In this thesis, we study two different kinds of approximation of plurisubharmonic functions. The first one is a Mergelyan type approximation for plurisubharmonic functions. That is, we study which domains in C^n have the property that every continuous plurisubharmonic function can be uniformly approximated with continuous and plurisubharmonic functions defined on neighborhoods of the domain. We will improve a result by Fornaess and Wiegerinck and show that domains with C^0-boundary have this property. We will also use the notion of plurisubharmonic functions on compact sets when trying to characterize those continuous and plurisubharmonic functions that can be approximated from outside. Here a new kind of convexity of a domain comes in handy, namely those domains in C^n that have a negative exhaustion function that is plurisubharmonic on the closure. For these domains, we prove that it is enough to look at the boundary values of a plurisubharmonic function to know whether it can be approximated from outside. The second type of approximation is the following: we want to approximate functions u that are defined on bounded hyperconvex domains Omega in C^n and have essentially boundary values zero and bounded Monge-Ampère mass, with increasing sequences of certain functions u_j that are defined on strictly larger domains. We show that for certain conditions on Omega, this is always possible. We also generalize this to functions with given boundary values. The main tool in the proofs concerning this second approximation is subextension of plurisubharmonic functions.

In this thesis we focus on Dirichlet's problem for the complex Monge-Ampère equation. That is, for a given non-negative Radon measure µ we are interested in the conditions under which there exists a plurisubharmonic function u such that (ddcu)n=µ, where (ddc)n is the complex Monge-Ampère operator. If this function u exists, then can it be chosen with given boundary values? Is this solution uniquely determined within a given class of functions?

This thesis consists of three papers concerning problems related to plurisubharmonic functions on bounded hyperconvex domains, in particular boundary values of such functions. The papers summarized in this thesis are:* Paper I Urban Cegrell and Berit Kemppe, Monge-Ampère boundary measures, Ann. Polon. Math. 96 (2009), 175-196.* Paper II Berit Kemppe, An ordering of measures induced by plurisubharmonic functions, manuscript (2009).* Paper III Berit Kemppe, On boundary values of plurisubharmonic functions, manuscript (2009).In the first paper we study a procedure for sweeping out Monge-Ampère measures to the boundary of the domain. The boundary measures thus obtained generalize measures studied by Demailly. A number of properties of the boundary measures are proved, and we describe how boundary values of bounded plurisubharmonic functions can be associated to the boundary measures.In the second paper, we study an ordering of measures induced by plurisubharmonic functions. This ordering arises naturally in connection with problems related to negative plurisubharmonic functions. We study maximality with respect to the ordering and a related notion of minimality for certain plurisubharmonic functions. The ordering is then applied to problems of weak*-convergence of measures, in particular Monge-Ampère measures.In the third paper we continue the work on boundary values in a more general setting than in Paper I. We approximate measures living on the boundary with measures on the interior of the domain, and present conditions on the approximation which makes the procedure suitable for defining boundary values of certain plurisubharmonic functions.

This work is concerned with the Almost Axisymmetric Flows with Forcing Terms which are derived from the inviscid Boussinesq equations. It is our hope that these flows will be useful in Meteorology to describe tropical cyclones. We show that these flows give rise to a collection of Monge-Ampere equations for which we prove an existence and uniqueness result. What makes these equations unusual is the boundary conditions they are expected to satisfy and the fact that the boundary is part of the unknown. Our study allows us to make inferences in a toy Almost Axisymmetric Flows with a forcing term model.

L'objet de cette thèse est de modéliser et d'étudier des structures d'irrigation telles les nervures des feuilles, réseau sanguin, poumons,etc. Un modèle généralisant le problème de Gilbert Steiner est introduit ; on étudie alors les propriétés d'existence, de stabilité et régularité. Des algorithmes sont alors proposés pour la simulation.

We present a numerical method to solve the optimal transport problem with a quadratic cost when the source and target measures are periodic probability densities. This method relies on a numerical resolution of the corresponding Monge-Ampère equation. We use an existing Newton-like algorithm that we generalize to the case of a non uniform final density. The main idea consists of designing an iterative scheme where the fully nonlinear equation is approximated by a non-constant coefficient linear elliptic PDE that we discretize and solve at each iteration, in two different ways: a second order finite difference scheme and a Fourier transform (FT) method. The FT method, made possible thanks to a preconditioning step based on the coefficient-averaged equation, results in an overall O(P LogP )-operations algorithm, where P is the number of discretization points. We prove that the generalized algorithm converges to the solution of the optimal transport problem, under suitable conditions on the initial and final densities. Numerical experiments demonstrating the robustness and efficiency of the method on several examples of image processing, including an application to multiple sclerosis disease detection, are shown. We also demonstrate by numerical tests that the method is competitive against some other methods available.

Abordamos o problema do transporte otimo de Monge-Kantorovich no caso em que o custo e dado pelo quadrado da distância. Tal custo tem uma estrutura que permite a obtenção de resultados mais ricos do que o caso geral. Nosso objetivo e determinar se h a soluções para tal problema e caracteriza-las. Al em disso, tratamos informalmente do problema de transporte otimo para um custo geral. / We analyze the Monge-Kantorovich optimal transportation problem in the case where the cost function is given by the square of the Euclidean norm. Such cost has a structure which allow us to get more interesting results than the general case. Our main purpose is to determine if there are solutions to such problem and characterize them. We also give an informal treatment to the optimal transportation problem in the general case.

El presente artículo analiza la relación entre ciencia, salud y nación a propósito de la construcción de una «patología de las alturas», categoría médica utilizada para agrupar a los distintos males producidos por una prolongada exposición a la hipoxia. Se sostiene que el incremento del interés de los médicos nacionales y extranjeros por las enfermedades de altura estuvo relacionado con el desarrollo minero-industrial de la sierra central del Perú a inicios del siglo XX, y con el establecimiento de hospitales y laboratorios financiados por la Cerro de Pasco Corporation, lo que permitió a los investigadores acceder a cientos de pacientes —que se convirtieron en sujetos de estudio— y a la tecnología necesaria para distinguir dichas enfermedades de males comunes hallados también a nivel del mar. Finalmente, el artículo hace hincapié en los impulsos nacionalistas que guiaron a gran parte de estos estudios, lo cual llevó a identificar a las «enferme­dades de altura» como «males peruanos».

Abordamos o problema do transporte otimo de Monge-Kantorovich no caso em que o custo e dado pelo quadrado da distância. Tal custo tem uma estrutura que permite a obtenção de resultados mais ricos do que o caso geral. Nosso objetivo e determinar se h a soluções para tal problema e caracteriza-las. Al em disso, tratamos informalmente do problema de transporte otimo para um custo geral. / We analyze the Monge-Kantorovich optimal transportation problem in the case where the cost function is given by the square of the Euclidean norm. Such cost has a structure which allow us to get more interesting results than the general case. Our main purpose is to determine if there are solutions to such problem and characterize them. We also give an informal treatment to the optimal transportation problem in the general case.

Esta tese trata da formação religiosa dos grupos populares no Sul do Brasil, em especial dos caboclos. Essa religiosidade está baseada em uma crença popular específica, que trata dos poderes de um andarilho que fez sua fama no século XIX entre os Estados de São Paulo e do Rio Grande do Sul. Outros andarilhos surgiram neste espaço até os anos 30 do século XX, e as narrativas de cada um deles fundiram-se em torno de um personagem, hoje cultuado como um santo, conhecido como Santo Monge ou São João Maria. Ao estudar a crença, demonstrase como a religião, neste caso, articulou-se às concepções de natureza e cultura desses grupos para configurar, nos termos de Elias, recursos e estratégias em torno do processo de modernização. Reconstruindo os personagens reconhecidos entre as populações que vivem no sul do Brasil como monges, em especial àqueles evocados como o Santo Monge ou São João Maria, a tese apresenta a formação religiosa de uma tradição que fora dos cânones institucionais do Catolicismo, forjou-se como uma religião.Demonstra que a partir dela os seus membros estabelecem relações que resultam na permanência dos valores que estruturam a sua sociabilidade. Demonstra também que nas relações em que se constata o fim da devoção, finda um formato específico de reprodução social e de identidade dos grupos. Dessa forma, defende-se aqui que esta religião, além das especificidades que a caracteriza, ocupa centralidade nas estratégias de sociabilidade no contexto de seus devotos / This thesis is about religion formation amoung popular groups in southern Brazilian, specially caboclos. This religiosity is based on a specific popular belief that treats the powers of a stroller who became famous in the 19th century in the states of São Paulo and Rio Grande do Sul. Others strollers appeared in that time until the 1930s of the 20th century and the narratives of each one of them were related to a character, today known as a saint, Saint Monge (monk) or Saint João Maria. Studying the belief, it is possible to show how religion connected to the concepts of nature and culture of these groups to configurate, according to the terms of Elias, resources and strategies around a process of modernization. Rebuilding the known characters, specially those related to Saint Monge or São João Maria, the thesis presents the religious formation of a tradition that out of the institutional canons of catholicism, was forged as a religion. It is shown that because of it the members stablished relationships that results in the permanence of the values that are the structures of their sociability. It is also shown that in the relationship that is possible to notice the end of devotion, it ends a specific form of social reproduction and identity of groups. So, it is defended that this religion, besides the specificities, has a central role in the strategies of sociability in the context of its devotees.