Em uma variedade riemanniana conexa e compacta introduziremos o conceito de espectro do operador laplaciano. Utilizando a existÃncia e a unicidade do nÃcleo do calor em uma variedade riemanniana,provaremos o teorema de decomposiÃÃo de Hodge. Este teorema afirma que o espaÃo de Hilbert L2(M, g) se decompÃe em uma soma direta de subespaÃos de dimensÃo finita, onde cada subespaÃo à o auto-espaÃo associado a um autovalor do laplaciano. AlÃm disso, os autovalores formam uma sequÃncia nÃo-negativa que acumula somente no infinito. Em seguida iniciaremos a construÃÃo do nÃcleo do calor e, por fim, mostraremos que se duas variedades riemannianas sÃo isospectrais entÃo elas possuem o mesmo volume. / In a connected and compact Riemannian Manifold we will introduce the concept of spectre of Laplace operator. Using the existence and unicity of the heat kernel in Riemannian manifold we proof the Hodge composition theorem. This theorem states that the Hilbert space L2(M, g) decompose in direct sum of subspaces with finite dimesion, where each subspace is the eigen-space relative of a eigenvalue of the laplacian. Furthermore, the eigenvalues form a nonnegative sequence the accumulate only in the infinity. After that we begin the construction of the heat kernel and, finally, we show that two isospetral Riemannian manifolds have the same volume.

Orientador: Marcelo Martins dos Santos / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-18T21:33:13Z (GMT). No. of bitstreams: 1 Dias_GilberlandioJesus_D.pdf: 729241 bytes, checksum: 697941f5eb00690299a087d1432b35cf (MD5) Previous issue date: 2011 / Resumo: Neste trabalho estudamos o escoamento de fluidos viscosos não Newtonianos, modelados pelo sistema estacionário incompressível de Navier-Stokes obedecendo a uma Lei de Potência, em domínios com canais infinitos. Tratamos basicamente de dois tipos de domínios: domínios com canais cuja seção transversal é limitada e domínios com canais possuindo seção transversal ilimitada. Tanto para domínios com seção transversal limitada quanto para domínios com seção transversal ilimitada, estudamos o problema proposto por Ladyzhenskaya e Solonnikov [Zap. Nauchn. Sem. Leningrad Otdel. Mat. Inst. Steklov (LOMI), 96(1980)117-160 (English Transl.: J. Soviet Math., 21, 1983, 728-761)]. Findamos nosso trabalho fazendo um estudo sobre estimativas em espaços de Sobolev com peso para soluções do sistema de Stokes com Lei de Potência / Abstract: In this work we study the flow of the viscous non-Newtonian fluids, modeled by the steady incompressible Navier-Stokes system obeying a power-law, in domains with infinite channels. We deal basically two types of domains: domains with channels whose cross section is limited and domains with channels having unlimited cross section. For both domains with limited cross section and for domains with unbounded cross section, we study the problem proposed by Ladyzhenskaya and Solonnikov [Zap. Nauchno. Sem Leningrad Otdel. Mat. Inst. Steklov (Lomi), 96 (1980) 117-160 (Portugu¿es Transl.: J. Soviet Math., 21, 1983, 728-761)]. We finished our work making a study of estimates in Sobolev weight spaces for solutions of the Stokes power-law system / Doutorado / Matematica / Doutor em Matemática

L’objet de cette thèse est l’étude du problème Δ 2uω = fω avec les conditions aux limites Uω = Δ uω = 0, le second membre étant supposé dépendre continûment de ω dans L2(ω), où ω = {(r, θ); 0 < r < 1, 0 < θ < ω} , 0 < ω ≤ π, est une famille de secteurs tronqués du plan. Si ω < π on sait d’après Blum et Rannacher (1980) que la solution de ce problème uω se décompose au voisinage de l’origine en uω = u1,ω + u2,ω + u3,ω, (1) où u1,ω, u2,ω sont les parties singulières de uω et u3,ω la partie régulière. En effet, au voisinage de l’origine u1,ω (resp. u2,ω, u3,ω) est de régularité H1+πω−ǫ (resp. H2+πω−ǫ, H4) pour tout Q > 0, tandis que la solution uπ appartient, au moins au voisinage de l’origine, à l’espace H4(π), où π est le demi-disque supérieur de centre 0 et de rayon r = 1. On voit clairement une résolution de la singularité près de l’angle π dont la description est l’objectif principal de ce travail. Le résultat obtenu est que la décomposition (1) de uω est uniforme par rapport à ω, lorsque ω → π, pour les meilleures topologies possibles pour chacun des termes, et converge terme à terme vers le développement limité de uπ au voisinage de 0. / In this work, we study the family of problems Δ 2uω = fω with boundary conditionuω = Δ uω = 0. There, the second member is assumed to depend smoothly on ω in L2(ω), where ω = {(r, θ); 0 < r < 1, 0 < θ < ω} , 0 < ω ≤ π, is a family of truncated sectors of the plane. If ω < π it is known from Blum et Rannacher (1980) that the solution uω decomposes as uω = u1,ω + u2,ω + u3,ω, (1) where u1,ω, u2,ω are singular and u3,ω is regular. Indeed, near the origin, u1,ω(resp. u2,ω, u3,ω) is of regularity H1+πω−ǫ (resp. H2+πω−ǫ, H4) for every Q > 0, while the solution uπ is, in the neighborhood of the origin again, of regularity H4. One clearly sees a resolution of the singularity near the angle π whose descriptionis the main objective of this work. The obtained result is that there exists a decomposition (1) of uω which is uniform with respect to ω, when ω → π, with the best possible topologies for each term, and which term by term convergestowards the Taylor expansion of uπ near 0.

京都大学 / 新制・課程博士 / 博士(情報学) / 甲第24262号 / 情博第806号 / 新制||情||136(附属図書館) / 京都大学大学院情報学研究科先端数理科学専攻 / (主査)教授 木上 淳, 教授 磯 祐介, 准教授 白石 大典 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

On s’intéresse aux équations stationnaires de Navier-Stokes linéarisées, il s'agit ici des équations d'Oseen et des équations de Stokes posées dans des domaines infinis, comme les domaines extérieurs, en dimension trois et l'espace tout entier. Le but est d'étudier l'existence de solutions généralisés et de solutions fortes dans un cadre général non nécessairement hilbertien. On s'intéresse aussi au cas des solutions très faibles. Dans ce travail, on considère aussi bien des conditions aux limites classiques de type Dirichlet que des conditions aux limites non standard portant sur certaines composantes du champ de vitesses, du tourbillon, voir du champ de pression. Les espaces de Sobolev classiques ne sont pas adaptés à l'étude de ces problèmes pour une telle géométrie. Pour une bonne analyse mathématique, nous avons choisi de travailler dans le cadre des espaces de Sobolev avec poids, ce qui permet en particulier de mieux contrôler le comportement à l'infini de la solution. / In this work, we study the linearized Navier-Stokes equations in an exterior domain or in the whole space at the steady state, that is, the Stokes equations and the Oseen equations. We give existence, uniqueness and regularity of solutions. The case of very weak solutions is also treated. We consider not only the Dirichlet boundary conditions but also the Non Standard boundary conditions, on some components of the velocity field, vorticity and also on the pressure. Since the domain is not bounded, the classical Sobolev spaces are not adequate. Therefore, a specific functional framework is necessary which also has to take into account the behaviour of the functions at infinity. Our approach rests on the use of weighted Sobolev spaces.

In this paper, several modifications of the quasi-interpolation operator of Scott and Zhang (Math. Comp. 54(1990)190, 483--493) are discussed. The modified operators are defined for non-smooth functions and are suited for the application on anisotropic meshes. The anisotropy of the elements is reflected in the local stability and approximation error estimates. As an application, an example is considered where anisotropic finite element meshes are appropriate, namely the Poisson problem in domains with edges.

For degenerate elliptic partial differential equations, it is often desirable to show that a weak solution is smooth. The first and most difficult step in this process is establishing local Hölder continuity. Sufficient conditions for establishing continuity have already been documented in [FP], [SW1], and [MRW], and their necessity in [R]. However, the complexity of the equations discussed in those works makes it difficult to understand the core structure of the arguments employed. Here, we present a harmonic-analytic method for establishing Hölder continuity of weak solutions in context of a simple linear equation div(Q?u) = f in a homogeneous space structure in order to showcase the form of the argument. Ad- ditionally, we correct an oversight in the adaptation of the John-Nirenberg inequality presented in [SW1], restricting it to a much smaller class of balls.

In this thesis we study Sobolev gradient flows for Perona-Malik style energy functionals and generalizations thereof. We begin with first order isotropic flows which are shown to be regularizations of the heat equation. We show that these flows are well-posed in the forward and reverse directions which yields an effective linear sharpening algorithm. We furthermore establish a number of maximum principles for the forward flow and show that edges are preserved for a finite period of time. We then go on to study isotropic Sobolev gradient flows with respect to higher order Sobolev metrics. As the Sobolev order is increased, we observe an increasing reluctance to destroy fine details and texture. We then consider Sobolev gradient flows for non-linear anisotropic diffusion functionals of arbitrary order. We establish existence, uniqueness and continuous dependence on initial data for a broad class of such equations. The well-posedness of these new anisotropic gradient flows opens the door to a wide variety of sharpening and diffusion techniques which were previously impossible under L2 gradient descent. We show how one can easily use this framework to design an anisotropic sharpening algorithm which can sharpen image features while suppressing noise. We compare our sharpening algorithm to the well-known shock filter and show that Sobolev sharpening produces natural looking images without the "staircasing" artifacts that plague the shock filter. / Thesis (Master, Mathematics & Statistics) -- Queen's University, 2010-08-25 10:44:12.23

We derive a lower bound for the Wehrl entropy in the setting of SU(1,1). For asymptotically high values of the quantum number k, this bound coincides with the analogue of the Lieb-Wehrl conjecture for SU(1,1) coherent states. The bound on the entropy is proved via a sharp norm bound. The norm bound is deduced by using an interesting identity for Fisher information of SU(1,1) coherent state transforms on the hyperbolic plane and a new family of sharp Sobolev inequalities on the hyperbolic plane. To prove the sharpness of our Sobolev inequality, we need to first prove a uniqueness theorem for solutions of a semi-linear Poisson equation (which is actually the Euler-Lagrange equation for the variational problem associated with our sharp Sobolev inequality) on the hyperbolic plane. Uniqueness theorems proved for similar semi-linear equations in the past do not apply here and the new features of our proof are of independent interest, as are some of the consequences we derive from the new family of Sobolev inequalities. We also prove Fisher information identities for the groups SU(n,1) and SU(n,n).

Made available in DSpace on 2014-06-11T19:26:55Z (GMT). No. of bitstreams: 0 Previous issue date: 2010-12-13Bitstream added on 2014-06-13T18:07:01Z : No. of bitstreams: 1 andrade_j_me_sjrp.pdf: 383087 bytes, checksum: ab50e9b76a6329cf8014c0127f5dc9ae (MD5) / Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) / Um problema de controle exato na fronteira para um sistema de equações de ondas acopladas e considerado em um retângulo do plano. Obtem-se controle de quadrado integrável para estados iniciais de energia finita. / We are concerned with a problem of exact boundary controllability for a coupled sistem of wave equations in a rectangle of the plane. We obtain square integrable control for initial state with nite energy.