Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-07-10T17:20:01Z No. of bitstreams: 1 ORLANDO BATISTA DE ALMEIDA - DISSERTAÇÃO PPGMAT 2006..pdf: 835416 bytes, checksum: ebd7a7b886fc9fa8eddfddb98da9aa05 (MD5) / Made available in DSpace on 2018-07-10T17:20:01Z (GMT). No. of bitstreams: 1 ORLANDO BATISTA DE ALMEIDA - DISSERTAÇÃO PPGMAT 2006..pdf: 835416 bytes, checksum: ebd7a7b886fc9fa8eddfddb98da9aa05 (MD5) Previous issue date: 2006-05 / Nesta dissertação, seguindo o trabalho do Berestycki [7] e idéias desenvolvidas por Alves & de Figueiredo [3] e Alves, Corrêa & Gonçalves [4], estudamos a Teoria do Grau de Brouwer e Leray & Schauder, bem como o Método de Galerkin para obter solução de alguns problemas elípticos. / In this of dissertation, motivated by work of Berestycki [7] and ideas conceived byAlves & from Figueiredo [3] andAlves, Corrêa & Gonçalves [4], we styding the theory of Degree fromBrouwer and Leray & Schauder, well how theMethod from Galerkin to obtain solution of some ellíptic problems.

Submitted by Erika Demachki (erikademachki@gmail.com) on 2015-01-16T17:13:32Z No. of bitstreams: 2 license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Tese - Marcos Roberto Marcial - 2014.pdf: 1577179 bytes, checksum: ac1649c996b2193bad6b704f05eca30c (MD5) / Approved for entry into archive by Erika Demachki (erikademachki@gmail.com) on 2015-01-16T17:40:15Z (GMT) No. of bitstreams: 2 license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Tese - Marcos Roberto Marcial - 2014.pdf: 1577179 bytes, checksum: ac1649c996b2193bad6b704f05eca30c (MD5) / Made available in DSpace on 2015-01-16T17:40:15Z (GMT). No. of bitstreams: 2 license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Tese - Marcos Roberto Marcial - 2014.pdf: 1577179 bytes, checksum: ac1649c996b2193bad6b704f05eca30c (MD5) Previous issue date: 2014-06-23 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work, we study the topological structure of the solution set for a class of problems −Δpu = λ f (u)+μg(u)|∇u|p+Ψ(x) in Ω, u > 0 in Ω, u = 0 on ∂Ω, where Ω ⊂ IRN is a bounded domain with ∂Ω smooth, p, λ, μ are constants with p > 1, λ ≥ 0, μ ∈ IR and f ,g : (0,∞)→IR Ψ : Ω→IR are continuous functions. We will use Variational and Topological Methods, which includes minimization of energy functional and building connected components of solutions in a sense that we will define. Also we will employ arguments about the theory of regularity for p-Laplacian operator, approach arguments , maximum principles, results about sub and supersolutions and also arguments including monotonic type operators. / Neste trabalho estudamos a estrutura topológica do conjunto de soluções da classe de problemas −Δpu = λ f (u)+μg(u)|∇u|p+Ψ(x) em Ω, u > 0 em Ω, u = 0 sobre ∂Ω, onde Ω⊂IRN é um domínio limitado com fronteira ∂Ω regular, p, λ, μ são constantes com p > 1, λ ≥ 0, μ ∈ IR e f ,g : (0,∞)→IR, Ψ : Ω→IR são funções contínuas. Utilizamos Métodos Variacionais e Topológicos, que incluem minimização de funcionais energia e construção de componentes conexas de soluções em um sentido que definiremos. Empregamos também argumentos sobre a teoria da regularidade para o operador p- Laplaciano, argumentos de aproximação, bem como princípios de máximo, resultados sobre sub e supersoluções e também argumentos com operadores tipo monotônico.

Suite aux travaux de Zbigniew Szafraniec et Nicolas Dutertre, je me suis intéressée aux calculs de caractéristiques d'Euler de certains espaces semi-algébriques. En particulier, ceux de laforme : $ {(-1)^{varepsilon_1} G_1geq 0 }cap...cap{(-1)^{varepsilon_l} G_lgeq 0}cap W$, où $epsilon=(epsilon_1,...,epsilon_l)in{0,1}^l$, $G=(G_1,...,G_l):R^nrightarrowR^l$ polynomiale et $W:=F^{-1}(0)subsetR^n$ où $F:R^nrightarrowR^k$ et $k+lleq n$. Une fois le cas lisse traité, on intersecte ces ensembles avec ${ fgeq 0}$ ou ${ fleq 0}$, où $f$ est polynomiale telle que $f^{-1}(0)$ admette un nombre fini de singularités. J'énonce alors un théorème reliant ces caractéristiques au degré d'applications faisant intervenir les fonctions $f$, $F$ et $G$. Pour finir, on s'intéresse au cas où l'ensemble $W$ possède un lieu critique compact.Dans une autre partie, je travaille sur l'indice radial, indice défini sur des variétés singulières. J'énonce un résultat faisant le lien entre l'indice radial d'un champ de vecteurs V en une singularité avec l'indice radial de son opposé -V. Finalement, je relie l'indice radial à un indice d'intersection. / After the works of Zbigniew Szafraniec and Nicolas Dutertre, we are interested in computing Euler characteristics of some particular semialgebraic sets. In particular, the ones of the form : $ {(-1)^{varepsilon_1} G_1geq 0 }cap...cap{(-1)^{varepsilon_l} G_lgeq 0}cap W$, where $varepsilon=(varepsilon_1,...,varepsilon_l)in{0,1}^l$, $G=(G_1,...,G_l):R^nrightarrowR^l$ polynomial and $W:=F^{-1}(0)subsetR^n$ where $F:R^nrightarrowR^k$ and $k+lleq n$. Once the smooth case is treated, we intersect these sets with ${ fgeq 0}$ or ${ fleq 0}$, where $f$ is polynomial such that $f^{-1}(0)$ contains a finite number of singularities. Then we state a theorem that makes a link between these caracteristics and some degrees of mappings involving the functions $f$, $F$ and $G$. Finally, we study the case where $W$ has a compact singular set.In another part, I work with the radial index, an index defined for singular manifolds. I have a result making a link between the radial index of a vector field V and its opposite -V at a singularity. Finally, I relate that radial index to an intersection index.

Abstract The dissertation considers a degree theory and the index of a critical point of demi-continuous, everywhere defined mappings of the monotone type. A topological degree is derived for mappings from a Banach space to its dual space. The mappings satisfy the condition (S+), and it is shown that the derived degree has the classical properties of a degree function. A formula for the calculation of the index of a critical point of a mapping A : X→X* satisfying the condition (S+) is derived without the separability of X and the boundedness of A. For the calculation of the index, we need an everywhere defined linear mapping A' : X→X* that approximates A in a certain set. As in the earlier results, A' is quasi-monotone, but our situation differs from the earlier results because A' does not have to be the Frechet or Gateaux derivative of A at the critical point. The theorem for the calculation of the index requires a construction of a compact operator T = (A' + Γ)-1Γ with the aid of linear mappings Γ : X→X and A'. In earlier results, Γ is compact, but here it need only be quasi-monotone. Two counter-examples show that certain assumptions are essential for the calculation of the index of a critical point.