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Demonstramos a Lei de Weyl sobre o comportamento assintótico dos autovalores do operador Laplaciano com condições de contorno de Dirichlet em domínios limitados e suaves com o auxílio do núcleo do calor. Para isso, fazemos um estudo dos operadores não-limitados, semigrupos e da transformada de Fourier. Por fim, expomos alguns resultados posteriores motivados pelo artigo de Mark Kac \"Can one hear the shape of a drum?\". / We prove the Weyl Law on the asymptotic behavior of eigenvalues of the Laplace operator with Dirichlet boundary conditions in smooth bounded domains with the help of the heat kernel. To that end, we study unbounded operators, semigroups and the Fourier transform. Lastly, we mention some further results motivated by Mark Kac\'s article \"Can one hear the shape of a drum?\".

This thesis suggests an approach to compute the short-time behaviour of the hypoelliptic heat kernel corresponding to sub-Riemannian structures on unimodular Lie groups of type I, without previously holding a closed form expression for this heat kernel. Our work relies on the use of classical non-commutative harmonic analysis tools, namely the Generalized Fourier Transform and its inverse, combined with the Trotter product formula from the theory of perturbation of semigroups. We illustrate our main results by computing, to our knowledge, a first expression in short-time for the hypoelliptic heat kernel on the Engel and the Cartan groups, for which there exist no closed form expression. / Thesis (Master, Mathematics & Statistics) -- Queen's University, 2011-10-08 01:32:32.896

Made available in DSpace on 2016-06-02T20:28:29Z (GMT). No. of bitstreams: 1 5895.pdf: 892631 bytes, checksum: 01dacb877c02765554b75042569f770c (MD5) Previous issue date: 2014-01-31 / Financiadora de Estudos e Projetos / In this work we seek to establish a first contact with Geometric Analysis, aiming the understanding of the Good Shadow Principle of Fontenele and Xavier ([FX11]), which is a generalization of the Omori-Yau Principle. We will expose the basic results that are needed for their comprehension, and extend the study to other topics of Geometric Analysis, as the heat kernel, the existence of exhaustion functions and estimates to the gradient of harmonic functions and subsolutions of the heat equation. Once understood the Good Shadow Principle, we intend to extend it by proving that the class of the second order uniformly bumpable manifolds, introduced by Azagra and Fry in [AF10], also satisfies this principle. / Neste trabalho, procuramos estabelecer um primeiro contato com a Análise Geométrica, tendo como objetivo a compreensão do Princípio da Boa Sombra de Fontenele e Xavier ([FX11]), que é uma generalização do Princípio de Omori-Yau. Apresentaremos resultados básicos necessários para sua compreensão, além de estender o estudo para outros tópicos da Análise Geométrica, como núcleo do calor, funções exaustão e estimativas do gradiente de funções harmônicas e de subsoluções da equação do calor. Uma vez compreendido o Princípio da Boa Sombra, visamos estende-lo provando que a classe de variedades introduzida por Azagra e Fry em [AF10] (second order uniformly bumpable manifolds) também satisfaz este princípio.

In this thesis, we prove the existence and uniqueness of the solution to a Neumann boundary problem for an elliptic differential operator with singular coefficients, and reveal the relationship between the solution to the partial differential equation (PDE in abbreviation) and the solution to a kind of backward stochastic differential equations (BSDE in abbreviation).This study is motivated by the research on the Dirichlet problem for an elliptic operator (\cite{Z}). But it turns out that different methods are needed to deal with the reflecting diffusion on a bounded domain. For example, the integral with respect to the boundary local time, which is a nondecreasing process associated with the reflecting diffusion, needs to be estimated. This leads us to a detailed study of the reflecting diffusion. As a result, two-sided estimates on the heat kernels are established. We introduce a new type of backward differential equations with infinity horizon and prove the existence and uniqueness of both L2 and L1 solutions of the BSDEs. In this thesis, we use the BSDE to solve the semilinear Neumann boundary problem. However, this research on the BSDEs has its independent interest. Under certain conditions on both the "singular" coefficient of the elliptic operator and the "semilinear coefficient" in the deterministic differential equation, we find an explicit probabilistic solution to the Neumann problem, which supplies a L2 solution of a BSDE with infinite horizon. We also show that, less restrictive conditions on the coefficients are needed if the solution to the Neumann boundary problem only provides a L1 solution to the BSDE.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第20887号 / 理博第4339号 / 新制||理||1623(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)准教授 福島 竜輝, 教授 中島 啓, 教授 牧野 和久 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

京都大学 / 新制・課程博士 / 博士(理学) / 甲第24391号 / 理博第4890号 / 新制||理||1699(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)准教授 梶野 直孝, 教授 並河 良典, 准教授 Croydon David Alexander / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

Nous revisitons la méthode classique des paramétrices pour en déduire une minoration et une majoration gaussiennes, pour la solution fondamentale d'un opérateur parabolique général sous forme non divergentielle. Nous utilisons ensuite le fait que la fonction de Neumann Green, d'un opérateur parabolique général sur un ouvert borné régulier, peut être construite comme somme de la solution fondamentale et d'une intégrale de type simple couche parabolique pour établir une minoration gaussienne pour cette fonction de Neumann Green. Le point clef de la preuve réside dans l'effet régularisant, en temps, de l'intégrale de type simple couche. Nous démontrons aussi que cette approche peut être adaptée pour démontrer une minoration gaussienne pour la fonction de Green-Neumann correspondante à l'opérateur de Laplace-Beltrami sur un ouvert régulier d'une variété riemannienne compacte sans bord. Nous démontrons ensuite une nouvelle majoration gaussienne pour la fonction de Neumann Green correspondante à l'opérateur de Laplace-Beltrami sur un ouvert Lipschitz d'une variété riemannienne complète. L'intérêt de cette nouvelle majoration est qu'elle ne contient pas le terme habituel d'une exponentielle en temps. Finalement, comme application des estimations gaussiennes, nous donnons un résultat de compacité des potentiels isospectraux en relation avec une formule asymptotique pour les noyaux de la chaleur / We revisit the parametrix method in order to obtain a gaussian two-sided bound for the fundamental solution of a general parabolic operator which is not in a divergence form. Then we use the fact that the Neumann Green function of a general parabolic operator on a regular bounded domain can be constructed as a perturbation of the fundamental solution by a simple-layer potential in order to establish a Gaussian lower bound for this Neumann Green function. The key point of the proof lies in the time-regularising effect of the single-layer potential. We also prove that this method can be adapted to get a lower Gaussian bound for the Neumann heat kernel of the Laplace-Beltrami operator on an open subset of a compact Riemannian manifold. In a second part, we prove a new Gaussian upper bound for the Neumann heat kernel of the Laplace-Beltrami operator on a Lipschitz domain of a complete Riemannian manifold. The principal interest of this new upper bound is that we do not have the usual exponentiel terme in time in this upper bound. In a last part, as an application of the Gaussian estimates, we give a compactness result of isospectral potentials which is in relation to an asymptotic formule for the heat kernels

Die perturbative Quantenfeldtheorie ist das am weitesten entwickelte Modell, zur präzisen Analyse vieler verschiedener physikalischer Phänomene. Das Standardwerkzeug der perturbativen QFT sind Feynman-Diagramme. Bei Rechnungen bis zu einer Schleife ist aber auch der Wärmekern eine mächtige Technik. Um in der QFT, über die Störungstheorie hinauszugehen ist Symmetrie von großer Bedeutung. Insbesondere die konforme Symmetrie schränkt die Korrelatoren stark ein. In dieser Arbeit wird sie mit der Average Null Energy Condition (ANEC) kombiniert, um die Hofman-Maldacena-Schranken für die Anomaliekoeffizienten in vier Dimensionen abzuleiten. Im Folgenden untersuchen wir verschiedene Probleme der perturbativen Quantenfeldtheorie. Zunächst studieren wir die Weyl-Anomalie für einen nicht-konformen freien Skalar in einer vierdimensionalen gekrümmten Raumzeit. Wir verstehen die Definition der Anomalie diagrammatisch ohne klassische Symmetrie und wir interpretieren die bekannte Wärmekernberechnung präzise. Dann untersuchen wir Eichtheorien mit höheren Ableitungen in sechs Dimensionen. Diese sind natürliche Kandidaten, um perturbativ nicht-unitäre konforme Theorien zu konstruieren. Die Berechnung erfolgt mit der Wärmekernmethode und wir leiten den allgemeinen Ausdruck des, zuvor nicht Bekannten, relevanten Koeffizienten her. Supersymmetrie sowie das Hinzufügen des Yang-Mills-Terms werden ebenfalls berücksichtigt. Schließlich beginnen wir die Untersuchung der Implikationen der ANEC auf nicht-konforme Feldtheorien, angefangen mit dem selbst-wechselwirkenden Skalar in vier Dimensionen. Der Energiefluss eines Zustands mit einer einzelnen Feldeinfügung wird berechnet. Ausgehend von den perturbativen Euklidischen Korrelatoren im Impulsraum konstruieren wir die relevante Wightman-Funktion, um den Energiefluss auszuwerten. Die Berechnung ist kompliziert, aber wir erhalten das erwartete Ergebnis und eröffnen so die Möglichkeit, interessantere Zustände zu untersuchen. / Perturbative quantum field theory is our most developed framework to accurately analyse many physical phenomena. The standard tool is Feynman diagrams, but at one-loop the heat kernel is also a powerful technique. It is however difficult to go beyond perturbation theory, and symmetry is a key factor. In particular, conformal symmetry strongly restricts the correlators, and have been combined with the average null energy condition (ANEC) to derive the Hofman-Maldacena bounds on the anomaly coefficients in four dimensions. In this thesis we study different problems in perturbative quantum field theory. First, we study the Weyl anomaly for a non-conformal free scalar in a four-dimensional curved spacetime. We diagrammatically understand the definition of the anomaly without classical symmetry, and we precisely interpret the well-known heat kernel calculation. Then, we study higher-derivative gauge theories in six dimensions. These theories are the natural candidate to perturbatively construct non-unitary conformal theories. The calculation is done with the heat kernel method and we derive the general expression of the relevant coefficient, which was previously unknown. Supersymmetry or the addition of a Yang-Mills term are also considered. Finally, we initiate the study of the consequence of the ANEC on non-conformal field theories with the example of the self-interacting scalar in four dimensions. The energy flux of a state with a single field insertion is computed. Starting from the perturbative momentum-space Euclidean correlators, we construct the relevant Wightman function to evaluate the energy flux. The calculation is considerably complicated, but we recover the expected result, opening the possibility of studying more interesting states.

Dans cette thèse nous étudions les transformées de Riesz et les espaces de Hardy associés à un opérateur sur un espace métrique mesuré. Ces deux sujets sont en lien avec des estimations du noyau de la chaleur associé à cet opérateur. Dans les Chapitres 1, 2 et 4, on étudie les transformées quasi de Riesz sur les variétés riemannienne et sur les graphes. Dans le Chapitre 1, on prouve que les quasi transformées de Riesz sont bornées dans Lp pour 1<p<2. Dans le Chapitre 2, on montre que les quasi transformées de Riesz est aussi de type faible (1,1) si la variété satisfait la propriété de doublement du volume et l'estimation sous-gaussienne du noyau de la chaleur. On obtient des résultats analogues sur les graphes dans le Chapitre 4. Dans le Chapitre 3, on développe la théorie des espaces de Hardy sur les espaces métriques mesurés avec des estimations différentes localement et globalement du noyau de la chaleur. On définit les espaces de Hardy par les molécules et par les fonctions quadratiques. On montre tout d'abord que ces deux espaces H1 sont les mêmes. Puis, on compare l'espace Hp défini par par les fonctions quadratiques et Lp. On montre qu'ils sont équivalents. Mais on trouve des exemples tels que l'équivalence entre Lp et Hp défini par les fonctions quadratiques avec l'homogénéité t2 n'est pas vraie. Finalement, comme application, on montre que les quasi transformées de Riesz sont bornées de H1 dans L1 sur les variétés fractales. Dans le Chapitre 5, on prouve des inégalités généralisées de Poincaré et de Sobolev sur les graphes de Vicsek. On montre aussi qu'elles sont optimales. / In this thesis, we mainly study Riesz transforms and Hardy spaces associated to operators. The two subjects are closely related to volume growth and heat kernel estimates. In Chapter 1, 2 and 4, we study Riesz transforms on Riemannian manifold and on graphs. In Chapter 1, we prove that on a complete Riemannian manifold, the quasi Riesz transform is always Lp bounded on for p strictly large than 1 and no less than 2. In Chapter 2, we prove that the quasi Riesz transform is also weak L1 bounded if the manifold satisfies the doubling volume property and the sub-Gaussian heat kernel estimate. Similarly, we show in Chapter 4 the same results on graphs. In Chapter 3, we develop a Hardy space theory on metric measure spaces satisfying the doubling volume property and different local and global heat kernel estimates. Firstly we define Hardy spaces via molecules and via square functions which are adapted to the heat kernel estimates. Then we show that the two H1 spaces via molecules and via square functions are the same. Also, we compare the Hp space defined via square functions with Lp. The corresponding Hp space for p large than 1 defined via square functions is equivalent to the Lebesgue space Lp. However, it is shown that in this situation, the Hp space corresponding to Gaussian estimates does not coincide with Lp any more. Finally, as an application of this Hardy space theory, we proved that quasi Riesz transforms are bounded from H1 to L1 on fractal manifolds. In Chapter 5, we consider Vicsek graphs. We prove generalised Poincaré inequalities and Sobolev inequalities on Vicsek graphs and we show that they are optimal.