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1
Fialho, João Manuel Ferrão. "Existence, localization and multiplicity results for nonlinear and functional". Doctoral thesis, Universidade de Évora, 2012. http://hdl.handle.net/10174/15248.
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In this thesis several problems are addressed. The problems considered vary
from second order problems up to high order problems where generaliza-
tions to nth order are studied. Such problems range from problems without
functional dependence up to problems where the functional dependence is
featured both in the equation and on the boundary conditions.
Functional boundary conditions include most of the classical conditions
as multipoint cases, conditions with delay and/or advances, nonlocal or in-
tegral, with maximum or minimum arguments,... Existence, nonexistence,
multiplicity and localization results are then discussed in accordance with
these conditions.
The method used is the lower and upper solutions combined with di¤erent
techniques (degree theory, Nagumo condition, iterative technique, Green s
function) to obtain such results.
Several applications are studied such as the periodic oscillations of the
axis of a satellite and conjugate boundary value problems, to emphasize the
applicability of the method used; RESUMO:Nesta tese, intitulada em português, Resultados de existência, localiza-
ção e multiplicidade para problemas não lineares e funcionais de ordem su-
perior com valores na fronteira , diferentes problemas são abordados. Estes
problemas variam desde problemas de segunda ordem até problemas de or-
dem superior, onde generalizações de ordem n são feitas e onde os problemas
apresentados variam desde o caso em que não existe dependência funcional
até aos em que esta dependência funcional está presente tanto na equação
como nas condições de fronteira.
Sobre estas condições, que incluem a maioria das condições clássicas, re-
sultados de existência, não existência, multiplicidade e localização de solução
são discutidos de acordo com estas condições.
O método utilizado é o método da sub e sobre-solução combinado com
diferentes técnicas.
Várias aplicações são estudadas, nomeadamente as oscilações periódicas
do eixo de um satélite e problemas conjugados, de forma a dar ênfase à
aplicabilidade do método utilizado.
2
Tian, Rushun. "Existence and Multiplicity Results on Standing Wave Solutions of Some Coupled Nonlinear Schrodinger Equations". DigitalCommons@USU, 2013. https://digitalcommons.usu.edu/etd/1484.
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Coupled nonlinear Schrodinger equations (CNLS) govern many physical phenomena, such as nonlinear optics and Bose-Einstein condensates. For their wide applications, many studies have been carried out by physicists, mathematicians and engineers from different respects. In this dissertation, we focused on standing wave solutions, which are of particular interests for their relatively simple form and the important roles they play in studying other wave solutions. We studied the multiplicity of this type of solutions of CNLS via variational methods and bifurcation methods.
Variational methods are useful tools for studying differential equations and systems of differential equations that possess the so-called variational structure. For such an equation or system, a weak solution can be found through finding the critical point of a corresponding energy functional. If this equation or system is also invariant under a certain symmetric group, multiple solutions are often expected. In this work, an integer-valued function that measures symmetries of CNLS was used to determine critical values. Besides variational methods, bifurcation methods may also be used to find solutions of a differential equation or system, if some trivial solution branch exists and the system is degenerate somewhere on this branch. If local bifurcations exist, then new solutions can be found in a neighborhood of each bifurcation point. If global bifurcation branches exist, then there is a continuous solution branch emanating from each bifurcation point.
We consider two types of CNLS. First, for a fully symmetric system, we introduce a new index and use it to construct a sequence of critical energy levels. Using variational methods and the symmetric structure, we prove that there is at least one solution on each one of these critical energy levels. Second, we study the bifurcation phenomena of a two-equation asymmetric system. All these bifurcations take place with respect to a positive solution branch that is already known. The locations of the bifurcation points are determined through an equation of a coupling parameter. A few nonexistence results of positive solutions are also given
3
Huang, Lirong. "Multiplicity results for some classes of Schrödinger-Poisson systems". Doctoral thesis, Universidade de Aveiro, 2014. http://hdl.handle.net/10773/12867.
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Doutoramento conjunto em Matemática - Matemática e Aplicações (PDMA)In this thesis, we study the existence and multiplicity of solutions of the following class of Schr odinger-Poisson systems: u + u + l(x) u = (x; u) in R3; = l(x)u2 in R3; where l 2 L2(R3) or l 2 L1(R3). And we consider that the nonlinearity satis es the following three kinds of cases: (i) a subcritical exponent with (x; u) = k(x)jujp2u + h(x)u (4 p < 2 ) under an inde nite case; (ii) a general inde nite nonlinearity with (x; u) = k(x)g(u) + h(x)u; (iii) a critical growth exponent with (x; u) = k(x)juj2 2u + h(x)jujq2u (2 q < 2 ). It is worth mentioning that the thesis contains three main innovations except overcoming several di culties, which are generated by the systems themselves. First, as an unknown referee said in his report, we are the rst authors concerning the existence of multiple positive solutions for Schr odinger- Poisson systems with an inde nite nonlinearity. Second, we nd an interesting phenomenon in Chapter 2 and Chapter 3 that we do not need the condition R R3 k(x)ep 1dx < 0 with an inde nite noncoercive case, where e1 is the rst eigenfunction of +id in H1(R3) with weight function h. A similar condition has been shown to be a su cient and necessary condition to the existence of positive solutions for semilinear elliptic equations with inde nite nonlinearity for a bounded domain (see e.g. Alama-Tarantello, Calc. Var. PDE 1 (1993), 439{475), or to be a su cient condition to the existence of positive solutions for semilinear elliptic equations with inde nite nonlinearity in RN (see e.g. Costa-Tehrani, Calc. Var. PDE 13 (2001), 159{189). Moreover, the process used in this case can be applied to study other aspects of the Schr odinger-Poisson systems and it gives a way to study the Kirchho system and quasilinear Schr odinger system. Finally, to get sign changing solutions in Chapter 5, we follow the spirit of Hirano-Shioji, Proc. Roy. Soc. Edinburgh Sect. A 137 (2007), 333, but the procedure is simpler than that they have proposed in their paper.
Nesta tese, estudamos a existência e a multiplicidade de soluções da seguinte classe de sistemas denominada de Schr odinger-Poisson: u + u + l(x) u = (x; u) in R3; = l(x)u2 in R3; onde l 2 L2(R3) ou l 2 L1(R3). Consideram-se não-linearidades que satisfazem um dos seguintes casos: (i) potências que envolvem um expoente sub-cr tico, da forma (x; u) = k(x)jujp2u + h(x)u, (4 p < 2 ), sendo k uma função com sinal indefinido e h uma função positiva; (ii) caso geral de uma não-linearidade indefi nida, da forma (x; u) = k(x)g(u) + h(x)u, sendo k uma função com sinal indefinido e h uma função positiva; (iii) potências que envolvem o expoente crí tico, da forma (x; u) = k(x)juj2 2u + h(x)jujq2u (2 q < 2 ). Convém salientar que esta tese tem três principais inovações, as quais ultrapassam dificuldades geradas pela natureza dos problemas estudados. Primeiro, como um relator anónimo referiu, este é o primeiro trabalho em que se trata a existência de várias soluções de sistemas de Schrödinger- Poisson com não-linearidade indefinida. Segundo, neste estudo encontrou-se um fen ómeno interessante, ver Capítulos 2 e 3, nomeadamente, não ser necess ária a condição R3 k(x)ep 1dx < 0 no caso indefinido e não-coercivo, sendo e1 a função associada ao primeiro valor próprio de + id em H1(R3) com peso h. Note-se que foi demonstrado que uma condi cão semelhante e condição necessária e suficiente na existência de solu cões positivas para equações elíticas semilineares com não-linearidades indefinidas em domínios limitados (ver e.g. Alama-Tarantello, Calc. Var. PDE 1 (1993), 439{475), ou ser uma condição suficiente na existência de soluções positivas para equações elíticas semilineares com não-linearidades indefinidas em RN (see e.g. Costa-Tehrani, Calc. Var. PDE 13 (2001), 159{189). Adicionalmente, o método utilizado pode ser utilizado para estudar outros aspetos dos sistemas de Schrodinger-Poisson, permite também estudar sistemas de Kirchho e sistemas de Schrodinger quasilineares. Por m, para obter soluções com mudança de sinal no Cap. 5, segue se a ideia de Hirano-Shioji, Proc. Roy. Soc. Edinburgh Sect. A 137 (2007), 333, mas o método utilizado é uma versão simplificada do método apresentado no artigo referido.
4
Woodward, Christopher Thomas. "Multiplicity-free Hamiltonian actions and existence of invariant Kähler structure". Thesis, Massachusetts Institute of Technology, 1996. http://hdl.handle.net/1721.1/38408.
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Gadam, Sudhasree. "Existence and Multiplicity of Solutions for Semilinear Elliptic Boundary Value Problems". Thesis, University of North Texas, 1992. https://digital.library.unt.edu/ark:/67531/metadc332520/.
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This thesis studies the existence, multiplicity, bifurcation and the stability of the solutions to semilinear elliptic boundary value problems. These problems are motivated both by the mathematical structure and the numerous applications in fluid mechanics chemical reactions, nuclear reactors, Riemannian geometry and elasticity theory. This study considers the problem for different classes of nonlinearities and obtain the existence and multiplicity of positive solutions.
6
GUARNOTTA, Umberto. "EXISTENCE RESULTS FOR SINGULAR CONVECTIVE ELLIPTIC PROBLEMS". Doctoral thesis, Università degli Studi di Palermo, 2021. http://hdl.handle.net/10447/524941.
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Zhang, Yanping. "Existence and multiplicity of positive solutions in semilinear elliptic boundary value problems". Thesis, Heriot-Watt University, 2003. http://hdl.handle.net/10399/414.
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Murillo, Kelly Patricia. "Existence results for elliptic equations with singular terms". Doctoral thesis, Universidade de Aveiro, 2013. http://hdl.handle.net/10773/9888.
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Doutoramento em Matemática e AplicaçõesEsta dissertação estuda em detalhe três problemas elípticos: (I) uma classe de equações que envolve o operador Laplaciano, um termo singular e nãolinearidade com o exponente crítico de Sobolev, (II) uma classe de equações com singularidade dupla, o expoente crítico de Hardy-Sobolev e um termo côncavo e (III) uma classe de equações em forma divergente, que envolve um termo singular, um operador do tipo Leray-Lions, e uma função definida nos espaços de Lorentz. As não-linearidades consideradas nos problemas (I) e (II), apresentam dificuldades adicionais, tais como uma singularidade forte no ponto zero (de modo que um "blow-up" pode ocorrer) e a falta de compacidade, devido à presença do exponente crítico de Sobolev (problema (I)) e Hardy-Sobolev (problema (II)). Pela singularidade existente no problema (III), a definição padrão de solução fraca pode não fazer sentido, por isso, é introduzida uma noção especial de solução fraca em subconjuntos abertos do domínio. Métodos variacionais e técnicas da Teoria de Pontos Críticos são usados para provar a existência de soluções nos dois primeiros problemas. No problema (I), são usadas uma combinação adequada de técnicas de Nehari, o princípio variacional de Ekeland, métodos de minimax, um argumento de translação e estimativas integrais do nível de energia. Neste caso, demonstramos a existência de (pelo menos) quatro soluções não triviais onde pelo menos uma delas muda de sinal. No problema (II), usando o método de concentração de compacidade e o teorema de passagem de montanha, demostramos a existência de pelo menos duas soluções positivas e pelo menos um par de soluções com mudança de sinal. A abordagem do problema (III) combina um resultado de surjectividade para operadores monótonos, coercivos e radialmente contínuos com propriedades especiais do operador de tipo Leray- Lions. Demonstramos assim a existência de pelo menos, uma solução no espaço de Lorentz e obtemos uma estimativa para esta solução.
This dissertation study mainly three elliptical problems: (I) a class of equations, which involves the Laplacian operator, a singular term and a nonlinearity with the critical Sobolev exponent, (II) a class of equations with double singularity, the critical Hardy-Sobolev exponent and a concave term and (III) a class of equations in divergent form, which involves a singular term, a Leray-Lions operator, and a function defined on Lorentz spaces. The nonlinearities considered in problems (I) and (II), bring additional difficulties which, as the strong singularity at zero (so blow-up may occur) and the lack of compactness due to the presence of a Sobolev critical exponent (problem (I)) and a Hardy-Sobolev critical exponent (problem (II)). In problem (III), the singularity implies that the standard definition of weak solution may not make sense. Therefore is necessary to introduce a special notion of weak solution on open subsets of the domain. Variational methods and Critical Point Theory techniques are used to prove the existence of solutions in the two first problems. In problem (I), our method combines Nehari's techniques, Ekeland's variational principle, minimax methods, a translation argument and integral estimates of the energy level. In this case, we prove the existence of (at least) four nontrivial solutions where at least one of them is sign-changing. In problem (II), we prove the existence of at least two positive solutions and a pair of sign-changing solutions, using the concentration-compactness method and the mountain pass theorem. The approach in problem (III) combines a surjectivity result for monotone, coercive and radially continuous operators with special properties of Leray-Lions operators. We prove the existence of at least one solution in a Lorentz space and obtain an estimative for the solution.
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Hata, Kazuya. "Multiplicity Results of Periodic Solutions for Two Classes of Nonlinear Problems". DigitalCommons@USU, 2014. https://digitalcommons.usu.edu/etd/4030.
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We investigate the existences and qualitative properties of periodic solutions of the following two classes of nonlinear differential equations:
I) (Special) Relativistic Pendulum Equations (RPEs);
II) (2-coupled) Gross-Pitaevskii Equations (GPEs).
The pendulum equation describes the motion of a pendulum. According to Special Relativity, which was published by A. Einstein in 1905, causality is more fundamental than constant time-space, thus time will ow slower and space will distort to keep causality if the speed of motion is near the speed of light. In such high speed situations, the pendulum equation needs to be revised due to Special Relativity. The revised equation is called RPE. Our result answers some open questions about the existence of multiple periodic solutions for RPEs.
GPEs are sometimes called coupled nonlinear schrodinger equations. the Schrodinger equation is the fundamental equation of Quantum Mechanics which is the \exotic" probabilistic fundamental physics law of the \micro" world { the world of atoms and molecules. A well-known physicist and Nobel laureate, R. Feynman, said \I think I can safely say that nobody understands quantum mechanics." which indicates the physical/ philosophical difficulty of interpretations. It raises paradoxical problems such the well-known Schrodinger's Cat. Setting aside these difficult, if we combine Special Relativity and Quantum Mechanics as a many-body system, then we have Quantum Field Theory (QFT) which is more deterministic, and governs even elementary particle physics. GPEs are also related to QFT. For example, superconductivity and Bose Einstein Condensates (BEC). These phenomena in condensed matter physics can be thought of as the emergence of the mysterious micro world physics at \macro" level.
We study these equations from the viewpoint of mathematical interest. It is generally difficult to solve nonlinear differential equations. It is also generally difficult even to prove the existence of solutions. Although we show there exist solutions, we still do not know how to solve the differential equations analytically.
Variational Methods (or Calculus of Variations) are useful tools to show there exist solutions of differential equations. The idea is to convert the problem of solving equations into the problem of finding critical points (i.e. minimum/maximum points or saddle points) of a functional, and each critical point can generally correspond to a weak solution. However, it is also generally difficult to find out such critical points because we look for critical points in an infinite-dimensional functions space. Thus many advanced mathematical theories or tools have been developed and used for decades in nonlinear analysis. We use some topological theories. From information of the functional's shape, these theories deduce if there exists a critical point, or how many critical points exist. The key of these theories is to use the symmetry of the equations.
We also investigate bifurcation structures for II), i.e. the connection structures between the solutions. By linearizations which look at the equations \locally," we reduce the problem in the infinite dimension to one in a finite dimension. Furthermore, it allows us to apply Morse Theory, which connects between local and global aspects of the functional's information. In several cases, we show that there are infinitely many bifurcation points that give rise to global bifurcation branches.
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Velichkov, Bozhidar. "Existence and regularity results for some shape optimization problems". Doctoral thesis, Scuola Normale Superiore, 2013. http://hdl.handle.net/11384/85690.
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Les problèmes d'optimisation de forme sont présents naturellement en physique, ingénierie, biologie, etc. Ils visent à répondre à différentes questions telles que:-A quoi une aile d'avion parfaite pourrait ressembler?-Comment faire pour réduire la résistance d'un objet en mouvement dans un gaz ou un fluide?-Comment construire une structure élastique de rigidité maximale?-Quel est le comportement d'un système de cellules en interaction?Pour des exemples précis et autres applications de l'optimisation de forme nous renvoyons à [20] et [69]. Ici, nous traitons les aspects mathématiques théoriques de l'optimisation de forme, concernant l'existence d'ensembles optimaux ainsi que leur régularité. Dans toutes les situations que l'on considère, la fonctionnelle dépend de la solution d'une certaine équation aux dérivées partielles posée sur la forme inconnue. Nous allons parfois se référer à cette fonction comme une fonction d'état.Les fonctions d'état les plus simples, mais qui apparaissent dans beaucoup de problèmes, sont données par les solutions des équations -Δw = 1 et -Δu = λu,qui sont liées à la torsion et aux modes d'oscillation d'un objet donné. Notre étude se concentrera principalement sur ces fonctionnelles de formes, impliquant la torsion et le spectre.[20] D. Bucur, G. Buttazzo: Variational Methods in Shape Optimization Problems. Progress in Nonlinear Differential Equations 65, Birkhauser Verlag, Basel (2005).[69] A. Henrot, M. Pierre: Variation et optimisation de formes: une analyse geometrique. Springer-Berlag, Berlin, 2005The shape optimization problems naturally appear in engineering and biology. They aim to answer questions as:-What a perfect wing may look like?-How to minimize the resistance of a moving object in a gas or a fluid?-How to build a rod of maximal rigidity?-What is the behaviour of a system of cells?The shape optimization appears also in physics, mainly in electrodynamics and in the systems presenting both classical and quantum mechanics behaviour. For explicit examples and furtheraccount on the applications of the shape optimization we refer to the books [20] and [69]. Here we deal with the theoretical mathematical aspects of the shape optimization, concerning existence of optimal sets and their regularity. In all the practical situations above, the shape of the object in study is determined by a functional depending on the solution of a given partial differential equation. We will sometimes refer to this function as a state function.The simplest state functions are provided by solutions of the equations−∆w = 1 and −∆u = λu,which usually represent the torsion rigidity and the oscillation modes of a given object. Thus our study will be concentrated mainly on the situations, in which these state functions appear,i.e. when the optimality is intended with respect to energy and spectral functionals. [20] D. Bucur, G. Buttazzo: Variational Methods in Shape Optimization Problems. Progress in Nonlinear Differential Equations 65, Birkhauser Verlag, Basel (2005).[69] A. Henrot, M. Pierre: Variation et optimisation de formes: une analyse geometrique. Springer-Berlag, Berlin, 2005
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Velichkov, Bozhidar. "Existence and regularity results for some shape optimization problems". Thesis, Grenoble, 2013. http://www.theses.fr/2013GRENM088/document.
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Les problèmes d'optimisation de forme sont présents naturellement en physique, ingénierie, biologie, etc. Ils visent à répondre à différentes questions telles que:-A quoi une aile d'avion parfaite pourrait ressembler?-Comment faire pour réduire la résistance d'un objet en mouvement dans un gaz ou un fluide?-Comment construire une structure élastique de rigidité maximale?-Quel est le comportement d'un système de cellules en interaction?Pour des exemples précis et autres applications de l'optimisation de forme nous renvoyons à [20] et [69]. Ici, nous traitons les aspects mathématiques théoriques de l'optimisation de forme, concernant l'existence d'ensembles optimaux ainsi que leur régularité. Dans toutes les situations que l'on considère, la fonctionnelle dépend de la solution d'une certaine équation aux dérivées partielles posée sur la forme inconnue. Nous allons parfois se référer à cette fonction comme une fonction d'état.Les fonctions d'état les plus simples, mais qui apparaissent dans beaucoup de problèmes, sont données par les solutions des équations -Δw = 1 et -Δu = λu,qui sont liées à la torsion et aux modes d'oscillation d'un objet donné. Notre étude se concentrera principalement sur ces fonctionnelles de formes, impliquant la torsion et le spectre.[20] D. Bucur, G. Buttazzo: Variational Methods in Shape Optimization Problems. Progress in Nonlinear Differential Equations 65, Birkhauser Verlag, Basel (2005).[69] A. Henrot, M. Pierre: Variation et optimisation de formes: une analyse geometrique. Springer-Berlag, Berlin, 2005The shape optimization problems naturally appear in engineering and biology. They aim to answer questions as:-What a perfect wing may look like?-How to minimize the resistance of a moving object in a gas or a fluid?-How to build a rod of maximal rigidity?-What is the behaviour of a system of cells?The shape optimization appears also in physics, mainly in electrodynamics and in the systems presenting both classical and quantum mechanics behaviour. For explicit examples and furtheraccount on the applications of the shape optimization we refer to the books [20] and [69]. Here we deal with the theoretical mathematical aspects of the shape optimization, concerning existence of optimal sets and their regularity. In all the practical situations above, the shape of the object in study is determined by a functional depending on the solution of a given partial differential equation. We will sometimes refer to this function as a state function.The simplest state functions are provided by solutions of the equations−∆w = 1 and −∆u = λu,which usually represent the torsion rigidity and the oscillation modes of a given object. Thus our study will be concentrated mainly on the situations, in which these state functions appear,i.e. when the optimality is intended with respect to energy and spectral functionals. [20] D. Bucur, G. Buttazzo: Variational Methods in Shape Optimization Problems. Progress in Nonlinear Differential Equations 65, Birkhauser Verlag, Basel (2005).[69] A. Henrot, M. Pierre: Variation et optimisation de formes: une analyse geometrique. Springer-Berlag, Berlin, 2005
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Khoury, Michael John Jr. "Multiplicity One Results and Explicit Formulas for Quasi-Split p-adic Unitary Groups". The Ohio State University, 2008. http://rave.ohiolink.edu/etdc/view?acc_num=osu1218567821.
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Platino, Vincenzo. "Existence, regularity and testability results in economic models with externalities". Doctoral thesis, Universita degli studi di Salerno, 2012. http://hdl.handle.net/10556/1312.
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2008 - 2009This thesis deals with economic models in the presence of externalities. The thesis consists of three chapters. In chapter 1, we consider a general model of production economies with consumption and production externalities. That is, the choices of all agents (households and firms) affect individual consumption sets, individual preferences and production technologies. Describing equlibria in terms of first order conditions and market clearing conditions, and using a homotopy, under differentiability and boundary conditions, we prove the non-emptiness and compactness of the set of competitive equilibria with consumptions and prices strictly positive. In chapter 2 we consider a general model of private ownership economies with consumption and production externalities. Showing by an example that basic assumptions are not enough to guarantee a regularity result in the space of initial endowments, we provide sufficient conditions for the regularity in the space of endowments and transformation functions. In chapter 3 we study the testability implications of public versus private consumption in collective models of group consumption. To the contrary at the previous literature, we find that assumptions regarding the public or private nature of specific goods do have testability implications, even if one only observes the aggregate group consumption. In fact, these testability implications apply as soon as the analysis includes three goods and four observations. In our opinion, our revealed preference approach obtains stronger testability conclusions because it focuses on conditions which involve personalized prices and personalized quantities, although we do not require that personalized prices and personalized quantities are observable. [edited by author]
VIII n.s.
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Juárez, Hurtado Elard. "Existence and multiplicity of solutions to a class of elliptic problems involving operators with variable exponent". Universidade Federal de São Carlos, 2016. https://repositorio.ufscar.br/handle/ufscar/8838.
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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)
We study the existence and multiplicity of nontrivial solutions for two classes of elliptic problems. The first problem covers a general class of operators with variables exponents where the nonlinearitv has subcritical growth. The second problem is a nonlocal elliptic problem where the nonlinearitv has critical growth. ... continua
Neste trabalho estudamos a existência e multiplicidade de soluções não triviais para duas classes de problemas elípticos. O primeiro problema elíptico que estudamos abrange uma classe geral de operadores com expoentes variáveis onde a não linearidade possui crescimento subcrítico. O segundo problema trata de uma equação não local envolvendo uma ampla classe de operadores onde a não linearidade possui crescimento sublinear/superlinear, mais um termo com crescimento crítico. ... continua
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Manicom, Gray Thomas. "Existence results for a class of semi-linear initial value problems". Diss., University of Pretoria, 2017. http://hdl.handle.net/2263/63288.
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The main result of this thesis is an existence result for parabolic semi-linear
problems. This is done by reformulating the semi-linear problem as an abstract
Cauchy problem
ut(t) = Au(t) + f(t; u(t)), t > 0
u(0) = u0 (1)
for u0 2 X, where X is a Banach space. We then develop and use the theory
of compact semigroups to prove an existence result.
In order to make this result applicable, we give a characterization of
compact semigroups in terms of its resolvent operator and continuity in the
uniform operator topology. Thus, using the theory of analytic semigroups,
we are able to determine under what conditions on A a solution to (1) exists.
Furthermore, we consider the asymptotic behaviour and regularity of such
solutions. By developing perturbation theory, we are easily able to apply our
existence result to a larger class of problems. We then demonstrate these
results with an example.
This work is signi cant in providing a novel approach to a group of previously
established results. The content can be considered pure mathematics,
but it is of signi cant importance in real world situations. The structure
of the thesis, and the choice of certain de nitions, lends itself to be easily
understood and interpreted in the light of these real world situations and
is intended to be easily followed by an applied mathematician. An important
part of this process is to develop the problem in a real Hilbert space
and then to consider the complexi cation of the problem in order to reset
it in a complex Hilbert space, in which we can apply the theory of analytic
semigroups. A large number of real world problems fall into the class of
problems discussed here, not only in biology as demonstrated, but also in
physics, chemistry, and elsewhere.Dissertation (MSc)--University of Pretoria, 2017.
Mathematics and Applied Mathematics
MSc
Unrestricted
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Sauer, Martin [Verfasser]. "Existence and Uniqueness Results for Randomly Forced Generalized Newtonian Fluids / Martin Sauer". München : Verlag Dr. Hut, 2013. http://d-nb.info/1034003283/34.
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Nerlich, Alexander [Verfasser]. "Adding randomness to nonlinear semigroups: existence, uniqueness and asymptotic results / Alexander Nerlich". Ulm : Universität Ulm, 2018. http://d-nb.info/1171900880/34.
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Luo, Yongming [Verfasser]. "Existence and Regularity Results of a Ferroelectric Phase-Field Model / Yongming Luo". Kassel : Universitätsbibliothek Kassel, 2019. http://d-nb.info/1201508339/34.
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Meinert, Melissa [Verfasser]. "Partial differential equations on fractals. Existence, Uniqueness and Approximation results / Melissa Meinert". Bielefeld : Universitätsbibliothek Bielefeld, 2020. http://d-nb.info/1214806538/34.
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Hebestreit, Niklas [Verfasser], Christiane [Gutachter] Tammer y Franco [Gutachter] Giannessi. "Existence results for vector quasi-variational problems / Niklas Hebestreit ; Gutachter: Christiane Tammer, Franco Giannessi". Halle (Saale) : Universitäts- und Landesbibliothek Sachsen-Anhalt, 2020. http://d-nb.info/122159981X/34.
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Araújo, Yane Lísley Ramos. "Existence results for some elliptic equations involving the fractional Laplacian operator and critical growth". Universidade Federal da Paraíba, 2015. http://tede.biblioteca.ufpb.br:8080/handle/tede/9252.
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In this work we prove some results of existence and multiplicity of solutions for equations of the type ( ) u + V (x)u = f(x; u) in RN; where 0 < < 1, N 2 , ( ) denotes the fractional Laplacian, V : RN ! R is a continuous function that satisfy suitable conditions and f : RN R ! R is a continuous function that may have critical growth in the sense of the Trudinger-Moser inequality or in the sense of the critical Sobolev exponent. In order to obtain our results we use variational methods combined with a version of the Concentration-Compactness Principle due to Lions.
Neste trabalho provamos alguns resultados de existência e multiplicidade de soluções para equações do tipo ( ) u + V (x)u = f(x; u) em RN; onde 0 < < 1, N 2 , ( ) denota o Laplaciano fracionário, V : RN ! R é uma função contínua que satisfaz adequadas condições e f : RN R ! R é uma função cont ínua que pode ter crescimento crítico no sentido da desigualdade de Trudinger-Moser ou no sentido do expoente crítico de Sobolev. A m de obter nossos resultados usamos métodos variacionais combinados com uma versão do Princípio de Concentração- Compacidade devido à Lions.
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Cabarrubias, Bituin C. "Existence, uniqueness and homogenization results for a class of nonlinear PDE in perforated domains". Rouen, 2012. http://www.theses.fr/2012ROUES046.
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This thesis is devoted to the existence, uniqueness and homogenization results for a quasilinear elliptic problem with oscillating coefficients and with nonlinear Robin boundary condition in a periodically perforated domain. A suitable frowth conditions are assumed on the nonlinear boundary term and on the quasilinear term, some assumptions on the modulus of continuity introduced in Chipot [17] and weaker than a Lipschitz condition, are prescribed. For the existence and uniqueness of a solution, we consider a more general framework which, in particular, will imply the existence and uniqueness of the solution of the problem. To deal with the existence of a solution, we prove first the weak continuity of the boundary nonlinear operator which is a difficult part. Together with this property, we use the Schauder's Fixed Point Theorem to show the existence. For the uniqueness, we adapt to our situation some arguments introduced in André-Chipot [5] (see also chapter 11 of [17] for Dirichlet conditions) and partially extended to linear Robin conditions in Bendib-Tcheugoué Tébou [11] and Bendib [10]. For the homogenization of the problem, we study the convergence to a limit problem using the Periodic Unfolding Method in perforated domains. Here, we proved related properties of the onfolding operators which are needed in the process. We also show the well-posedness of the limit system by proving that the homogenized operator inherits the modulus of continuity of the initial problem. As a consequence, the uniqueness of a solution of the homogenized quasilinear problem follows. A corrector result is also obtained using this method.
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Galstyan, Anahit. "Existence and Number of Global Solutions to Model Nonlinear Partial Differential Equations". University of Cincinnati / OhioLINK, 2005. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1115841126.
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Yusuf, Owolabi. "On models of Kirchhoff Equations with damping terms: existence results and asymptotic behaviour of solutions". Master's thesis, University of Cape Town, 2018. http://hdl.handle.net/11427/29370.
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MONTANARI, Piera. "Local and Global Existence results for the Characteristic Problem for Linear and Semi-linear Wave Equations". Doctoral thesis, Università degli studi di Ferrara, 2010. http://hdl.handle.net/11392/2389334.
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The thesis concerns the well posedness of the Characteristic Initial
Value Problem for the Semilinear Wave Equation, with initial data on
a light cone. In the first part of the thesis, an explicit representation
formula for the solution of the linear equation is given, extending the
results known for the homogeneous equation and the trace on the time
axis of the solution.
Further, Energy Estimates are derived. In constructing such Estimates
one encounters several difficulties due to the presence of a
geometrical singularity at the tip of the cone. To manage the construction
of the Energy Estimate, one introduces suitable Sobolev-like
norms characterized by weights, which mitigates the difficulties in the
origin. These Estimates are well posed only for functions which vanish
of order high enough at the origin. This fact brings us to split the initial
data in the sum of two terms. The first term consists of the Taylor
polynomial of the initial datum, the second one consist of remainder
regular function with the required vanishing order at the origin.
An interesting phenomenon observed here is a gap of differentiability
between the solution and the initial data.
The solution obtained using the Energy method is still incomplete,
because of the splitting of the initial data. This fact brings us to solve
the problem for purely polynomial data. For this purpose, it is used a
generalization of the well-known harmonic polynomials.
The last part of the thesis is devoted to the semi-linear problem,
for which the tools developed in the previous chapters are generalized.
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Azizieh, Céline. "A priori estimates, continuation methods and existence results for positive solutions of p-Laplace eauqations and systems". Doctoral thesis, Universite Libre de Bruxelles, 2001. http://hdl.handle.net/2013/ULB-DIPOT:oai:dipot.ulb.ac.be:2013/211660.
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Mazzoleni, Dario [Verfasser], Aldo [Akademischer Betreuer] Pratelli, Dorin [Akademischer Betreuer] Bucur y Giuseppe [Akademischer Betreuer] Buttazzo. "Existence and regularity results for solutions of spectral problems / Dario Mazzoleni. Gutachter: Aldo Pratelli ; Dorin Bucur ; Giuseppe Buttazzo". Erlangen : Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), 2014. http://d-nb.info/1065270380/34.
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Luo, Tingjian. "Existence non existence et multiplicité d'ondes stationnaires normalisées pour quelques équations non linéaires elliptiques". Phd thesis, Université de Franche-Comté, 2013. http://tel.archives-ouvertes.fr/tel-01061670.
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Dans cette thèse, nous étudions l'existence, non existence et multiplicité des ondes stationnairesavec les normes prescrites pour deux types d'équations aux dérivées partiellesnon linéaires elliptiques découlant de différents modèles physiques. La stabilité orbitale desondes stationnaires est également étudiée dans certains cas. Les principales méthodes denos preuves sont des arguments variationnels. Les solutions sont obtenues comme pointscritiques de fonctionnelle associée sur une contrainte.La thèse se compose de sept chapitres. Le Chapitre 1 est l'introduction de la thèse. Dansles Chapitres 2 à 4, nous étudions une classe d'équations de Schrödinger-Poisson-Slaternon linéaires. Nous établissons dans le Chapitre 2 des résultats optimaux non existencede solutions d'énergie minimale ayant une norme L2 prescrite. Dans le Chapitre 3, nousmontrons un résultat d'existence de solutions L2 normalisées, dans une cas où la fonctionnelleassociée n'est pas bornée inférieurement sur la contrainte. Nos solutions sonttrouvées comme des points de selle de la fonctionnelle, mais ils correspondent à des solutionsd'énergée minimale. Nous montrons également que les ondes stationnaires associéessont orbitalement instables. Ici, puisque nos points critiques présumés ne sont pas desminimiseurs globaux, il n'est pas possible d'utiliser de façon systématique les méthodesde compacité par concentration développées par P. L. Lions. Ensuite, dans le Chapitre4, nous montrons que sous les hypothèses du Chapitre 3, il existe une infinité de solutionsayant une norme L2 prescrite. Dans les deux chapitres suivants, nous étudions uneclasse d'équations de Schrödinger quasi-linéaires. Des résultats optimaux non existence desolutions d'énergie minimale sont donnés dans le Chapitre 5. Dans le Chapitre 6, nousprouvons l'existence de deux solutions positives ayant une norme donnée. L'une d'elles,relativement à la contrainte L2, est de type point selle. L'autre est un minimum, soit localou global. Le fait que la fonctionnelle naturelle associée à cette équation n'est pas biendéfinie nécessite l'utilisation d'une méthode de perturbation pour obtenir ces deux pointscritiques. Enfin, au Chapitre 7, nous mentionnons quelques questions que cette thèse asoulevées.
29
Hauser, Carlos [Verfasser] y W. [Akademischer Betreuer] Reichel. "Existence Results and A Priori Bounds for Positive Solutions of Discrete Nonlinear Elliptic Equations / Carlos Hauser ; Betreuer: W. Reichel". Karlsruhe : KIT-Bibliothek, 2019. http://d-nb.info/1190178923/34.
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Herán, Andreas [Verfasser], Jens [Akademischer Betreuer] Habermann y Jens [Gutachter] Habermann. "Existence and Regularity Results for Parabolic Problems on Metric Measure Spaces / Andreas Herán ; Gutachter: Jens Habermann ; Betreuer: Jens Habermann". Erlangen : Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), 2020. http://d-nb.info/1218785721/34.
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Schätzler, Leah [Verfasser], Frank [Akademischer Betreuer] Duzaar y Frank [Gutachter] Duzaar. "Existence and Stability Results for Nonlinear and Doubly Nonlinear Evolutionary Problems / Leah Schätzler ; Gutachter: Frank Duzaar ; Betreuer: Frank Duzaar". Erlangen : Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), 2020. http://d-nb.info/1216704287/34.
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Piovano, Paulo. "Evolution and Regularity Results for Epitaxially Strained Thin Films and Material Voids". Research Showcase @ CMU, 2012. http://repository.cmu.edu/dissertations/96.
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In this dissertation we study free boundary problems that model the evolution of interfaces in the presence of elasticity, such as thin film profiles and material void boundaries. These problems are characterized by the competition between the elastic bulk energy and the anisotropic surface energy.
First, we consider the evolution equation with curvature regularization that models the motion of a two-dimensional thin film by evaporation-condensation on a rigid substrate. The film is strained due to the mismatch between the crystalline lattices of the two materials and anisotropy is taken into account. We present the results contained in [62] where the author establishes short time existence, uniqueness and regularity of the solution using De Giorgi’s minimizing movements to exploit the L2 -gradient flow structure of the equation. This seems to be the first analytical result for the evaporation-condensation case in the presence of elasticity.
Second, we consider the relaxed energy introduced in [20] that depends on admissible pairs (E, u) of sets E and functions u defined only outside of E. For dimension three this energy appears in the study of the material voids in solids, where the pairs (E, u) are interpreted as the admissible configurations that consist of void regions E in the space and of displacements u of the atoms of the crystal. We provide the precise mathematical framework that guarantees the existence of minimal energy pairs (E, u). Then, we establish that for every minimal configuration (E, u), the function u is C 1,γ loc -regular outside an essentially closed subset of E. No hypothesis of starshapedness is assumed on the voids and all the results that are contained in [18] hold true for every dimension d ≥ 2.
33
Palmieri, Alessandro [Verfasser], Michael [Akademischer Betreuer] Reissig, Michael [Gutachter] Reissig y Vladimir [Gutachter] Georgiev. "Global in time existence and blow-up results for a semilinear wave equation with scale-invariant damping and mass / Alessandro Palmieri ; Gutachter: Michael Reissig, Vladimir Georgiev ; Betreuer: Michael Reissig". Freiberg : Technische Universität Bergakademie Freiberg, 2018. http://d-nb.info/1226100945/34.
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Al, Zohbi Maryam. "Contributions to the existence, uniqueness, and contraction of the solutions to some evolutionary partial differential equations". Thesis, Compiègne, 2021. http://www.theses.fr/2021COMP2646.
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Dans cette thèse, nous nous sommes principalement intéressés à l’étude théorique et numérique de quelques équations qui décrivent la dynamique des densités des dislocations. Les dislocations sont des défauts microscopiques qui se déplacent dans les matériaux sous l’effet des contraintes extérieures. Dans un premier travail, nous démontrons un résultat d’existence globale en temps des solutions discontinues pour un système hyperbolique diagonal qui n’est pas nécessairement strictement hyperbolique, dans un espace unidimensionnel. Ainsi dans un deuxième travail, nous élargissons notre portée en démontrant un résultat similaire pour un système d’équations de type eikonal non-linéaire qui est en fait une généralisation du système hyperbolique déjà étudié. En effet, nous prouvons aussi l’existence et l’unicité d’une solution continue pour le système eikonal. Ensuite, nous nous sommes intéressés à l’analyse numérique de ce système en proposant un schéma aux différences finies, par lequel nous montrons la convergence vers le problème continu et nous consolidons nos résultats avec quelques simulations numériques. Dans une autre direction, nous nous sommes intéressés à la théorie de contraction différentielle pour les équations d’évolutions. Après avoir introduit une nouvelle distance, nous construisons une nouvelle famille des solutions contractantes positives pour l’équation d’évolution p-LaplaceIn this thesis, we are mainly interested in the theoretical and numerical study of certain equations that describe the dynamics of dislocation densities. Dislocations are microscopic defects in materials, which move under the effect of an external stress. As a first work, we prove a global in time existence result of a discontinuous solution to a diagonal hyperbolic system, which is not necessarily strictly hyperbolic, in one space dimension. Then in another work, we broaden our scope by proving a similar result to a non-linear eikonal system, which is in fact a generalization of the hyperbolic system studied first. We also prove the existence and uniqueness of a continuous solution to the eikonal system. After that, we study this system numerically in a third work through proposing a finite difference scheme approximating it, of which we prove the convergence to the continuous problem, strengthening our outcomes with some numerical simulations. On a different direction, we were enthused by the theory of differential contraction to evolutionary equations. By introducing a new distance, we create a new family of contracting positive solutions to the evolutionary p-Laplacian equation
35
Tavares, Lucas Alves. "O envolvimento da proteína adaptadora 1 (AP-1) no mecanismo de regulação negativa do receptor CD4 por Nef de HIV-1". Universidade de São Paulo, 2016. http://www.teses.usp.br/teses/disponiveis/17/17136/tde-06012017-113215/.
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O Vírus da Imunodeficiência Humana (HIV) é o agente etiológico da Síndrome da Imunodeficiência Adquirida (AIDS). A AIDS é uma doença de distribuição mundial, e estima-se que existam atualmente pelo menos 36,9 milhões de pessoas infectadas com o vírus. Durante o seu ciclo replicativo, o HIV promove diversas alterações na fisiologia da célula hospedeira a fim de promover sua sobrevivência e potencializar a replicação. A rápida progressão da infecção pelo HIV-1 em humanos e em modelos animais está intimamente ligada à função da proteína acessória Nef. Dentre as diversas ações de Nef está a regulação negativa de proteínas importantes na resposta imunológica, como o receptor CD4. Sabe-se que esta ação resulta da indução da degradação de CD4 em lisossomos, mas os mecanismos moleculares envolvidos ainda são totalmente elucidados. Nef forma um complexo tripartite com a cauda citosólica de CD4 e a proteína adaptadora 2 (AP-2), em vesículas revestidas por clatrina nascentes, induzindo a internalização e degradação lisossomal de CD4. Pesquisas anteriores demonstraram que o direcionamento de CD4 aos lisossomos por Nef envolve a entrada do receptor na via dos corpos multivesiculares (MVBs), por um mecanismo atípico, pois, embora não necessite da ubiquitinação de carga, depende da ação de proteínas que compõem os ESCRTs (Endosomal Sorting Complexes Required for Transport) e da ação de Alix, uma proteína acessória da maquinaria ESCRT. Já foi reportado que Nef interage com subunidades dos complexos AP-1, AP-2, AP-3 e Nef não parece interagir com subunidades de AP-4 e AP-5. Entretanto, o papel da interação de Nef com AP-1 e AP-3 na regulação negativa de CD4 ainda não está totalmente elucidado. Ademais, AP-1, AP-2 e AP-3 são potencialmente heterogêneos devido à existência de isoformas múltiplas das subunidades codificadas por diferentes genes. Todavia, existem poucos estudos para demonstrar se as diferentes combinações de isoformas dos APs são formadas e se possuem propriedades funcionais distintas. O presente trabalho procurou identificar e caracterizar fatores celulares envolvidos na regulação do tráfego intracelular de proteínas no processo de regulação negativa de CD4 induzido por Nef. Mais especificamente, este estudo buscou caracterizar a participação do complexo AP-1 na modulação negativa de CD4 por Nef de HIV-1, através do estudo funcional das duas isoformas de ?-adaptina, subunidades de AP-1. Utilizando a técnica de Pull-down demonstramos que Nef é capaz de interagir com ?2. Além disso, nossos dados de Imunoblot indicaram que a proteína ?2-adaptina, e não ?1-adaptina, é necessária no processo de degradação lisossomal de CD4 por Nef e que esta participação é conservada para degradação de CD4 por Nef de diferentes cepas virais. Ademais, por citometria de fluxo, o silenciamento de ?2, e não de ?1, compromete a diminuição dos níveis de CD4 por Nef da membrana plasmática. A análise por imunofluorêsncia indireta também revelou que a diminuição dos níveis de ?2 impede a redistribuição de CD4 por Nef para regiões perinucleares, acarretando no acúmulo de CD4, retirados por Nef da membrana plasmática, em endossomos primários. A depleção de ?1A, outra subunidade de AP-1, acarretou na diminuição dos níveis celulares de ?2 e ?1, bem como, no comprometimento da eficiente degradação de CD4 por Nef. Além disso, foi possível observar que, ao perturbar a maquinaria ESCRT via super-expressão de HRS (uma subunidade do complexo ESCRT-0), ocorreu um acumulo de ?2 em endossomos dilatados contendo HRS-GFP, nos quais também detectou-se CD4 que foi internalizado por Nef. Em conjunto, os resultados indicam que ?2-adaptina é uma importante molécula para o direcionamento de CD4 por Nef para a via ESCRT/MVB, mostrando ser uma proteína relevante no sistema endo-lisossomal. Ademais, os resultados indicaram que as isoformas ?-adaptinas não só possuem funções distintas, mas também parecem compor complexos AP-1 com diferentes funções celulares, já que apenas a variante AP-1 contendo ?2, mas não ?1, participa da regulação negativa de CD4 por Nef. Estes estudos contribuem para o melhor entendimento dos mecanismos moleculares envolvidos na atividade de Nef, que poderão também ajudar na melhor compreensão da patogênese do HIV e da síndrome relacionada. Em adição, este trabalho contribui para o entendimento de processos fundamentais da regulação do tráfego de proteínas transmembrana no sistema endo-lisossomal.The Human Immunodeficiency Virus (HIV) is the etiologic agent of Acquired Immunodeficiency Syndrome (AIDS). AIDS is a disease which has a global distribution, and it is estimated that there are currently at least 36.9 million people infected with the virus. During the replication cycle, HIV promotes several changes in the physiology of the host cell to promote their survival and enhance replication. The fast progression of HIV-1 in humans and animal models is closely linked to the function of an accessory protein Nef. Among several actions of Nef, one is the most important is the down-regulation of proteins from the immune response, such as the CD4 receptor. It is known that this action causes CD4 degradation in lysosome, but the molecular mechanisms are still incompletely understood. Nef forms a tripartite complex with the cytosolic tail of the CD4 and adapter protein 2 (AP-2) in clathrin-coated vesicles, inducing CD4 internalization and lysosome degradation. Previous research has demonstrated that CD4 target to lysosomes by Nef involves targeting of this receptor to multivesicular bodies (MVBs) pathway by an atypical mechanism because, although not need charging ubiquitination, depends on the proteins from ESCRTs (Endosomal Sorting Complexes Required for Transport) machinery and the action of Alix, an accessory protein ESCRT machinery. It has been reported that Nef interacts with subunits of AP- 1, AP-2, AP-3 complexes and Nef does not appear to interact with AP-4 and AP-5 subunits. However, the role of Nef interaction with AP-1 or AP-3 in CD4 down-regulation is poorly understood. Furthermore, AP-1, AP-2 and AP-3 are potentially heterogeneous due to the existence of multiple subunits isoforms encoded by different genes. However, there are few studies to demonstrate if the different combinations of APs isoforms are form and if they have distinct functional properties. This study aim to identify and characterize cellular factors involved on CD4 down-modulation induced by Nef from HIV-1. More specifically, this study aimed to characterize the involvement of AP-1 complex in the down-regulation of CD4 by Nef HIV-1 through the functional study of the two isoforms of ?-adaptins, AP-1 subunits. By pull-down technique, we showed that Nef is able to interact with ?2. In addition, our data from immunoblots indicated that ?2- adaptin, not ?1-adaptin, is required in Nef-mediated targeting of CD4 to lysosomes and the ?2 participation in this process is conserved by Nef from different viral strains. Furthermore, by flow cytometry assay, ?2 depletion, but not ?1 depletion, compromises the reduction of surface CD4 levels induced by Nef. Immunofluorescence microscopy analysis also revealed that ?2 depletion impairs the redistribution of CD4 by Nef to juxtanuclear region, resulting in CD4 accumulation in primary endosomes. Knockdown of ?1A, another subunit of AP-1, resulted in decreased cellular levels of ?1 and ?2 and, compromising the efficient CD4 degradation by Nef. Moreover, upon artificially stabilizing ESCRT-I in early endosomes, via overexpression of HRS, internalized CD4 accumulates in enlarged HRS-GFP positive endosomes, where co-localize with ?2. Together, the results indicate that ?2-adaptin is a molecule that is essential for CD4 targeting by Nef to ESCRT/MVB pathway, being an important protein in the endo-lysosomal system. Furthermore, the results indicate that ?-adaptins isoforms not only have different functions, but also seem to compose AP-1 complex with distinct cell functions, and only the AP-1 variant comprising ?2, but not ?1, acts in the CD4 down-regulation induced by Nef. These studies contribute to a better understanding on the molecular mechanisms involved in Nef activities, which may also help to improve the understanding of the HIV pathogenesis and the related syndrome. In addition, this work contributes with the understanding of primordial process regulation on intracellular trafficking of transmembrane proteins.
36
Baldelli, Laura. "Existence and multiplicity results for nonlinear elliptic problems". Doctoral thesis, 2022. http://hdl.handle.net/2158/1261959.
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In this work of thesis, we investigate existence and multiplicity results for
a class of nonlinear elliptic problems. First, we deal with problems involving
the p-Laplacian operator on bounded smooth domains, where a diffusion term
appears into the nonlinearity. For this reason, variational methods cannot be
used. Secondly, we treat existence and multiplicity of weak solutions for (p; q)-
Laplacian equations, as well as for singular p-Laplacian Schrodinger equations, in
the entire R^N whose nonlinearity combines a power-type term at critical level with
a subcritical term, involving also nontrivial weights and a positive parameter.
This latter case, considered also in a symmetric setting, allows us to use variational
methods, but in the delicate situation of lack of compactness, so that classical
results cannot be directly used, they need to be adapted.
37
Feltrin, Guglielmo. "Positive solutions to indefinite problems: a topological approach". Doctoral thesis, SISSA, 2016. http://hdl.handle.net/2318/1655560.
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The present Ph.D. thesis is devoted to the study of positive solutions to indefinite problems. In particular, we deal with the second order nonlinear differential equation u'' + a(t) g(u) = 0, where g : [0,+∞[→[0,+∞[ is a continuous nonlinearity and a : [0,T]→R is a Lebesgue integrable sign-changing weight. We analyze the Dirichlet, Neumann and periodic boundary value problems on [0,T] associated with the equation and we provide existence, nonexistence and multiplicity results for positive solutions. In the first part of the manuscript, we investigate nonlinearities g(u) with a superlinear growth at zero and at infinity (including the classical superlinear case g(u)=u^p, with p>1). In particular, we prove that there exist 2^m-1 positive solutions when a(t) has m positive humps separated by negative ones and the negative part of a(t) is sufficiently large. Then, for the Dirichlet problem, we solve a conjecture by Gómez‐Reñasco and López‐Gómez (JDE, 2000) and, for the periodic problem, we give a complete answer to a question raised by Butler (JDE, 1976). In the second part, we study the super-sublinear case (i.e. g(u) is superlinear at zero and sublinear at infinity). If a(t) has m positive humps separated by negative ones, we obtain the existence of 3^m-1 positive solutions of the boundary value problems associated with the parameter-dependent equation u'' + λ a(t) g(u) = 0, when both λ>0 and the negative part of a(t) are sufficiently large. We propose a new approach based on topological degree theory for locally compact operators on open possibly unbounded sets, which applies for Dirichlet, Neumann and periodic boundary conditions.
As a byproduct of our method, we obtain infinitely many subharmonic solutions and globally defined positive solutions with complex behavior, and we deal with chaotic dynamics. Moreover, we study positive radially symmetric solutions to the Dirichlet and Neumann problems associated with elliptic PDEs on annular domains. Furthermore, this innovative technique has the potential and the generality needed to deal with indefinite problems with more general differential operators. Indeed, our approach apply also for the non-Hamiltonian equation u'' + cu' + a(t) g(u) = 0. Meanwhile, more general operators in the one-dimensional case and problems involving PDEs will be subjects of future investigations.
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Cheng, Yi-Hsin y 鄭益新. "Existence, multiplicity of positive solutions for elliptic equations". Thesis, 2015. http://ndltd.ncl.edu.tw/handle/42781816877433940597.
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博士國立高雄大學
應用數學系博士班
103
In this thesis, we study a class of indefinite semilinear elliptic equations and the Kirchhoff type equations by using the variational methods to obtain the existence and multiplicity of positive solutions. Roughly speaking, we use the mountain pass theorem, the Palais - Smale theory and study properties of the Nehari manifold to investigate the existence and multiplicity.
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Tsing-san, Hsu y 許清山. "Existence and Multiplicity of Solutions of Semilinear Elliptic Equations". Thesis, 1995. http://ndltd.ncl.edu.tw/handle/10411383007821343383.
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Wu, Tsung-fang y 吳宗芳. "Existence and Multiplicity of Positive Solutions of Semilinear Elliptic Equations". Thesis, 2002. http://ndltd.ncl.edu.tw/handle/72879607002399242158.
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Shih, Yi-Wen y 施逸文. "Existence and Multiplicity of Solutions for Semi-linear Elliptic Equations". Thesis, 1999. http://ndltd.ncl.edu.tw/handle/18903138654509963764.
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博士國立交通大學
應用數學系
87
In part 1 we study the problem of symmetry-breaking of positive symmetric solutions of a semi-linear elliptic equation on finite cylinders with mixed type boundary conditions in two dimentions. Taking the length as a bifurcation parameter, we prove that there are asymmetric bifurcations at certain critical numbers, and obtain some global results. In part 2 we consider some semi-linear elliptic equations with asymptotic linear non-linearity and show the existence, uniqueness, and asymptotic behavior of these solutions.
42
Chan, Justin. "Asymptotic existence results on specific graph decompositions". Thesis, 2010. http://hdl.handle.net/1828/2909.
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This work examines various asymptotic edge-decomposition problems on graphs. A G-group divisible design (G-GDD) of type [g_1, ..., g_u] and index lambda is a decomposition of the edges of the complete lambda-fold multipartite graph H, with groups (maximal independent sets) G_1, ..., G_n, |G_i| = g_i, into graphs (blocks) isomorphic to G. We shall also examine special types of G-GDDs (such as G-frames) and prove that, given all parameters except u, these structures exist for all asymptotically large u satisfying the necessary conditions. Our primary technique is to invoke a useful theorem of Lamken and Wilson on edge-colored
graph decompositions. The basic construction for k-RGDDs shall be outlined at the end of the thesis.
43
Mou, Libin H. "Some existence and uniqueness results of harmonic maps". Thesis, 1990. http://hdl.handle.net/1911/16374.
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This thesis discusses some existence and uniqueness problems of harmonic maps. It consists of two parts: Part I. Existence of harmonic maps with prescribed finite singularities. Here we address the question of existence of a harmonic map from a spatial domain to the sphere S$\sp2$ which has a prescribed finite set of singularities.
Part II. Uniqueness of energy minimizing harmonic maps for almost all smooth boundary data. Suppose $\Omega$ is a smooth domain in R$\sp{m}$ and N is a compact smooth manifold. Here we show roughly that almost all smooth maps from $\partial\Omega$ to N serve as boundary values for a unique energy minimizing map u from $\Omega$ to N. This involves constructing a finite measure on a suitable (infinite dimensional) space of smooth boundary values.
44
Xu, Jia-Ren y 許嘉仁. "Existence results of capillary surfaces over convex domains". Thesis, 1989. http://ndltd.ncl.edu.tw/handle/47886333637563213405.
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45
"Ergodic control of multidimensional diffusions I. : the existence results". Laboratory for Information and Decision Systems, Massachusetts Institute of Technology], 1986. http://hdl.handle.net/1721.1/2951.
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46
謝宗憲. "Some existence results for soluations of traveling wave type". Thesis, 2006. http://ndltd.ncl.edu.tw/handle/31120718744331384489.
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47
Chen, Show-Ching y 陳秀青. "EXISTENCE RESULTS FOR HAMMERSTEIN INTEGRAL EQUATIONS AND RELATED TOPICS". Thesis, 2002. http://ndltd.ncl.edu.tw/handle/99734197373023885880.
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碩士國立台北師範學院
數理教育研究所
90
Abstract We shall provide conditions on non-positive function f (t, u), the Hammerstein integral equation and the weighted Hammerstein integral equation have at least one solution.
48
BAI, FENG-MENG y 白豐銘. "Existence and multiplicity of positive radial solutions for semilinear elliptic equations in annular domains". Thesis, 1992. http://ndltd.ncl.edu.tw/handle/31271202693393080477.
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49
Lin, Ya-Ping y 林雅萍. "Some existence results for steady states of reaction-diffusion systems". Thesis, 2006. http://ndltd.ncl.edu.tw/handle/93457131788327469250.
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博士國立彰化師範大學
數學系所
94
Abstract In this thesis, we are interested in the existence of steady states of reaction-di.usion systems with skew-gradient structure. We use two di.erent types of variational arguments to study the existence of steady states. In addition, we obtain a standing wave solution, by making use of ordered methods for quasi-monotone systems.
50
YU, HUI-ZHEN y 余慧真. "Existence results of strong solutions for certain quasilinear elliptic problems". Thesis, 1992. http://ndltd.ncl.edu.tw/handle/27518013844424526130.
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