[pt] O estudo de equaçõe semi-lineares do tipo Ambrosetti-Prodi frequentemente usa regularidade da não linearidade. Nesse texto, consideramos nãoo linearidades Lipschitz. Os argumentos geométricos baseados em teoremas de função implícita são substituidos pelo uso de contrações adequadas. / [en] The study of semi-linear equations of Ambrosetti-Prodi type frequently makes use of some smoothness of the nonlinearity. In this text, we consider Lipschitz nonlinearities. The geometric arguments based on implicit functions thoerems are replaced by appropriate contractive mappings.

Cette thèse est consacrée à l’analyse numérique en particulier aux estimations a posteriori de l’erreur dans la méthode de décomposition asymptotique partielle de domaine. Il s’agit de problèmes au dérivées partielles elliptiques linéaires et semi- linéaires avec une source qui ne dépend que d’une seule variable dans une partie du domaine. La MAPDD - Méthod of Asymptotic Partial Domain Decomposition - est une méthode inventée par Grigori . Panasenko et développée dans les références [G.P98, G.P99]. L’aidée principale est de remplacer un problème 3D ou 2D par un problème hybride combinée 3D−1D, 3D−2D ou 2D−1D, ou la dimension du problème diminue dans une partie du domaine. Des méthodes de calcul efficaces de solution pour le problème hybride en résultant sont récemment devenues disponibles pour plusieurs systèmes (linéaires/non linéaires, fluide/solide, etc.) ainsi chaque sous-problème est calcul ́ avec un code indépendant de type boîte noire [PBB10, JLB09, JLB11]. La position de la jonction entre les problèmes hétérogènes est asymptotiquement estimée dans les travaux de G. Panasenko [G.P98]. La méthode MAPDD a été conçu pour traiter des problèmes ou un petit paramètre apparaître, et fournit un développement en série de la solution avec des solutions de problèmes simplifiées à l’égard de ce petit paramètre. Dans le problème considéré dans les chapitres 3 et 4, aucun petit paramètre n’existe, mais en raison de considérations géométriques concernant le domaine on suppose que la solution ne diffère pas significativement d’une fonction qui dépend seulement d’une variable dans une partie du domaine Ω. La théorie de MAPDD n’est pas adaptée pour une telle situation, et si cette théorie est appliquée formellement elle ne fournit pas d’estimation d’erreur. / This thesis is devoted to numerical analysis in particular a postoriori estimates of the error in the method of asymptotic partial domain decomposition. There are problems in linear elliptic partial and semi-linear with a source which depends only of one variable in a portion of domain. Method of Asymptotic Partial Decomposition of a Domain (MAPDD) originates from the works of Grigori.Panasonko [12, 13]. The idea is to replace an original 3D or 2D problem by a hybrid one 3D − 1D; or 2D − 1D, where the dimension of the problem decreases in part of domain. Effective solution methods for the resulting hybrid problem have recently become available for several systems (linear/nonlinear, fluid/solid, etc.) which allow for each subproblem to be computed with an independent black-box code [21, 17, 18]. The location of the junction between the heterogeneous problems is asymptotically estimated in the works of Panasenko [12]. MAPDD has been designed for handling problems where a small parameter appears, and provides a series expansion of the solution with solutions of simplified problems with respect to this small parameter. In the problem considered in chapter 3 and 4, no small parameter exists, but due to geometrical considerations concerning the domain Ω it is assumed that the solution does not differ very much from a function which depends only on one variable in a part of the domain. The MAPDD theory is not suited for such a context, but if this theory is applied formally it does not provide any error estimate. The a posteriori error estimate proved in this chapter 3 and 4, is able to measure the discrepancy between the exact solution and the hybrid solution which corresponds to the zero-order term in the series expansion with respect to a small parameter when it exists. Numerically, independently of the existence of an asymptotical estimate of the location of the junction, it is essential to detect with accuracy the location of the junction. Let us also mention the interest of locating with accuracy the position of the junction in blood flows simulations [23]. Here in this chapter 3,4 the method proposed is to determine the location of the junction (i.e. the location of the boundary Γ in the example treated) by using optimization techniques. First it is shown that MAPDD can be expressed with a mixed domain decomposition formulation (as in [22]) in two different ways. Then it is proposed to use an a posteriori error estimate for locating the best position of the junction. A posteriori error estimates have been extensively used in optimization problems, the reader is referred to, e.g. [1, 11].

Šiame darbe nagrinėjame elipsinės lygties stačiakampėje srityje su nelokaliąja sąlyga sprendimą baigtinių skirtumų metodu. Sprendžiame dvimates skirtumines lygčių sistemas, jas gavome pakeitę diferencialinę lygtį skirtumine. Trumpai apžvelgtas maksimumo principas ir sprendinio radimas iteraciniais metodais bei tikrinių reikšmių radimas dvimačiu atveju. Įvertinta skirtuminės lygčių sistemos paklaida, kuri gaunama sprendžiant elipsinę lygtį skirtuminiu metodu. Darbo pabaigoje išspręstas konkretus uždavinys. / In this work we consider two dimensional elliptic equation on the rectangle with non local condition by finite difference method. We solve two dimensional equations instead one intricate differential equation. A short review of maximum principle and solution finding with iteration method, and the proper account finding with two dimensional case. Estimated differential equationerror, this making calculate elliptic equation difference method. Finally we solve particilar example with different steps.

[pt] Um resultado clássico no âmbito de equações diferenciais parciais e de geometria diferencial é o seguinte: Dada uma constante a existe uma condição da fronteira do domínio (Omega) de maneira que o problema de Dirichlet para a equação da curvatura média a no espaço Euclidiano é sempre solúvel. Este é um teorema devido a Serrin (1969). Além disso, se a condição de Serrin não for satisfeita, há um resultado de não-existência. A partir disso foi perguntado se um resultado similar valeria no espaço Hiperbólico. A finalidade desta tese é dar uma resposta afirmativa a esta pergunta, exibindo uma condição tipo Serrin. De maneira que obtém-se existência de superfícies cujo gráfico tenha curvatura média hiperbólica pré-determinada H(x) no espaço hiperbólico. O resultado é sharp no sentido que se tal condição for negada então não-existência pode ser estabelecida. O ponto central é uma estimativa a priori do gradiente de uma tal solução. / [en] A classical result in Partial Differential Equations and Differential Geometrydue to Serrin (1969) is the following: Given a constant (alfa) there exists a condition on the boundary of the domain (omega)such that the Dirichlet problem for the mean equation (alfa)is solvable. Besides, if Serrin's condition fails there is a non-existence result. Taking into account this classical result one may ask if a similar theorem holds in hyperbolic space. The goal of this thesis is to give a positive answer to this question establishing a certain Serrin type condition. Thus we obtain existence of surfaces whose graphs has prescribed mean curvature H(x) in hyperbolic space. This result is sharp because if the condition is not satisfied then a non- existence result can be inferred. The main point of the argument is some a priori gradient estimate and degree theory.

Submitted by Letícia dos Santos Silva null (leticiadstos@gmail.com) on 2018-03-04T13:10:40Z No. of bitstreams: 1 leticia_dissertacao.pdf: 941545 bytes, checksum: 75b9baf79f051810ab82bd9bb946dd83 (MD5) / Approved for entry into archive by Claudia Adriana Spindola null (claudia@fct.unesp.br) on 2018-03-05T11:45:13Z (GMT) No. of bitstreams: 1 silva_ls_me_prud.pdf: 941545 bytes, checksum: 75b9baf79f051810ab82bd9bb946dd83 (MD5) / Made available in DSpace on 2018-03-05T11:45:13Z (GMT). No. of bitstreams: 1 silva_ls_me_prud.pdf: 941545 bytes, checksum: 75b9baf79f051810ab82bd9bb946dd83 (MD5) Previous issue date: 2018-02-15 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Neste trabalho mostra-se a existência de solução de variação limitada para um problema envolvendo o operador 1− Laplaciano em um domínio exterior com condição de fronteira de Dirichlet. Para isso, será usada uma versão do Teorema do Passo da Montanha adequada a funcionais localmente lipschitzianos. As dificuldades na implementação de métodos variacionais no espaço das funções de variação limitada são múltiplas, entre elas, a falta de reflexividade, dificuldade de se usar condições de compacidade como a de Palais-Smale e ainda a falta de regularidade do funcional energia. / In this work we prove existence of bounded variation solution for a problem involving the 1-Laplacian operator in an exterior domain with Dirichlet boundary condition. For this, a version of the Mountain Pass Theorem to locally Lipschitz functionals is used. There are many difficulties in implementing variational methods in the space of limited variation functions, among them, lack of reflexivity, difficulty in using compactness conditions such as Palais-Smale and the lack of regularity of the functional energy.

The thesis consists of four papers on numerical complexityanalysis of weak approximation of ordinary and partialstochastic differential equations, including illustrativenumerical examples. Here by numerical complexity we mean thecomputational work needed by a numerical method to solve aproblem with a given accuracy. This notion offers a way tounderstand the efficiency of different numerical methods. The first paper develops new expansions of the weakcomputational error for Itˆo stochastic differentialequations using Malliavin calculus. These expansions have acomputable leading order term in a posteriori form, and arebased on stochastic flows and discrete dual backward problems.Beside this, these expansions lead to efficient and accuratecomputation of error estimates and give the basis for adaptivealgorithms with either deterministic or stochastic time steps.The second paper proves convergence rates of adaptivealgorithms for Itˆo stochastic differential equations. Twoalgorithms based either on stochastic or deterministic timesteps are studied. The analysis of their numerical complexitycombines the error expansions from the first paper and anextension of the convergence results for adaptive algorithmsapproximating deterministic ordinary differential equations.Both adaptive algorithms are proven to stop with an optimalnumber of time steps up to a problem independent factor definedin the algorithm. The third paper extends the techniques to theframework of Itˆo stochastic differential equations ininfinite dimensional spaces, arising in the Heath Jarrow Mortonterm structure model for financial applications in bondmarkets. Error expansions are derived to identify differenterror contributions arising from time and maturitydiscretization, as well as the classical statistical error dueto finite sampling. The last paper studies the approximation of linear ellipticstochastic partial differential equations, describing andanalyzing two numerical methods. The first method generates iidMonte Carlo approximations of the solution by sampling thecoefficients of the equation and using a standard Galerkinfinite elements variational formulation. The second method isbased on a finite dimensional Karhunen- Lo`eve approximation ofthe stochastic coefficients, turning the original stochasticproblem into a high dimensional deterministic parametricelliptic problem. Then, adeterministic Galerkin finite elementmethod, of either h or p version, approximates the stochasticpartial differential equation. The paper concludes by comparingthe numerical complexity of the Monte Carlo method with theparametric finite element method, suggesting intuitiveconditions for an optimal selection of these methods. 2000Mathematics Subject Classification. Primary 65C05, 60H10,60H35, 65C30, 65C20; Secondary 91B28, 91B70. / QC 20100825

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This master thesis is concerned to nonlinear elliptic problem with mono-polar anisotropic potential u + u|u|p−1 + v (x)u + f(x) = 0 in Rn u(x) - 0, as |x| - 00 provided n > 3 and p > n n−2 . These results, between others things, deals with sub-critical, critical and super-critical nonlinearity. We obtain well-posedness of solutions, regularity in c2(Rn), symmetries and asymptotic behavior of solutions in singular spaces Hk. We employ Banach fixed technique and a theorem of regularity elliptic to get those results, this technique does not need of the Hardy type inequalities and variational methods. / Nesta dissertação estudamos o problema elíptico u + u|u|p−1 + v (x)u + f(x) = 0 em Rn u(x) - 0, quando |x| - 00 sujeito a restrições n > 3 e p > n n−2 , cobrindo os casos sub-críticos, críticos e super-críticos. Obtemos boa-colocação de soluções, regularidade, simetrias de soluções e comportamento assintótico em espaços singulares Hk. Empregamos um argumento de ponto fixo em Hk e Ek ao invés de usar desigualdades do tipo Hardy e métodos variacionais.

This thesis is in the field of non-linear partial differential equations (PDE), focusing on problems which show some type of phase-transition. A single phase Hele-Shaw flow models a Newtoninan fluid which is being injected in the space between two narrowly separated parallel planes. The time evolution of the space that the fluid occupies can be modelled by a semi-linear PDE. This is a problem within the field of free boundary problems. In the multi-phase problem we consider the time-evolution of a system of phases which interact according to the principle that the joint boundary which emerges when two phases meet is fixed for all future times. The problem is handled by introducing a parameterized equation which is regularized and penalized. The penalization is non-local in time and tracks the history of the system, penalizing the joint support of two different phases in space-time. The main result in the first paper is the existence theory of a weak solution to the parameterized equations in a Bochner space using the implicit function theorem. The family of solutions to the parameterized problem is uniformly bounded allowing us to extract a weakly convergent subsequence for the case when the penalization tends to infinity. The second problem deals with a parameterized highly oscillatory quasi-linear elliptic equation in divergence form. As the regularization parameter tends to zero the equation gets a jump in the conductivity which occur at the level set of a locally periodic function, the obstacle. As the oscillations in the problem data increases the solution to the equation experiences high frequency jumps in the conductivity, resulting in the corresponding solutions showing an effective global behaviour. The global behavior is related to the so called homogenized solution. We show that the parameterized equation has a weak solution in a Sobolev space and derive bounds on the solutions used in the analysis for the case when the regularization is lost. Surprisingly, the limiting problem in this case includes an extra term describing the interaction between the solution and the obstacle, not appearing in the case when obstacle is the zero level-set. The oscillatory nature of the problem makes standard numerical algorithms computationally expensive, since the global domain needs to be resolved on the micro scale. We develop a multi scale method for this problem based on the heterogeneous multiscale method (HMM) framework and using a finite element (FE) approach to capture the macroscopic variations of the solutions at a significantly lower cost. We numerically investigate the effect of the obstacle on the homogenized solution, finding empirical proof that certain choices of obstacles make the limiting problem have a form structurally different from that of the parameterized problem. / <p>QC 20180222</p>

Ce mémoire de thèse est consacrée à l'étude des problèmes d'évolution où la dynamique est régi par l'opérateur de diffusion de sous-gradient. Nous nous intéressons à deux types de problèmes d'évolution. Le premier problème est régi par un opérateur local de type Leray-Lions avec un domaine borné. Dans ce problème, l'opérateur est maximal monotone et ne satisfait pas la condition standard de contrôle de la croissance polynomiale. Des exemples typiques apparaît dans l'étude de fluide non-Neutonian et aussi dans la description de la dynamique du flux de sous-gradient. Pour étudier le problème nous traitons l'équation dans le contexte de l'EDP non linéaire avec le flux singulier. Nous utilisons la théorie de gradient tangentiel pour caractériser l'équation d'état qui donne la relation entre le flux et le gradient de la solution. Dans le problème stationnaire, nous avons l'existence de la solution, nous avons également l'équivalence entre le problème minimisation initial, le problème dual et l'EDP. Dans l'équation de l'évolution, nous proposons l'existence, l'unicité de la solution. Le deuxième problème est régi par un opérateur discret. Nous étudions l'équation d'évolution discrète qui décrivent le processus d'effondrement du tas de sable. Ceci est un exemple typique de phénomènes auto-organisés critiques exposées par une slope critique. Nous considérons l'équation d'évolution discrète où la dynamique est régie par sous-gradient de la fonction d'indicateur de la boule unité. Nous commençons par établir le modèle, nous prouvons existence et l'unicité de la solution. Ensuite, en utilisant arguments de dualité nous étudions le calcul numérique de la solution et nous présentons quelques simulations numériques. / This thesis is devoted to the study of evolution problems where the dynamic is governed by sub-gradient diffusion operator. We are interest in two kind of evolution problems. The first problem is governed by local operator of Leray-Lions type with a bounded domain. In this problem, the operator is maximal monotone and does not satisfied the standard polynomial growth control condition. Typical examples appears in the study of non-Neutonian fluid and also in the description of sub-gradient flows dynamics. To study the problem we handle the equation in the context of nonlinear PDE with singular flux. We use the theory of tangential gradient to characterize the state equation that gives the connection between the flux and the gradient of the solution. In the stationary problem, we have the existence of solution, we also get the equivalence between the initial minimization problem, the dual problem and the PDE. In the evolution one, we provide the existence, uniqueness of solution and the contractions. The second problem is governed by a discrete operator. We study the discrete evolution equation which describe the process of collapsing sandpile. This is a typical example of Self-organized critical phenomena exhibited by a critical slop. We consider the discrete evolution equation where the dynamic is governed by sub-gradient of indicator function of the unit ball. We begin by establish the model, we prove existence and uniqueness of the solution. Then by using dual arguments we study the numerical computation of the solution and we present some numerical simulations.