The spectral theory of the Laplace operator has long been studied in connection with physics. It appears in the wave equation, the heat equation, Schroedinger's equation and in the expression of quantum effects such as the Casimir force. The Casimir effect can be studied in terms of spectral invariants computed entirely from the spectrum of the Laplace operator. It is these spectral invariants and their computation that are the object of study in the present work. The objective of this thesis is to present a computational framework for the spectral zeta function $\zeta(s)$ and its derivative on a Euclidean domain in $\mathbb{R}^2$, with rigorous theoretical error bounds when this domain is polygonal. To obtain error bounds that remain practical in applications an improvement to existing heat trace estimates is necessary. Our main result is an original estimate and proof of a heat trace estimate for polygons that improves the one of van den Berg and Srisatkunarajah, using finite propagation speed of the corresponding wave kernel. We then use this heat trace estimate to obtain a rigorous error bound for $\zeta(s)$ computations. We will provide numerous examples of our computational framework being used to calculate $\zeta(s)$ for a variety of situations involving a polygonal domain, including examples involving cutouts and extrusions that are interesting in applications. Our second result is the development a new eigenvalue solver for a planar polygonal domain using a partition of unity decomposition technique. Its advantages include multiple precision and ease of use, as well as reduced complexity compared to Finite Elemement Method. While we hoped that it would be able to contend with existing packages in terms of speed, our implementation was many times slower than MPSPack when dealing with the same problem (obtaining the first 5 digits of the principal eigenvalue of the regular unit hexagon). Finally, we present a collection of numerical examples where we compute the spectral determinant and Casimir energy of various polygonal domains. We also use our numerical tools to investigate extremal properties of these spectral invariants. For example, we consider a square with a small square cut out of the interior, which is allowed to rotate freely about its center.

In this work we study the Generalized Lane-Emden equation and the interplay between the exponents involved and their consequences on the existence and non existence of radial solutions on a unit ball in n dimensions. We extend the analysis to the phase plane for a clear understanding of the behavior of solutions and the relationship between their existence and the growth of nonlinear terms, where we investigate the critical exponent p and a sub-critical exponent, which we refer to as ^p. We discover a structural change of solutions due the existence of this sub-critical exponent which we relate to the same change in behavior of the Lane- Emden equation solutions, for ; = 0; andp = 2, due to the same sub-critical exponent. We hypothesize that this sub-critical exponent may be related to a weighted trace embedding.

by He Zhaokui. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2000. / Includes bibliographical references (leaves 81-82). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.5 / Chapter 2 --- Preliminaries --- p.8 / Chapter 2.1 --- The Laplacian of a compact manifold --- p.8 / Chapter 2.2 --- The Laplacian of a graph --- p.9 / Chapter 2.3 --- Some basic facts about the eigenvalues of a graph --- p.13 / Chapter 3 --- Bound of the first non-zero eigenvalue in terms of Cheeger constant --- p.18 / Chapter 3.1 --- The Cheeger constant --- p.18 / Chapter 3.2 --- The Cheeger inequality of a compact manifold --- p.19 / Chapter 3.3 --- The Cheeger inequality of a graph --- p.23 / Chapter 4 --- Diameters and eigenvalues --- p.27 / Chapter 4.1 --- Some facts --- p.27 / Chapter 4.2 --- Estimate the eigenvalues of graphs --- p.29 / Chapter 4.3 --- The heat kernel of compact manifolds --- p.34 / Chapter 4.4 --- Estimate the eigenvalues of manifolds --- p.35 / Chapter 5 --- Harnack inequality and eigenvalues on homogeneous graphs --- p.40 / Chapter 5.1 --- Preliminaries --- p.40 / Chapter 5.2 --- The Neumann eigenvalue of a subgraph --- p.41 / Chapter 5.3 --- The Harnack inequality --- p.44 / Chapter 5.4 --- A lower bound of the first non-zero eigenvalue --- p.52 / Chapter 6 --- Harnack inequality and eigenvalues on compact man- ifolds --- p.54 / Chapter 6.1 --- Gradient estimate --- p.54 / Chapter 6.2 --- Lower bounds for the first non-zero eigenvalue --- p.59 / Chapter 7 --- Heat kernel and eigenvalues of graphs --- p.63 / Chapter 7.1 --- The heat kernel of a graph --- p.54 / Chapter 7.2 --- Lower bounds for eigenvalues --- p.70 / Chapter 8 --- Estimate the eigenvalues of a compact manifold --- p.73 / Chapter 8.1 --- An isoperimetric constant --- p.75 / Chapter 8.2 --- A lower estimate for the (m + l)-st eigenvalue --- p.77

by Chow Ha Tak. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (leaves 96-97). / Abstract also in Chinese. / Chapter 1 --- The Laplacian on a Riemannian Manifold --- p.5 / Chapter 1.1 --- Riemannian metrics --- p.5 / Chapter 1.2 --- L2 Spaces of Functions and Forms --- p.6 / Chapter 1.3 --- The Laplacian on Functions and Forms --- p.8 / Chapter 2 --- Hodge Theory for Functions and Forms --- p.14 / Chapter 2.1 --- Analytic Preliminaries --- p.14 / Chapter 2.2 --- The Hodge Theorem for Functions --- p.20 / Chapter 2.3 --- The Hodge Theorem for Forms --- p.27 / Chapter 2.4 --- Regularity Results --- p.29 / Chapter 2.5 --- The Kernel of the Laplacian on Forms --- p.33 / Chapter 3 --- Fermion Calculus and Weitzenbock Formula --- p.36 / Chapter 3.1 --- The Levi-Civita Connection --- p.36 / Chapter 3.2 --- Fermion calculus --- p.39 / Chapter 3.3 --- "Weitzenbock Formula, Bochner Formula and Garding's Inequality" --- p.53 / Chapter 3.4 --- The Laplacian in Exponential Coordinates --- p.59 / Chapter 4 --- The Construction of the Heat Kernel --- p.63 / Chapter 4.1 --- Preliminary Results for the Heat Kernel --- p.63 / Chapter 4.2 --- Construction of the Heat Kernel --- p.66 / Chapter 4.2.1 --- Construction of the Parametrix --- p.66 / Chapter 4.2.2 --- The Heat Kernel for Functions --- p.70 / Chapter 4.2.3 --- The Heat Kernel for Forms --- p.76 / Chapter 4.3 --- The Asymptotics of the Heat Kernel --- p.77 / Chapter 5 --- The Heat Equation Approach to the Chern-Gauss- Bonnet Theorem --- p.82 / Chapter 5.1 --- The Heat Equation Approach --- p.82 / Chapter 5.2 --- Proof of the Chern-Gauss-Bonnet Theorem --- p.85 / Chapter 5.3 --- Introduction to Atiyah-Singer Index Theorem --- p.87 / Chapter 5.3.1 --- A Survey of Characteristic Forms --- p.87 / Chapter 5.3.2 --- The Hirzenbruch Signature Theorem --- p.90 / Chapter 5.3.3 --- The Atiyah-Singer Index Theorem --- p.93 / Bibliography / Notation index

Orientador: Simone Mazzini Bruschi / Banca: Cláudia Buttarello Gentile / Banca: Marta Cilene Gadoti / Resumo: Neste trabalho caracterizamos o conjunto dos pontos de equilíbrio de um problema parabólico quasilinear governado pelo p-Laplaciano, p > 2, e do problema parabólico governado pelo Laplaciano / Abstract: In this work we give a characterization set of the equilibrium points of a parabolic problem quasi-linear governed by the p-Laplacian, p > 2, and the a parabolic problem governed by the Laplacian / Mestre

Cette thèse est constituée de trois chapitres. Le premièr chapitre porte sur l’examen desconditions de validité du principe d’équipartition de l’énergie totale de la solution de l’équationdes ondes associée au laplacien de Dunkl trigonométrique. Enfin, nous établissons lecomportement asymptotique de l’équipartition dans le cas général. Les résultats de cettepartie ont fait l’objet de la publication [8]. Le deuxième chapitre, publié avec J.Ph. Ankeret M. Sifi [6], montre que les fonctions d’Opdam dans le cas de rang 1 satisfont à uneformule produit. Cela nous a permis de définir une structure de convolution du genre hypergroupe.En particulier, on montre que cette convolution satisfait l’analogue du phénomènede Kunze-Stein. Le dernier chapitre est consacrée à l’étude des propriétés dispersives et estimationsde Strichartz pour la solution de l’équation de Schrödinger associée au laplaciende Dunkl trigonométrique unidimensionnel [7]. Cette étude commence par des estimationsoptimales du noyau de la chaleur et de Schrödinger. À l’aide de ces résultats, ainsi que lesoutils d’analyse harmonique dévellopée dans le chapitre 2, on montre des éstimées de typeStrichartz qui permettent de trouver des conditions d’admissibilité pour des équations deSchrödinger semi-linéaires. / This thesis consists of three chapters. The first one is concerned with energy properties of the wave equation associated with the trigonometric Dunkl Laplacian. We establish the conservation of the total energy, the strict equipartition of energy under suitable assumptions and the asymptotic equipartition in the general case. These results were published in [8]. The second chapter, in collaboration with J.Ph. Anker and M. Sifi [6], shows that Opdam’s functions in the rank one case satisfy a product formula. We then define and study a convolution structure related to Opdam’s functions. In particular, we prove that this convolution fulfills a Kunze-Stein type phenomena. The last chapter deals with dispersive and Strichartz estimates for the linear Schrödinger equation associated with the one dimensional trigonometric Dunkl Laplacian [7]. We establish sharp estimates for the heat kernel in complex time, and therefore for the Schrödinger kernel. We then use these estimates together with tools from chapter 2 to deduce dispersive and Strichartz inequalities for the linear Schrödinger equation and apply them to well–posedness in the nonlinear case.

Nous étudions des équations relatives au p-Laplacien anisotrope lorsque certaines composantes du vecteur p sont égales à 1. / We study anisotropic p-Laplacian equations when some components of p are equal to 1.

In the first part of this dissertation, we give a new definition of a Laplace operator for Finsler metric as an average, with regard to an angle measure, of the second directional derivatives. This operator is elliptic, symmetric with respect to the Holmes-Thompson volume, and coincides with the usual Laplace--Beltrami operator when the Finsler metric is Riemannian. We compute explicit spectral data for some Katok-Ziller metrics. When the Finsler metric is negatively curved, we show, thanks to a result of Ancona that the Martin boundary is H��lder-homeomorphic to the visual boundary. This allow us to deduce the existence of harmonic measures and some ergodic preoperties. In the second part of this dissertation, we study Anosov flows in 3-manifolds, with leaf-spaces homeomorphic to .... When the manifold is hyperbolic, Thurston showed that the (un)stable foliations induces an "orthogonal" flow. We use this second flow to study isotopy class of periodic orbits of the Anosov flow and existence of embedded cylinders.

In the first part of this dissertation, we give a new definition of a Laplace operator for Finsler metric as an average, with regard to an angle measure, of the second directional derivatives. This operator is elliptic, symmetric with respect to the Holmes-Thompson volume, and coincides with the usual Laplace--Beltrami operator when the Finsler metric is Riemannian. We compute explicit spectral data for some Katok-Ziller metrics. When the Finsler metric is negatively curved, we show, thanks to a result of Ancona that the Martin boundary is H��lder-homeomorphic to the visual boundary. This allow us to deduce the existence of harmonic measures and some ergodic preoperties. In the second part of this dissertation, we study Anosov flows in 3-manifolds, with leaf-spaces homeomorphic to .... When the manifold is hyperbolic, Thurston showed that the (un)stable foliations induces an "orthogonal" flow. We use this second flow to study isotopy class of periodic orbits of the Anosov flow and existence of embedded cylinders.

The function $sin x$ as one of the six trigonometric functions is fundamental in nearly every branch of mathematics, and its applications. In this thesis, we study an integral equation related to that of $sin x$: $mbox{~for~}xin[-frac{hat{pi}_{p}}{2},~frac{hat{pi}_{p}}{2}] mbox{~and~} p>1$ $$x=int_0^{S_{p}(x)}(1-|t|^{p})^{-frac{1}{p}}dt.$$ Here $hat{pi}_{p}=frac{2pi}{psin(frac{pi}{p})}=2int_0^1(1-t^{p})^{-frac{1}{p}}dt.$ We find that the function $S_{p}(x)$ is well defined. Its properties are also similar to those of $sin x$ : differentiation, identities, periodicity, asymptotic expansions, $cdots$, etc. For example, we have $$|S_{p}(x)|^{p}+|S'_{p}(x)|^{p}=1mbox{~~and~~}frac{d}{dx}(|S'_{p}(x)|^{p-2}S'_{p}(x))=-(p-1)|S_{p}(x)|^{p-2}S_{p}(x).$$ We call $S_{p}(x)$ the generalized sine function. Similarly, we define the generalized cosine function $C_{p}(x)$ by $|x|=int_{C_{p}(x)}^{1}(1- t^{p})^{-frac{1}{p}}dt$ for $xin[-frac{hat{pi}_{p}}{2}$,~$frac{hat{pi}_{p}}{2}]$ and derive its properties. Thus we obtain two sets of trigonometric functions: egin{itemize} item[(i)]$~S_{p}(x),~ S'_{p}(x),~ T_{p}(x)=frac{S_{p}(x)}{S'_{p}(x)},~RT_{p}(x)=frac{S'_{p}(x)}{S_{p}(x)},~ SE_{p}(x)=frac{1}{S'_{p}(x)},~ RS_{p}(x)=frac{1}{S_{p}(x)}~;$ item[(ii)]$~C_{p}(x),~ C'_{p}(x),~RCT_{p}(x)=-frac{C'_{p}(x)}{C_{p}(x)},~ CT_{p}(x)=-frac{C_{p}(x)}{C'_{p}(x)},~RC_{p}(x)=frac{1}{C_{p}(x)},~ CS_{p}(x)=-frac{1}{C'_{p}(x)}mbox{~¡C~}$ end{itemize}These two sets of functions have similar differentiation formulas, identities and periodic properties as the classical trigonometric functions. They coincide when $p=2$. Their graphs and asymptotic expansions are also interesting. Through this study, we understand more about the theoretical framework of trigonometric functions.