Doutorado em Matemática (IME)
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Item Álgebras associadas a grafos orientados em níveis e a propriedade da Koszulidade(Universidade Federal de Goiás, 2014-11-20) Vasconcelos, José Eder Salvador de; Lima, Aline de Souza; Silva, Jhone Caldeira; http://lattes.cnpq.br/6848751340618892; Rodrigues, Paulo Henrique de Azevedo; Lima, Aline de Souza; Guerreiro, Marinês; Lemos, Manoel José Machado Soares; Castonguay, DianeIn this work we study classes of algebras associated with layered directed graphs. Let be a layered directed graph. We determine the algebra Apq; generated by the edges of the graph, satisfying a set of quadratic relations R; and the dual algebra Apq!, associated with grpApqq. For each P Autpq we determine: the algebra Ap q; where is the subgraph of whose vertices are xed by ; the graded trace generating functions Tr pApq; tq and Tr pApq!; tq: We also determine the multiplicities of the irreducible representations of AutpApqq acting on Apq and Apq!: We show that for a layered directed graph , satisfying some hypotheses, AutpApqq K Autpq. Finally, we verify the property Tr pApq; tq Tr pApq!; tq 1 for all P Autpq, called koszulity property. We consider two classes of algebras, the algebra associated to the Hasse graph of the partially ordered set of faces of a star polygon, Ap q; and the algebra associated with the Hasse graph of the lattice of subespaces of a nite dimensional vector space over Fq; ApLpn; qqq:Item Bifurcações de campos vetoriais em duas zonas com simetria(Universidade Federal de Goiás, 2017-11-28) Castro, Ubirajara José Gama de; Medrado, João Carlos da Rocha; http://lattes.cnpq.br/5021927574622286; Tonon, Durval José; Pessoa, Cláudio Gomes; Martins, Ricardo Miranda; Oliveira, Regilene Delazari dos SantosIn this work we study reversible vector fields in two zones and equivariant vector fields in two zones. Our main result is the classification of the symmetric singularities of codimensions 0,1 and 2 of such vector fields. More precisely, in the reversible case in R3, where the dimension of the fixed points variety of the involution associated to the vector field is 2, we present all bifurcation diagram of the codimensions 1 and 2 singularities, describing the changes in the behavior of the symmetric singularities and tangents of the vector field with the transition manifold, S, according to the variation of the bifucartion parameter. We also show the existence of invariant cylinders and, in this case, doing small perturbations we determine invariant manifolds that persisted and we determine the number of limit cycles that were born. When the vector field defined on two zones is equivariant, the dynamic is enriched with the emergence of the sliding vector field and we also do a local study and the classification of singularities (and pseudo-singularities) of codimensions 0,1 and 2. We show the existence of homoclinic sliding orbit and that it is a codimension one phenomenon. Moreover, provided the symmetry we get a double Shilnikov sliding orbit.Item Campos descontínuos com chaveamento no Rn(Universidade Federal de Goiás, 2016-05-13) Silva , Tharsis Souza; Jaquemard , Alain Guy; Garcia , Ronaldo Alves; http://buscatextual.cnpq.br/buscatextual/visualizacv.do?id=K4787335Z3; Garcia , Ronaldo Alves; Buzzi , Claudio Aguinaldo; Teixeira , Marco Antonio; Jaquemard , Alain Guy; Medrado , João Carlos da RochaIn this work we _rstly study a relay system X on the Rn that, under certain conditions, it has a one parameter family of 1-periodic orbits that arises in the origin and increase inde_nitely. We study yet another relay system class X_, that it is formed from the initial relay system by aditions of nilpotent parameters that, under certain conditions, it has the same result of the previous, and yet family of periodic orbits that arises in the origin and ends in a loop, or family that bifurcate of a loop and arise inde_nitelly. Furthermore the periodic solutions are explicitely given by Euler polynomials. Finally we study a third order di_erential equation with relay looking for periodic orbits of di_erent degrre of di_erentiability and this is done by the associated vector _eld with jump.Item Ciclos limite e singularidades típicas de sistemas de equações diferenciais suaves por partes(Universidade Federal de Goiás, 2017-03-07) Cespedes, Oscar Alexander Ramírez; Medrado, João Carlos da Rocha; http://lattes.cnpq.br/5021927574622286; Tonon, Durval José; Buzzi, Claudio Aguinaldo; Silva, Paulo Ricardo da; Martins, Ricardo MirandaIn this work, we analize the version of Hilbert’s 16th problem for a piecewise linear differential system, PWLS, in R2. More precisely,we determinete the maximum number of certain types of limit cycles when the system is define in two zones separated by a straight line. Some results on the maximum number of cycles of a PWLS defined in two sectors were established. In addition, we classify typical singularities of a piecewise smooth systemin R3, taking into account the behavior of the associated sliding field.Item Ciclos limite e superfícies invariantes em sistemas diferenciais(Universidade Federal de Goiás, 2016-05-13) Freitas, Bruno Rodrigues de; Medrado, João Carlos da Rocha; http://lattes.cnpq.br/5021927574622286; Medrado, João Carlos da Rocha; http://lattes.cnpq.br/5021927574622286; Buzzi, Claudio Aguinaldo; Carvalho, Tiago de; garcia, Ronaldo Alves; Teixeira, Marco AntonioWe consider a class of piecewise linear di erential systems in R3 separated by a plane and we study its global and local dynamics. More precisely, we give conditions to the existence of invariant surfaces and limit cycles, presenting the maximum number of limit cycles and characterizing these invariant surfaces. Also, we obtain results about the T-singularity obtained by a perturbation of piecewise linear di erential systems. In our approach, we use many techniques, as an extension of the theorem’s Rolle for vector fields, Theory of Sturm’s sequence, extendedcomplete Tchebyche systems and extensions of Averaging theory.Item Classes de hipersuperfícies Weingarten generalizada no espaço euclidiano(Universidade Federal de Goiás, 2014-09-29) Dias, D. G.; Corro, Armando M.V.; http://lattes.cnpq.br/4498595305431615; Corro, Armando M.V.; Piccione, Paolo; Dorea, Chang C.Y.; Ferreira, W.; Adriano, LeviWe present hypersurfaces with prescribed normal Gauss map. These surfaces are obtained as the envelope of a sphere congruence where the other envelope is contained in a plane. We introduce classes of surfaces that generalize linear Weingarten surfaces, where the coefficients are functions that depend on the support function and the distance function from a fixed point (in short, DSGW-surfaces). The linear Weingarten surfaces, the Appell’s surfaces and the Tzitzeica’s surfaces are all DSGW-surfaces. From them we obtain new classes of DSGW-surfaces applying inversions and dilatations. For a special class of DSGW-surfaces, which is invariant under dilatations and inversions, we obtain a Weierstrass type representation (in short, EDSGW-surfaces). As application we classify the EDSGW-surfaces of rotation and present a 4-parameter family of complete cyclic EDSGW-surfaces with an isolated singularity and foliated by non-parallel planes. We generalized the EDSGW-surfaces for the case of hypersurfaces in Rn+1, n ≥ 2. We present a representation for these hypersurfaces in the case where the stereographic projection of the normal Gauss map N is given by the identity application. As an application, we will characterize the rotational examples.Item Classes de hipersuperfícies Weingarten generalizadas tipo Laguerre(Universidade Federal de Goiás, 2017-12-07) Ruys, Wesley da Silva; Corro, Armando Mauro Vasquez; http://lattes.cnpq.br/4498595305431615; Corro, Armando Mauro Vasquez; Riveros, Carlos Maber Carrion; Adriano, Levi Rosa; Velásquez, Marco Antonio Lázaro; Pina, Romildo da SilvaIn this work we present a classification of the Laguerre minimal surfaces with flat curvature lines. We introduce three classes of hypersurfaces that generalize the Laguerre minimal surfaces with the prescribed Gaussian normal application. The first class is associated to biharmonic applications and is related by a Legendre transformation to hypersurfaces that in the isotropic model has harmonic isotropic mean curvature. As an application, we classify the hypersurfaces of rotation and we present examples of these hypersurfaces parameterized by flat curvature lines. We obtain a characterization of the other two classes of hypersurfaces, we study the rotation ones and we present examples.Item Computing inexact K-steepest descent directions and a new line search procedure for vector optimization(Universidade Federal de Goiás, 2022-03-24) Vieira, Flávio Pinto; Pérez, Luis Román Lucambio; http://lattes.cnpq.br/6532280983965503; Ferreira, Orizon Pereira; http://lattes.cnpq.br/0201145506453251; Ferreira, Orizon Pereira; Pérez, Luis Román Lucambio; Prudente, Leandro da Fonseca; Fukuda, Ellen Hidem; Iusem, Alfredo NoelNeste trabalho, propomos uma nova busca linear para otimização vetorial e uma forma de calcular a direção σ-aproximada de máxima descida. Yunda Dong, em 2010 e 2012, introduziu um procedimento de busca linear para o método de Gradiente Conjugado usando apenas informações de primeira ordem, ou seja, sem utilizar valores funcionais. Estenderemos seus trabalhos para Otimização Vetorial. Estudaremos o método de gradiente conjugado, mostrando a convergência quando são utilizados os seguintes βk's: Fletcher-Reeves, conjugate descent, Dai-Yuan, Polak-Ribière-Polyak e Hestenes-Stiefel. Também usamos essa mesma busca linear para o método tipo-gradiente, mostrando sua convergência. Em 2004, Iusem e Graña Drummond introduziram o conceito de σ-aproximada K-diereção de máxima descida. Eles mostraram que ao substituir a direção de Cauchy por essas direções, o resultado de convergência da sequência gerada é o mesmo: todo ponto de acumulação é crítico. Apresentaremos um procedimento eficiente para calcular essas direções quando o cone K for finitamente gerado.Item Conditional gradient methods for multiobjective optimization(Universidade Federal de Goiás, 2021-08-06) Assunção Filho, Pedro Bonfim de; Prudente, Leandro da Fonseca; http://lattes.cnpq.br/4573611419840935; Ferreira, Orizon Pereira; http://lattes.cnpq.br/0201145506453251; Ferreira, Orizon Pereira; Prudente, Leandro da Fonseca; Melo, Jefferson Divino Gonçalves de; Bento, Glaydston de Carvalho; Souza, João Carlos de OliveiraNeste trabalho, analisamos o método do gradiente condicional, também conhecido como método de Frank-Wolfe, para resolver problemas de otimização multiobjetivo restrita. Também propomos e analisamos uma versão generalizada deste método para resolver problemas de otimização composta multiobjetivo que consistem em minimizar simultaneamente várias funções objetivo. Cada função objetiva é a soma de duas funções, uma é considerada continuamente diferenciável e a outra não é necessariamente diferenciável. Ambos os métodos são analisados com três estratégias de obtenção dos tamanhos dos passos, a saber: tipo Armijo, adaptativos e tamanhos decrescentes dos passos. Propriedades de convergência assintótica e limites de complexidade de iteração com e sem suposições de convexidade na função objetivo são estabelecidas. Experimentos numéricos para o método do gradiente condicional são fornecidos para ilustrar a eficácia do método e certificar os resultados teóricos obtidos.Item Conjuntos limite e transitividade de campos vetoriais suaves por partes em variedades Riemannianas bi-dimensionais(Universidade Federal de Goiás, 2020-12-14) Jucá, Joaby de Souza; Varão Filho, José Régis Azevedo; http://lattes.cnpq.br/9729493483105088; Euzébio, Rodrigo Donizete; http://lattes.cnpq.br/9213320273714493; Euzébio, Rodrigo Donizete; Varão Filho, José Régis Azevedo; Oliveira, Regilene Delazari dos Santos; Buzzi, Claudio Aguinaldo; Tonon, Durval JoséIn this work we study piecewise-smooth vector fields defined on a two-di\-men\-sio\-nal differential manifold M, according to the Filippov convention. In the first part, M is considered as being any Riemannian manifold and we present a classification of the possible limit sets for a maximal trajectory whose its positive branch is contained on a compact subset $K\subset M$ (Theorem 3.1). We consider the occurrence of sliding motion and we verify the presence of limit sets with non-empty interior that can present a non-deterministic chaotic behavior. Moreover, we provide some examples and classes of systems satisfying the hypotheses of the main results. In the second part, we study the topological transitivity of piecewise-smooth vector fields defined on the two-dimensional sphere $S^2$. We guarantee the existence of an one-parameter family of topologically transitive piecewise-smooth vector fields on $S^2$ (Theorem 4.1), which does not happen for continuous vector fields on $S^2$. We prove that the occurrence of transitivity on $S^2$ implies the existence of escaping and sliding regions. We also prove they connect to each other through infinitely many Filippov trajectories. Moreover, we prove that there exist no robustly transitive piecewise-smooth vector fields on $S^2$.Item Crossing periodic orbits in some piecewise smooth vector fields: local and global approach(Universidade Federal de Goiás, 2021-11-09) Velter, Mariana Queiroz; Tonon, Durval José; http://lattes.cnpq.br/3688981956532711; Tonon, Durval José; Medrado, João Carlos da Rocha; Freitas, Bruno Rodrigues de; Buzzi, Claudio Aguinaldo; Oliveira, Regilene Delazare dos SantosNeste trabalho estudaremos as órbitas periódicas costurantes de algumas famílias de campos de vetores por partes definidos em e em segundo a convenção de Filippov. Inicialmente daremos a cota máxima para o número de ciclos limite costurante para campos de vetores linear por partes formados por centros definidos em onde a variedade de descontinuidade é dada pela união de duas semirretas que determinam um ângulo , onde 0 < < . Posteriormente descreveremos todas as órbitas periódicas costurantes e as órbitas homoclínicas de dois campos de vetores por partes em ambos formados por campos de vetores completamente integráveis, sendo um linear por partes e outro não linear. Finalmente, estudaremos a existência de ciclos limite costurantes em duas famílias de campos de vetores linear por partes que possuem duas retas de tangência na variedade de descontinuidade. A primeira família possui um ponto de cúspide para cada reta de tangência e a matriz associada a cada campo de vetores que a formam é a mesma. Para essa família, mostramos a existência de dois ciclos limite costurante que bifurcam da origem através da análise dos pontos fixos da aproximação em série de Taylor da aplicação de primeiro retorno em uma vizinhança da origem. O campo deslizante associado a essa família também é estudado. A segunda família possui somente tangências do tipo dobra invisível e a matriz associada a cada campo de vetores que a formam pode ser diferente. Para essa família, descreveremos as bifurcações que ocorrem no campo deslizante associado e mostraremos que este não possui ciclo limite. Mostraremos também a não existência de órbitas periódicas costurantes para essa família quando ambas matrizes associadas aos campos de vetores que a formam possuem autovalores complexos com parte real nula através de suas integrais primeiras.Item Desigualdade de Caffarelli-Kohn-Nirenberg e solitons de Yamabe gradiente(Universidade Federal de Goiás, 2019-05-31) Tokura, Willian Isao; Adriano, Levi Rosa; http://lattes.cnpq.br/3206466156270217; Adriano, Levi Rosa; Silva, Edcarlos Domingos da; Pina, Romildo da Silva; Sousa, Paulo Alexandre Araújo; Ribeiro Junior, Ernani de SousaThis thesis deals with two distinct problems. Namely, we study [(P1)] Rigidity of metric spaces that support CKN inequality; [(P2)] Gradient Yamabe solitons on top of warped product manifolds B x f F. For the first problem, we prove that the metric measure spaces that support the CKN inequality have n-dimensional volume growth, that is, there exists a universal constant C 0gt; 0 such that, m(B x (ρ)) ≥ C 0 ρ n , ∀x ∈ M, ρ gt; 0. As application, some rigidity theorems are obtained in the following spaces: Riemannian manifolds, Finsler manifolds and Alexandrov spaces. For the second problem, taking a gradient Yamabe soliton (B x f F, g, h, ρ), we obtain triviality results for h and f by means of some hypotheses on the base B. Furthermore, under a hypothesis involving the Ricci curvature of the base Ric gB , we prove estimates for h, f and for scalar curvature scal g , in addition, by means of a warping gradient estimates, we present a beautiful obstruction in the construction of gradient Yamabe solitons on warped product manifolds. Finally, by making use of invariant solution techniques, we classify all steady gradient Yamabe solitons with a conformally flat base that is invariant by the action of a codimension 1 translation group.Item Uma desigualdade de Minkowski e soluções exatas para o espaço-tempo estático de Einstein-Maxwell(Universidade Federal de Goiás, 2023-02-24) Costa, Ana Paula de Melo da; Leandro Neto, Benedito; http://lattes.cnpq.br/3393448440968708; Leandro Neto, Benedito; Cruz, Cícero Tiarlos Nogueira; Adriano, Levi Rosa; Silva, Maria de Andrade Costa e; Batista, Rondinelle MarcolinoWe study the static Einstein-Maxwell space when it is conformal to an n-dimensional pseudo-Euclidean space, which is invariant under the action of an (n−1)-dimensional translation group. We also provide a complete classification of such space. Moreover, we prove a Minkowski-like inequality for an asymptotically flat static Einstein- Maxwell (electrovacuum) space-time using as an approach the inverse mean curvature flow (IMCF).Item Desigualdades universais para autovalores do polidrifting laplaciano em dominios compactos do R^n e S^n(Universidade Federal de Goiás, 2016-03-08) Pereira, Rosane Gomes; Adriano, Levi Rosa; http://buscatextual.cnpq.br/buscatextual/visualizacv.do?id=K4768692Y1; Adriano, Levi Rosa; Pina, Romildo da Silva; Changyu, Xia; Gonçalves, José Vlado Abreu; Barbosa, Ezequiel RodriguesIn this work, we study eigenvalues of poly-drifting laplacian on compact Riemannian manifolds with boundary (possibly empty). Here, we bring a universal inequality for the eigenvalues of the poly-drifting operator on compact domains in an Euclidean spaceRn. Besides,weintroduce universal inequalities for eigenvalues of poly-drifting operator on compact domains in a unit n-sphere Sn. We give an universal inequality for lower order eigenvalues of the poly-drifting operator inRn and Sn. Moreover, we prove an universal inequality type Ashbaugh and Benguria for the drifting Laplacian on Riemannian manifold immersed in an unit sphere or a projective space. Let be a bounded domain in a n-dimensional Euclidean space Rn. We study eigenvalues of an eigenvalue problem of a system of elliptic equations of the drifting laplacian 8>><>>: L u+ (r(divu)r divu) = ¯ u; in ; uj@ = 0 Estimates for eigenvalues of the above eigenvalue problem are obtained. Furthermore, a universal inequality for lower order eigenvalues of the problem is also derived.Item Electrostatic system and divergence formulas(Universidade Federal de Goiás, 2023-02-16) Santos, Róbson Lousa dos; Leandro Neto, Benedito; http://lattes.cnpq.br/3393448440968708; Leandro Neto, Benedito; Cederbaum, Rafaela Carla Deborah; Silva, Maria de Andrade Costa e; Baltazar, Halyson Irene; Santos, João Paulo dosUma questão clássica em relatividade geral é a classificação de soluções de buracos negros regulares estáticos das equações Eisntein-Maxwell (ou sistema eletrovácuo). Nós provamos alguns resultados de classificação para um sistema eletrovácuo tal que o potencial elétrico é uma função diferenciável da função lapso. Nós, particularmente, mostramos que um espaço ndimensional eletrovácuo localmente conformemente plano satisfazendo algumas condições deve estar na classe Majumdar-Papapetrou. Além disso, nós provamos que qualquer espaço eletrovácuo de dimensão 3 ou 4 em que algumas condições são satisfeitas deve ser localmente conformemente plano. Mais ainda, nós demonstramos que um espaço electrovácuo ndimensional satisfazendo algumas condições, sem divergência de quarta ordem do tensor de Weyl e curvatura radial de Weyl zero tal que o potencial elétrico está na classe ReissnerNordström é localmente uma variedade produto torcido com fibra Einstein de dimensão n − 1. Finalmente, um espaço electrovácuo tridimensional satisfazendo algumas condições, sem divergência de terceira ordem do tensor de Cotton, também é classificado. Nós também provamos que variedades eletrostáticas (ou eletrovácuos) tridimensional com constante cosmológica não nula e tensor de Bach livre de divergência são localmente conformemente planos, desde que o campo elétrico e o gradiente da função lapso sejam linearmente dependentes. Consequentemente, uma variedade eletrostática tridimensional admite uma estrutura local de produto torcido com uma base unidimensional e fibra uma superfície de curvatura constante.Item Equações diferenciais parciais elípticas multivalentes: crescimento crítico, métodos variacionais(Universidade Federal de Goiás, 2013-09-27) Carvalho, Marcos Leandro Mendes; Gonçalves, José Valdo Abreu; http://lattes.cnpq.br/5148611284176776; Gonçalves, José Valdo Abreu; Mota, Jesus Carlos da; Silca, EdCarlos Domingos da; Alves, Claudianor Oliveira; Santos, Carlos Alberto Pereira dosIn this work we develop arguments on the critical point theory for locally Lipschitz functionals on Orlicz-Sobolev spaces, along with convexity, minimization and compactness techniques to investigate existence of solution of the multivalued equation −∆Φu ∈ ∂ j(.,u) +λh in Ω, where Ω ⊂ RN is a bounded domain with boundary smooth ∂Ω, Φ : R → [0,∞) is a suitable N-function, ∆Φ is the corresponding Φ−Laplacian, λ > 0 is a parameter, h : Ω → R is a measurable and ∂ j(.,u) is a Clarke’s Generalized Gradient of a function u %→ j(x,u), a.e. x ∈ Ω, associated with critical growth. Regularity of the solutions is investigated, as well.Item Estrutura topológica do conjunto de soluções de perturbações não lineares do p-laplaciano(Universidade Federal de Goiás, 2014-06-23) Marcial, Marcos Roberto; Gonçalves, José Valdo Abreu; http://lattes.cnpq.br/5148611284176776; Gonçalves, José Valdo Abreu; Mota, Jesus Carlos da; Miyagaki, Olimpio Hiroshi; Santos, Carlos Alberto Pereira dos; Silva, Maxwell Lizete daIn 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.Item Estudo qualitativo de equações diferenciais binárias cúbicas(Universidade Federal de Goiás, 2022-12-05) Maranhão Neto, Raimundo Cavalcante; Garcia, Ronaldo Alves; http://lattes.cnpq.br/5680428710939826; Garcia, Ronaldo Alves; Freitas, Bruno Rodrigues de; Martins, Luciana de Fátima; Euzébio, Rodrigo Donizete; Buzzi, Claudio AguinaldoIn this work we present a qualitative study for two classes of differential equations. The first of these is of the form Im[(a + i b)(du + i dv)3 ] = 0 (0-1) where a, b : R 2 → R are functions of class C∞ and the second is from the implicit differential equation of the Laguerre lines of a surface of class C6 . This second class, as proved in [5], has the shape A3(u, v) dv3 + 3 A2(u, v) dv2 du + 3 A1(u, v) dv du2 + A0(u, v) du3 = 0. With regard to equations of the form (0-2), we perform a local study, express the derivative of the application of the first return, we classify the singularities at infinity and present a global result for the case where a and b are polynomials of degree one. For the differential equation of the Laguerre lines, we studied the qualitative behavior close to the discriminant curve, we made a partial study of the singularities, we presented an expression for the derivative of the application of the first return, we carried out a study of structural stability and we studied the particular cases for surfaces of rotation , ruled surfaces and quadric surfaces.Item Existência e multiplicidade de soluções de problemas de autovalor não lineares elípticos(Universidade Federal de Goiás, 2015-07-03) Silva, Kaye Oliveira da; Gonçalves, José Valdo Abreu; http://buscatextual.cnpq.br/buscatextual/visualizacv.do?id=K4781975P5; Gonçalves, José Valdo Abreu; Correa, Francisco Julio Sobreira de Araujo; Rodrigues, Rodrigo da Silva; Mota, Jesus Carlos da; Silva, Edcarlos Domingos daIn this work, we study two problems in partial differential equations. The first one is a nonlinear eigenvalue problem given by: ( div( (jruj)ru) = f(x; u) em , u = 0 em @ , where the nonlinearity f is oscilatory. By using Orlicz-Sobolev spaces and techniques of minimization, degree theory, lower and upper solutions and regularization of solutions, we show that for each sufficiently big, there is a family of solutions, which is finite when f oscillates a finite number of times (with respect to the second variable) and it is infinite when f oscillates infinitely many times. On the second problem, we use the shooting method, to show that the problem: ( (r (ju0(r)j)u0(r))0 = r f(u(r)); 0 < r < R; u(R) = u0(0) = 0; has for each sufficiently small, a family fukg1k =1 of solutions, where for each positive integer k, uk has exactly k roots in the interval (0;R).Item Fluidos perfeitos estáticos com simetrias(Universidade Federal de Goiás, 2019-04-25) Barboza, Marcelo Bezerra; Pina, Romildo da Silva; http://lattes.cnpq.br/2675728978857991; Corro, Armando Mauro Vasquez; Leandro Neto, Benedito; Manfio, Fernando; Marrocos, Marcus Antônio Mendonça; Pina, Romildo da SilvaThis work presents a two step procedure that is virtually capable of producing an infinite number of exact solutions to Einstein's equation of a perfect fluid on a static manifold. These steps could roughly be described as: 1) classifying the symmetries of the referred equation that convert it into a second order non linear ordinary differential equation of very specific nature -- whose solutions are a whole lot easier to come up with than those of the original problem, and 2) solving this ordinary equation -- which quite explains the need for the word `virtually' above, since not all solutions of the ordinary equation are known to its exact form. Finally, in the last chapter, we utilize a Theorem due to Liouville to determine the rigid motions of Riemannian metrics on euclidean space that do admit symmetries in a translational group and also belong to the conformal class of the flat metric.