Made available in DSpace on 2015-05-15T11:46:05Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 813665 bytes, checksum: 8aa09df2661d8ea4c0561ebad8cd9584 (MD5) Previous issue date: 2012-07-31 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / The goal of our work is to prove the existence of solutions to a class of semilinear elliptic equations in a bounded domain, involving concave-convex type nonlinearities. We use a variety of methods to and these solutions, such as Mountain Pass Theorem, Ekeland's Variational Principle, Lagrange Multipliers Theorem, Nehari Manifold and sub and supersolution method. / O objetivo da nossa dissertação é provar a existência de soluções para uma classe de equações elípticas semilineares em um domínio limitado, envolvendo não linearidades do tipo côncavo-convexas. Mostraremos alguns casos diferentes e métodos diversificados para encontrar tais soluções, usando o Teorema do Passo da Montanha, o Princípio Variacional de Ekeland, Teorema dos Multiplicadores de Lagrange, a Variedade de Nehari e sub e supersolução.

Submitted by Luciana Ferreira (lucgeral@gmail.com) on 2018-09-06T10:34:36Z No. of bitstreams: 2 Dissertação - Lucas Menezes de Brito - 2018.pdf: 2914034 bytes, checksum: 600a20e123b6c9b15b12092b1a8071c8 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2018-09-06T10:35:12Z (GMT) No. of bitstreams: 2 Dissertação - Lucas Menezes de Brito - 2018.pdf: 2914034 bytes, checksum: 600a20e123b6c9b15b12092b1a8071c8 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2018-09-06T10:35:12Z (GMT). No. of bitstreams: 2 Dissertação - Lucas Menezes de Brito - 2018.pdf: 2914034 bytes, checksum: 600a20e123b6c9b15b12092b1a8071c8 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2018-08-10 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work we study a singular partial differential problem in a bounded domain with smoth boundary. We have two main cases, one superlinear with weak singularity, and the other one sublinear with strong songularity. We use Variational Methods, such as the Ekeland Variational Principle and the Nehari Manifolds, to solve this problem, finding weak solutions and proving the multiplicity of solutions in one of the cases. / Neste trabalho estudaremos um problema diferencial parcial singular em um domínio limitado com bordo suave. Temos dois casos principais, um superlinear com singularidade fraca e um sublinear com singularidade forte. Usaremos Métodos Variacionais, como o Princípio Variacional de Ekeland e as Variedades de Nehari, para resolver este problema, encontrando soluções fracas e provando a multiplicidade das mesmas em um dos casos.

Submitted by JÚLIO HEBER SILVA (julioheber@yahoo.com.br) on 2017-03-03T18:04:36Z No. of bitstreams: 2 Dissertação - Karla Carolina Vicente de Sousa 2017.pdf: 802534 bytes, checksum: b021fd17684c91eaed58191b3674afd7 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2017-03-06T10:40:35Z (GMT) No. of bitstreams: 2 Dissertação - Karla Carolina Vicente de Sousa 2017.pdf: 802534 bytes, checksum: b021fd17684c91eaed58191b3674afd7 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2017-03-06T10:40:35Z (GMT). No. of bitstreams: 2 Dissertação - Karla Carolina Vicente de Sousa 2017.pdf: 802534 bytes, checksum: b021fd17684c91eaed58191b3674afd7 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2017-03-01 / Conselho Nacional de Pesquisa e Desenvolvimento Científico e Tecnológico - CNPq / In this work we study the existence of positive solutions for the following semilinear elliptic problem with concave-convex nonlinearities    −∆u = λa(x)u q +b(x)u p , x ∈ Ω u = 0, x ∈ ∂Ω where Ω is a bounded domain in R N with smooth boundary and 0 < q < 1 < p < 2 ∗−1 (where 2∗−1 = +∞, if N = 1 or N = 2 and 2∗−1 = N+2 N−2 , where N ≥ 3). Furthermore, λ > 0 is a parameter and a,b : Ω → R are continuous functions which are somewhere positives, however, such functions may change sign in Ω. / Neste trabalho estudaremos a existência de soluções positivas para o seguinte problema elíptico semilinear com não linearidades do tipo côncavo-conexo    −∆u = λa(x)u q +b(x)u p , x ∈ Ω u = 0, x ∈ ∂Ω onde Ω é uma domínio limitado de R N , com bordo regular e 0 < q < 1 < p < 2 ∗ −1 (onde 2∗ −1 = +∞, se N = 1 ou N = 2 e 2∗ −1 = N+2 N−2 , quando N ≥ 3). Além disso, λ > 0 é um parâmetro e a,b : Ω → R são funções contínuas que assumem valores positivos, porém, tais funções podem mudar de sinal em Ω.

L'objectif de cette thèse est d'étudier l'existence, la multiplicité et le comportement des solutions positives de systèmes d'équations aux dérivées <br />partielle faisant intervenir le (p,q)-Laplacien ou des opérateurs anisotropiques dans les cas sous-critique et critique.<br /> Dans le 1er chapitre on s' intéresse au système suivant (S):<br />\begin{eqnarray}<br />\left\{\begin{array}{lll}-\Delta_p u&=&\lambda f(x,u,v)\quad\mbox{dans}\quad\Omega,\\<br />-\Delta_q v&=&\mu g(x,u,v)\quad\mbox{dans}\quad\Omega,<br />\end{array}<br />\right.<br />\end{eqnarray}<br />avec $f$ et $g$ présentent des termes sous-critiques en u et v . On a pu construire deux suites de Palais-Smale sur la variété de Nehari convergeant <br />fortement dans $W{1,p}(\Omega)\times W{1,q}(\Omega)$ vers deux solutions distinctes.<br /> Dans le 2ème chapitre, on considère la même classe du système (S) dans le cas critique et dans $\mathbb{R}^N$. A la différence du chapitre 1, dans <br />ce cas on retrouve qu'une seule solution positive et pour $p=q$ on retrouve une seconde solution.<br /> Dans le chapitre 3, on généralise l'étude de Brézis-Nirenberg à une équation et puis à un système critique du type (S). On donne une définition plus générale de la notion de niveau critique.<br /> Le Dernier chapitre traîte d'une nouvelle classe de systèmes d'équations elliptiques anisotropiques (puissance dépend de la direction) avec des termes de réaction de type puissance de façon que l'espace fonctionnel naturel devient un espace de Sobolev anisotrope. On démontre l'existence ainsi que la régularité des solutions faibles du système puis l'existence d'une solution dans le cas où on a une sous et une sur-solution du système.

Submitted by Erika Demachki (erikademachki@gmail.com) on 2018-03-23T22:13:05Z No. of bitstreams: 2 Tese - Thiago Rodrigues Cavalcante - 2018.pdf: 2200622 bytes, checksum: 39118adda6b7ceff14825da442b5be57 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2018-03-26T12:16:44Z (GMT) No. of bitstreams: 2 Tese - Thiago Rodrigues Cavalcante - 2018.pdf: 2200622 bytes, checksum: 39118adda6b7ceff14825da442b5be57 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2018-03-26T12:16:44Z (GMT). No. of bitstreams: 2 Tese - Thiago Rodrigues Cavalcante - 2018.pdf: 2200622 bytes, checksum: 39118adda6b7ceff14825da442b5be57 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2018-02-27 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In the first two chapters, we consider the following problem \begin{equation*} \left \{ \begin{array}{rcll} \alpha \Delta^{2} u + \beta \Delta u & = & f(x,u)\, & \mbox{in}\,\, \Omega \\ u = \Delta u & = & 0 \, &\mbox{on } \,\,\, \partial \Omega, \end{array} \right. \end{equation*} where $\displaystyle{\Delta^{2} u = \Delta(\Delta u)-\,\mbox{biharmonic (fourth-order operator)}}$, $\alpha > 0$ and $ \beta \in \R.$ The subset $\displaystyle{ \Omega \subset \mathbb{R}^{N}\, (N \geq 4)}$ is as somooth bounded domain and $\displaystyle{ f \in C(\overline{\Omega} \times \mathbb{R},\mathbb{R}) }.$ In each of the results obtained, we will consider different technical hypotheses and characteristics for the nonlinear function $f$ e for the value of the constant $ \beta. $ In the third chapter, we study an equation of the concave type super linear, of the form: \begin{equation} \left \{ \begin{array}{rcll} \alpha \Delta^{2} u + \beta \Delta u & = & a(x)|u|^{s-2}u + f(x,u)\, & \mbox{in}\,\, \Omega \\ u = \Delta u & = & 0 \, &\mbox{on} \,\,\, \partial \Omega, \end{array} \right. \end{equation} where $\beta \in (-\infty, \alpha \lambda_{1}).$ We consider that the function $a \in L^{\infty} (\Omega)$ and $s \in (1,2).$ Finally, in the last chapter we will consider a fourth order problem in which nonlinearity is also of the convex concave type. More precisely, we study the following class of equations: \begin{equation} \left\{ \begin{aligned} \alpha \Delta^{2} u + \beta \Delta u & = \mu a(x)|u|^{q-2}u + b(x)|u|^{p-2}u&\,\,\,\,\ &\mbox{in}\,\, \Omega \\ u = \Delta u & = 0 & \,\,\,\,&\mbox{on} \,\, \partial \Omega, \end{aligned} \right. \end{equation} where the parameter $ \mu > 0 $, the powers $ 1 <q <2 <p <2 N / (N - 4) $. In addition we assume that the functions $ \displaystyle {a, b: \Omega \rightarrow \mathbb {R}}$ are continuous that can change signal and, $ a ^{+}, b ^{+} \neq 0. $ / Nos dois primeiros Capítulos, consideramos a seguinte classe de problemas: \begin{equation*} \left \{ \begin{array}{rcll} \alpha \Delta^{2} u + \beta \Delta u & = & f(x,u)\, & \mbox{em}\,\, \Omega \\ u = \Delta u & = & 0 \, &\mbox{sobre } \,\,\, \partial \Omega, \end{array} \right. \end{equation*} onde $\displaystyle{\Delta^{2} u = \Delta(\Delta u)-\,\mbox{biharmônico},}$ $\alpha > 0$ e $ \beta \in \R.$ O subconjunto $\displaystyle{ \Omega \subset \mathbb{R}^{N}\,(N \geq 4)}$ será um domínio limitado e a não linearidade $\displaystyle{ f \in C(\overline{\Omega} \times \mathbb{R},\mathbb{R}) }.$ Em cada um dos resultados obtidos, consideraremos hipóteses técnicas e características diferentes para a função não linear $f$ e para o valor da constante $\beta.$ No terceiro Capítulo, estudamos uma equação do tipo côncavo super linear, da forma: \begin{equation*} \left \{ \begin{array}{rcll} \alpha \Delta^{2} u + \beta \Delta u & = & a(x)|u|^{s-2}u + f(x,u)\, & \mbox{em}\,\, \Omega \\ u = \Delta u & = & 0 \, &\mbox{sobre } \,\,\, \partial \Omega, \end{array} \right. \end{equation*} onde $\alpha > 0$ e $\beta \in (-\infty, \alpha \lambda_{1})$. Consideramos que a função $a \in L^{\infty}(\Omega)$ e que $s \in (1,2).$ Por fim, no último Capítulo vamos considerar um problema de quarta ordem no qual a não linearidade é do tipo côncavo-convexa. Mais precisamente, estudamos a seguinte classe de equações: \begin{equation*} \left\{ \begin{aligned} \alpha \Delta^{2} u + \beta \Delta u & = \mu a(x)|u|^{q-2}u + b(x)|u|^{p-2}u&\,\,\,\,\ &\mbox{em}\,\, \Omega \\ u = \Delta u & = 0 & \,\,\,\,&\mbox{sobre} \,\, \partial \Omega, \end{aligned} \right. \end{equation*} onde o parâmetro $\mu > 0$ e as potências $ 1 < q < 2 < p < 2 N /(N - 4)$. Adicionalmente supomos que as funções $\displaystyle{a, b : \Omega \rightarrow \mathbb{R} }$ sejam contínuas podendo trocar de sinal em $\Omega$ e que $a^{+},b^{+} \neq 0.$

Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-08-06T15:14:18Z No. of bitstreams: 1 ROMILDO NASCIMENTO DE LIMA - DISSERTAÇÃO PPGMAT 2013..pdf: 632336 bytes, checksum: 5661cad2fea6b9bb474c05bca0983c4b (MD5) / Made available in DSpace on 2018-08-06T15:14:18Z (GMT). No. of bitstreams: 1 ROMILDO NASCIMENTO DE LIMA - DISSERTAÇÃO PPGMAT 2013..pdf: 632336 bytes, checksum: 5661cad2fea6b9bb474c05bca0983c4b (MD5) Previous issue date: 2013-02 / Capes / Neste trabalho estaremos interessados em estudar resultados de existência e não existência de solução, comportamento do funcional energia e condição de Palais-Smale para sistemas de equações do tipo Schrödinger-Poisson; usaremos o método variacional. E, as soluções são pontos críticos do funcional energia associado ao problema. Para alcançar nossos objetivos, será fundamental o estudo das variedades de Ruiz e de Nehari, o Princípio Variacional de Ekeland, o teorema do Passo da Montanha, e o lema Concentração de Compacidade. / In this work we are interested in studying the results of existence and nonexistence of solution, behavior of the energy functional and Palais-Smale condition for systems of equations of the type Schrödinger-Poisson; by using variational approach. In fact the solutions are critical points of the energy functional associated with the problem. To achieve our goals, it is essential to study the Manifolds of Ruiz and Nehari, the Ekeland Variational Principle, the Mountain Pass theorem, and the Concentration-Compactness argument.

Ce mémoire de thèse est formé de quatre chapitres. Un premier chapitre présente les différentes notions et outils mathématiques utilisés dans le corps de la thèse ainsi qu’une description des résultats principaux que nous avons obtenus. Le second chapitre présente une généralisation d’un résultat obtenu par Rousselet-Chénais en 1990 qui décrit la sensibilité des sous-espaces propres d’opérateurs compacts auto-adjoints. Rousselet-Chénais se sont limités aux sous-espaces propres de dimension 1 et nous avons étendu leur résultat aux dimensions supérieures. Nous avons appliqué nos résultats à la Décomposition par Projection Orthogonale (POD) dans le cas de variation paramétrique, temporelle ou spatiale (Gappy-POD). Le troisième chapitre traite de l’estimation du flot optique avec des énergies quadratiques ou linéaires à l’infini. On montre des résultats mathématiques de convergence de la méthode de Décomposition Progressive Généralisée (PGD) dans le cas des énergies quadratiques. Notre démonstration est basée sur la décomposition de Brézis-Lieb via la convergence presque-partout de la suite gradient PGD. Une étude numérique détaillée est faite sur différents type d’images : sur les équations de transport de scalaire passif, dont le champ de déplacement est solution des équations de Navier-Stokes. Ces équations présentent un défi pour l’estimation du flot optique à cause du faible gradient dans plusieurs régions de l’image. Nous avons appliqué notre méthode aux séquences d’images IRM pour l’estimation du mouvement des organes abdominaux. La PGD a présenté une supériorité à la fois au niveau du temps de calcul (même en 2D) et au niveau de la représentation correcte des mouvements estimés. La diffusion locale des méthodes classiques (Horn & Schunck, par exemple) ralentit leur convergence contrairement à la PGD qui est une méthode plus globale par nature. Le dernier chapitre traite de l’application de la méthode PGD dans le cas d’équations elliptiques variationnelles dont l’énergie présente tous les défis aux méthodes variationnelles classiques : manque de convexité, manque de coercivité et manque du caractère borné de l’énergie. Nous démontrons des résultats de convergence, pour la topologie faible, des suites PGD (lorsqu’elles sont bien définies) vers deux solutions extrémales sur la variété de Nehari. Plusieurs questions mathématiques concernant la PGD restent ouvertes dans ce chapitre. Ces questions font partie de nos perspectives de recherche. / This thesis is formed of four chapters. The first one presents the mathematical notions and tools used in this thesis and gives a description of the main results obtained within. The second chapter presents our generalization of a result obtained by Rousselet-Chenais in 1990 which describes the sensitivity of eigensubspaces for self-adjoint compact operators. Rousselet-Chenais were limited to sensitivity for specific subspaces of dimension 1, we have extended their result to higher dimensions. We applied our results to the Proper Orthogonal Decomposition (POD) in the case of parametric, temporal and spatial variations (Gappy- POD). The third chapter discusses the optical flow estimate with quadratic or linear energies at infinity. Mathematical results of convergence are shown for the method Progressive Generalized Decomposition (PGD) in the case of quadratic energies. Our proof is based on the decomposition of Brézis-lieb via the convergence almost everywhere of the PGD sequence gradients. A detailed numerical study is made on different types of images : on the passive scalar transport equations, whose displacement fields are solutions of the Navier-Stokes equations. These equations present a challenge for optical flow estimates because of the presence of low gradient regions in the image. We applied our method to the MRI image sequences to estimate the movement of the abdominal organs. PGD presented a superiority in both computing time level (even in 2D) and accuracy representation of the estimated motion. The local diffusion of standard methods (Horn Schunck, for example) limits the convergence rate, in contrast to the PGD which is a more global approach by construction. The last chapter deals with the application of PGD method in the case of variational elliptic equations whose energy present all challenges to classical variational methods : lack of convexity, lack of coercivity and lack of boundedness. We prove convergence results for the weak topology, the PGD sequences converge (when they are well defined) to two extremal solutions on the Nehari manifold. Several mathematical questions about PGD remain open in this chapter. These questions are part of our research perspectives.