In this thesis, we study homogenization of optimal control problems in various oscillatory domains. Specifically, we consider four types of domains given in Figure 1 below. Figure 1: Oscillating Domains The thesis is organized into six chapters. Chapter 1 provides an introduction to our work and the rest of the thesis. The main contributions of the thesis are contained in Chapters 2-5. Chapter 6 presents the conclusions of the thesis and possible further directions. A brief description of our work (Chapters 2-5) follows: Chapter 2: Asymptotic behaviour of a fourth order boundary optimal control problem with Dirichlet boundary data posed on an oscillating domain as in Figure 1(A) is analyzed. We use the unfolding operator to study the asymptotic behavior of this problem. Chapter 3: Homogenization of a time dependent interior optimal control problem on a branched structure domain as in Figure 1(B) is studied. Here we pose control on the oscillating interior part of the domain. The analysis is carried out by appropriately defined unfolding operators suitable for this domain. The optimal control is characterized using various unfolding operators defined at each branch of every level. Chapter 4: A new unfolding operator is developed for a general oscillating domain as in Figure 1(C). Homogenization of a non-linear elliptic problem is studied using this new un-folding operator. Using this idea, homogenization of an optimal control problem on a circular oscillating domain as in Figure 1(D) is analyzed. Chapter 5: Homogenization of a non-linear optimal control problem posed on a smooth oscillating domain as in Figure 1(C) is studied using the unfolding operator.

Yes / Seeking efficient solutions to nonlinear boundary value problems is a crucial challenge in the mathematical modelling of many physical phenomena. A well-known example of this is solving the Biharmonic equation relating to numerous problems in fluid and solid mechanics. One must note that, in general, it is challenging to solve such boundary value problems due to the higher-order partial derivatives in the differential operators. An artificial neural network is thought to be an intelligent system that learns by example. Therefore, a well-posed mathematical problem can be solved using such a system. This paper describes a mesh free method based on a suitably crafted deep neural network architecture to solve a class of well-posed nonlinear boundary value problems. We show how a suitable deep neural network architecture can be constructed and trained to satisfy the associated differential operators and the boundary conditions of the nonlinear problem. To show the accuracy of our method, we have tested the solutions arising from our method against known solutions of selected boundary value problems, e.g., comparison of the solution of Biharmonic equation arising from our convolutional neural network subject to the chosen boundary conditions with the corresponding analytical/numerical solutions. Furthermore, we demonstrate the accuracy, efficiency, and applicability of our method by solving the well known thin plate problem and the Navier-Stokes equation.

Estudamos o problema de valores de contorno {\'DELTA POT. 2\' u = f em \'OMEGA\', \'BETA\' u = 0 em \'PARTIAL OMEGA\', um aberto limitado \'OMEGA\' \'ESTÁ CONTIDO\' \'R POT. N\' , sob diferentes condições de contorno. As questões de existência e positividade de soluções para este problema são abordadas com condições de contorno de Dirichlet, Navier e Steklov. Deduzimos condições de contorno naturais através do estudo de um modelo para uma placa com carga estática. Estudamos ainda propriedades do primeiro autovalor de \'DELTA POT. 2\' e o problema semilinear {\'DELTA POT. 2\' u = F (u) em \'OMEGA\' u = \'PARTIAL\'u SUP . \'PARTIAL\' v = 0 em \'PARTIUAL\' \'OMEGA\', para não-linearidades do tipo F(t) = \'l t l POT. p-1\', p \' DIFERENTE\' t, p > 0. Para tal problema estudamos existência e não-existência de soluções e positividade / We study the boundary value problem {\'DELTA POT. 2\' u = f in \'OMEGA\', \'BETA\' u = 0 in \'PARTIAL OMEGA\', in a bounded open \'OMEGA\'\'THIS CONTAINED\' \'R POT. N\' , under different boundary conditions. The questions of existence and positivity of solutions for this problem are addressed with Dirichlet, Navier and Steklov boundary conditions. We deduce natural boundary conditions through the study of a model for a plate with static load. We also study properties of the first eigenvalue of \'DELTA POT. 2\' and the semi-linear problem { \'DELTA POT. 2\' e o problema semilinear {\'DELTA POT. 2\' u = F (u) in \'OMEGA\' u = \'PARTIAL\'u SUP . \'PARTIAL\' v = 0 in \'PARTIUAL\' \'OMEGA\', for non-linearities like F(t) = \'l t l POT. p-1\', p \' DIFFERENT\' t, p > 0. For such problem we study existence and non-existence of solutions and its positivity

Neste trabalho estudamos a existência e a multiplicidade de soluções clássicas positivas para uma classe de problemas de quarta-ordem sob a condição de fronteira de Navier, relacionando o número de soluções com a topologia do domínio, mais precisamente, com sua categoria de Lusternik-Schnirelman. Introduzimos também uma noção de regiões crítica e não-crítica associadas a um de nossos problemas, a fim de garantir condições para existência de solução / In this work we study the existence and multiplicity of positive classical solutions for a class of fourth-order problems under Navier boundary condition, relating the number of solutions to the domain topology, more specifically, to its Lusternik-Schnirelman category. We also introduce the notion of critical and noncritical regions related to one of our problems, in order to ensure conditions to existence of solutions

Submitted by ANA KARLA PEREIRA RODRIGUES (anakarla_@hotmail.com) on 2017-08-14T11:52:04Z No. of bitstreams: 1 arquivototal.pdf: 1035035 bytes, checksum: 24db9b859fa0c32ac6b5b442ef6e12fa (MD5) / Made available in DSpace on 2017-08-14T11:52:04Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 1035035 bytes, checksum: 24db9b859fa0c32ac6b5b442ef6e12fa (MD5) Previous issue date: 2016-11-28 / In this work we study the existence of multi-bump solutions to a certain class of elliptic problems involving biharmonic problems. Moreover, we apply the method developed to biharmonic for study the existence of multi-bump solutions to Choquard Equation. / Neste trabalho estudamos a existência de soluções multi-bump para uma determinada classe de problemas elípticos que envolvem o operador Biharmônico. Além disso, aplicamos o método desenvolvido para o biharmônico no estudo da existência de solução multi-bump para equação de Choquard.

Made available in DSpace on 2014-07-29T16:02:19Z (GMT). No. of bitstreams: 1 Dissertacao Kaye O da Silva.pdf: 935849 bytes, checksum: 3342ffadf63161660c1795053815a170 (MD5) Previous issue date: 2012-02-24 / In this work, we employ topological methods in order to study existence and multiplicity of solutions, of nonlinear boundary value problems of the fourth order. More precisely, we make use of results on connected components of fixed points, as well as global bifurcation, to show existence and multiplicity of weak solutions of Partial Differential Equations, involving the Biharmonic operator under Navier boundary conditions. Proofs of the abstract results used, are presented in detail. / Neste trabalho, utilizamos métodos topológicos para estudar existência e multiplicidade de soluções de Problemas de Contorno Elípticos Não Lineares de 4a ordem. Mais precisamente, utilizamos resultados sobre componentes conexas de pontos fixos e tambem bifurcação global, para provar existência e multiplicidade de soluções fracas de Equações Diferenciais Parciais, envolvendo o Operador Binarmônico, sob condições de fronteira de Navier. As demonstrações dos resultados abstratos que utilizamos, são apresentadas em detalhes.

Submitted by Franciele Moreira (francielemoreyra@gmail.com) on 2017-12-27T13:50:44Z No. of bitstreams: 2 Tese - Wesley da Silva Ruys - 2017.pdf: 2660976 bytes, checksum: 3c6402ac0974e65da560b50e1f65b52e (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2017-12-28T09:42:07Z (GMT) No. of bitstreams: 2 Tese - Wesley da Silva Ruys - 2017.pdf: 2660976 bytes, checksum: 3c6402ac0974e65da560b50e1f65b52e (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2017-12-28T09:42:07Z (GMT). No. of bitstreams: 2 Tese - Wesley da Silva Ruys - 2017.pdf: 2660976 bytes, checksum: 3c6402ac0974e65da560b50e1f65b52e (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2017-12-07 / Fundação de Amparo à Pesquisa do Estado de Goiás - FAPEG / In 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. / Neste trabalho apresentamos uma classificação das superfícies mínimas de Laguerre com linhas de curvatura planas. Introduzimos três classes de hipersuperfícies que generalizam as superfícies mínimas de Laguerre com aplicação normal de Gauss prescrita. A primeira classe está associada a aplicações biharmônicas e está relacionada por uma transformação de Legendre a hipersuperfícies que no modelo isotrópico tem curvatura média isotrópica harmônica. Como aplicação, classificamos as hipersuperfícies de rotação e apresentamos exemplos destas hipersuperfícies parametrizadas por linhas de curvatura planas. Obtemos uma caracterização das outras duas classes de hipersuperfícies, estudamos as de rotação e apresentamos exemplos.

Estudamos o problema de valores de contorno {\'DELTA POT. 2\' u = f em \'OMEGA\', \'BETA\' u = 0 em \'PARTIAL OMEGA\', um aberto limitado \'OMEGA\' \'ESTÁ CONTIDO\' \'R POT. N\' , sob diferentes condições de contorno. As questões de existência e positividade de soluções para este problema são abordadas com condições de contorno de Dirichlet, Navier e Steklov. Deduzimos condições de contorno naturais através do estudo de um modelo para uma placa com carga estática. Estudamos ainda propriedades do primeiro autovalor de \'DELTA POT. 2\' e o problema semilinear {\'DELTA POT. 2\' u = F (u) em \'OMEGA\' u = \'PARTIAL\'u SUP . \'PARTIAL\' v = 0 em \'PARTIUAL\' \'OMEGA\', para não-linearidades do tipo F(t) = \'l t l POT. p-1\', p \' DIFERENTE\' t, p > 0. Para tal problema estudamos existência e não-existência de soluções e positividade / We study the boundary value problem {\'DELTA POT. 2\' u = f in \'OMEGA\', \'BETA\' u = 0 in \'PARTIAL OMEGA\', in a bounded open \'OMEGA\'\'THIS CONTAINED\' \'R POT. N\' , under different boundary conditions. The questions of existence and positivity of solutions for this problem are addressed with Dirichlet, Navier and Steklov boundary conditions. We deduce natural boundary conditions through the study of a model for a plate with static load. We also study properties of the first eigenvalue of \'DELTA POT. 2\' and the semi-linear problem { \'DELTA POT. 2\' e o problema semilinear {\'DELTA POT. 2\' u = F (u) in \'OMEGA\' u = \'PARTIAL\'u SUP . \'PARTIAL\' v = 0 in \'PARTIUAL\' \'OMEGA\', for non-linearities like F(t) = \'l t l POT. p-1\', p \' DIFFERENT\' t, p > 0. For such problem we study existence and non-existence of solutions and its positivity

Neste trabalho estudamos a existência e a multiplicidade de soluções clássicas positivas para uma classe de problemas de quarta-ordem sob a condição de fronteira de Navier, relacionando o número de soluções com a topologia do domínio, mais precisamente, com sua categoria de Lusternik-Schnirelman. Introduzimos também uma noção de regiões crítica e não-crítica associadas a um de nossos problemas, a fim de garantir condições para existência de solução / In this work we study the existence and multiplicity of positive classical solutions for a class of fourth-order problems under Navier boundary condition, relating the number of solutions to the domain topology, more specifically, to its Lusternik-Schnirelman category. We also introduce the notion of critical and noncritical regions related to one of our problems, in order to ensure conditions to existence of solutions

The plumage colour and patterns of birds have interested biologists for a long time.Why are some bird species all black while others have a multitude of colours? Does ithave anything to do with sexual selection, predator avoidance or social signalling?Many questions such as these have been asked and as many hypotheses about thefunctional role of the plumage have been formed. The problem, however, has been toprove any of these. To test these hypotheses you need to analyse the bird plumagesand today such analyses are still rather subjective. Meaning the results could varydepending on the individual performing the analysis. Another problem that stemsfrom this subjectiveness is that it is difficult to make quantitative measurements of theplumage colours. Quantitative measurements would be very useful since they couldbe related to other statistical data like speciation rates, sexual selection and ecologicaldata. This thesis aims to assist biologists with the analysis and measurement of birdplumages by developing a MATLAB toolbox for this purpose. The result is a wellstructured and user friendly toolbox that contains functions for segmenting, resizing,filtering and warping, all used to prepare the images for analysis. It also containsfunctions for the actual analysis such as basic statistical measurements, principalcomponent analysis and eigenvector projection.