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Normally solvable nonlinear boundary value problemsAlsaedy, Ammar, Tarkhanov, Nikolai January 2013 (has links)
We study a boundary value problem for an overdetermined elliptic system of nonlinear first order differential equations with linear boundary operators.
Such a problem is solvable for a small set of data, and so we pass to its variational formulation which consists in minimising the discrepancy. The Euler-Lagrange equations for the variational problem are far-reaching analogues of the classical Laplace equation. Within the framework of Euler-Lagrange equations we specify an operator on the boundary whose zero set consists precisely of those boundary data for which the initial problem is solvable. The construction of such operator has much in common with that of the familiar Dirichlet to Neumann operator. In the case of linear problems we establish complete results.
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Restriktionsätze für getwistete Sub-Laplace-Operatoren und Anwendungen auf RieszmittelKempe, Michael. Unknown Date (has links) (PDF)
Universiẗat, Diss., 2002--Kiel.
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NÃo existÃncia de autovalores do operador de Laplace-Beltrami em grÃficos radiais / Nonexistence of eigenvalues of the Laplace-Beltrami operator in radial graphsFrancisca Damiana Vieira 16 June 2014 (has links)
Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / Neste trabalho estudamos o operador de Laplace-Beltrami definido em variedades Riemannianas. AlÃm do espectro de tal operador, apresentamos tambÃm algumas de suas propriedades, como o fato deste operador ser auto adjunto e nÃo negativo. Nosso objetivo principal consiste em analisar a existÃncia de autovalores para o operador de Laplace-Beltrami, sob determinadas condiÃÃes, em superfÃcies que sÃo grÃficos de funÃÃes radiais, definida sobre todo o plano, ou seja, superfÃcies nÃo compactas de revoluÃao. Esta dissertaÃÃo se baseia no artigo On the spectrum of the Laplace-Beltrami Operator on a Non-Compact Surface" de Takao Tayoshi ( Comm. By Kinjir^o Kunugi, M. J. A., Feb. 12,1971). Para realizaÃÃo desse trabalho foram introduzidos conceitos bÃsicos de anÃlise funcional com destaque para o estudo de espaÃos de Hilbert e a teoria espectral de operadores auto adjuntos, geometria riemanniana em superfÃcies e equaÃÃes diferenciais parciais, em particular resultados para operadores elÃpticos de segunda ordem. AlÃm disso, se fizeram necessÃrios alguns resultados de matemÃtica avanÃada. / In this work we study the Laplace-Beltrami operator defined on Riemannian manifolds. In addition to the spectrum of such an operator, we also present some of its properties,
such as the fact that this operator is self-adjoint and non-negative. Our main goal is to analyze the existence of eigenvalues for the Laplace-Beltrami operator, under certain
conditions, for exemple, surfaces that are complete graphs of radial functions, which is a revolution non-compact surfaces. This dissertation is based on the article "On the spectrum of the Laplace-Beltrami Operator on the Non-Compact Surface"of Takao Tayoshi(Comm. By Kinjiro Kunugi, MJA, Feb. 12, 1971). To perform this work were introduced basics concepts of functional analysis, with emphasis on the study of Hilbert spaces and the spectral theory of self-adjoint operators, Riemannian Geometry in surfaces and Partial Differential Equations, in particular results for elliptic operators of second order.In addition, were needed some results for advanced mathematics.
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Das Gitterpunktproblem in der hyperbolischen EbeneThirase, Jan 01 November 2000 (has links)
Es wird sowohl das klassischen Kreisproblem als auch dessen Verallgemeinerung auf geometrisch endliche Fuchssche Gruppen betrachten. Insbesondere werden obere und untere Schranken für die jeweiligen Zählfunktionen angeben. Mit einem Computerprogramm wird die Zählfunktion einer Heckegruppe bestimmt und damit eine Abschätzung ihres Konvergenzexponenten gegeben.
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Spectra of Normalized Laplace Operators for Graphs and HypergraphsMulas, Raffaella 25 June 2020 (has links)
In this thesis, we bring forward the study of the spectral properties of graphs and we extend this theory for chemical hypergraphs, a new class of hypergraphs that model chemical reaction networks.
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The Eigenvalue Problem of the 1-Laplace Operator: Local Perturbation Results and Investigation of Related Vectorial QuestionsLittig, Samuel 23 January 2015 (has links)
As a first aspect the thesis treats existence results of the perturbed eigenvalue problem of the 1-Laplace operator. This is done with the aid of a quite general critical point theory results with the genus as topological index. Moreover we show that the eigenvalues of the perturbed 1-Laplace operator converge to the eigenvalues of the unperturebed 1-Laplace operator when the perturbation goes to zero. As a second aspect we treat the eigenvalue problems of the vectorial 1-Laplace operator and the symmetrized 1-Laplace operator. And as a third aspect certain related parabolic problems are considered.
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Isospectral nearly Kaehler manifoldsVasquez, Jose J. 04 September 2017 (has links)
We give an Ansatz to construct pairs of locally homogeneous nearly Kaehler manifolds that are isospectral for the Dirac and the Hodge Laplace operator
in dimensions higher than six and investigate the existence of generic isospectral pairs in dimension six.
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Processus de diffusion discret : opérateur laplacien appliqué à l'étude de surfaces / Digital diffusion processes : discrete Laplace operator for discrete surfacesRieux, Frédéric 30 August 2012 (has links)
Le contexte est la géométrie discrète dans Zn. Il s'agit de décrire les courbes et surfaces discrètes composées de voxels: les définitions usuelles de droites et plans discrets épais se comportent mal quand on passe à des ensembles courbes. Comment garantir un bon comportement topologique, les connexités requises, dans une situation qui généralise les droites et plans discrets?Le calcul de données sur ces courbes, normales, tangentes, courbure, ou des fonctions plus générales, fait appel à des moyennes utilisant des masques. Une question est la pertinence théorique et pratique de ces masques. Une voie explorée, est le calcul de masques fondés sur la marche aléatoire. Une marche aléatoire partant d'un centre donné sur une courbe ou une surface discrète, permet d'affecter à chaque autre voxel un poids, le temps moyen de visite. Ce noyau permet de calculer des moyennes et par là, des dérivées. L'étude du comportement de ce processus de diffusion, a permis de retrouver des outils classiques de géométrie sur des surfaces maillées, et de fournir des estimateurs de tangente et de courbure performants. La diversité du champs d'applications de ce processus de diffusion a été mise en avant, retrouvant ainsi des méthodes classiques mais avec une base théorique identique.} motsclefs{Processus Markovien, Géométrie discrète, Estimateur tangentes, normales, courbure, Noyau de diffusion, Analyse d'images / The context of discrete geometry is in Zn. We propose to discribe discrete curves and surfaces composed of voxels: how to compute classical notions of analysis as tangent and normals ? Computation of data on discrete curves use average mask. A large amount of works proposed to study the pertinence of those masks. We propose to compute an average mask based on random walk. A random walk starting from a point of a curve or a surface, allow to give a weight, the time passed on each point. This kernel allow us to compute average and derivative. The studied of this digital process allow us to recover classical notions of geometry on meshes surfaces, and give accuracy estimator of tangent and curvature. We propose a large field of applications of this approach recovering classical tools using in transversal communauty of discrete geometry, with a same theorical base.
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Generalizations of a Laplacian-Type Equation in the Heisenberg Group and a Class of Grushin-Type SpacesChilders, Kristen Snyder 01 January 2011 (has links)
In [2], Beals, Gaveau and Greiner find the fundamental solution to a 2-Laplace-type equation in a class of sub-Riemannian spaces. This fundamental solution is based on the well-known fundamental solution to the p-Laplace equation in Grushin-type spaces [4] and the Heisenberg group [6]. In this thesis, we look to generalize the work in [2] for a p-Laplace-type equation. After discovering that the "natural" generalization fails, we find two generalizations whose solutions are based on the fundamental solution to the p-Laplace equation.
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Asymptotic bounds and values for the norm of the Laplace operator and other partial differential operators on spaces of polynomialsRebs, Christian 09 December 2020 (has links)
In der vorliegenden Dissertation werden endlichdimensionale Räume multivariater Polynome in N Variablen mit der Laguerre-, Hermite- bzw. Legendrenorm versehen.
Dabei sei der Höchstgrad der Polynome oder die Summe der Grade der Variablen durch eine natürliche Zahl n nach oben beschränkt. Wir betrachten auf diesen Räumen den Laplaceoperator und zwei weitere partielle Differentialoperatoren
und interessieren uns für das Verhalten der von den Polynomnormen induzierten Operatornormen dieser Operatoren, wenn n gegen unendlich strebt.
Im Fall der Laguerre- und Legendrenorm werden asymptotische obere und untere Schranken der Operatornormen hergeleitet. Im Fall der Hermitenorm kann sogar eine asymptotische Formel gezeigt werden, wenn man voraussetzt, dass der Höchstgrad der Poynome duch n beschränkt ist.
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