.

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