Orientador: Luiz Antonio Barreira San Martin, Marco Antonio Teixeira / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Científica / Made available in DSpace on 2018-08-05T23:39:22Z (GMT). No. of bitstreams: 1 Patrao_MauroMoraesAlves_D.pdf: 1206233 bytes, checksum: 5ea367b133840cc39b04e05a28c59d28 (MD5) Previous issue date: 2006 / Resumo: A presente tese fornece uma abordagem que estabelece conexões entre dinâmica e teoria de semigrupos. Esta abordagem, denominada de teoria de semigrupos de sombreamento, é aplicada com sucesso no estudo de semifluxos de endomorfismos de uma classe bastante ampla de fibrados, que inclui a classe dos fibrados projetivos. Os semifluxos de endomorfismos de um fibrado generalizam, por meio da linguagem geométrica de fibrados, os semifluxos de produto cruzado associados a um cociclo, definidos em fibrados triviais / Abstract: The present thesis provides an approach which establishes connections between dynamics and the theory of semigroup. This approach, named theory of shadowing semigroups, is successfully applied to study semiflows of endomorphisms of a wide class of fiber bundles, which includes the class of the projective bundles. The semiflows of endomorphisms of a fiber bundle generalize, by using the geometric concept of fiber bundle, the skew-product semiflows associated to a cocycle, which are defined in trivial bundles / Doutorado / Geometria e Topologia/Sistemas Dinamicos / Doutor em Matemática

Orientadores: Paulo Roberto Brumatti, Fernando Eduardo Torres Orihuela / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-05T23:44:44Z (GMT). No. of bitstreams: 1 Melo_Nolmar_M.pdf: 705654 bytes, checksum: b8ecfe0cc3ffd2dd2f63bc813a9c4a8d (MD5) Previous issue date: 2006 / Resumo: O objetivo central desta dissertação foi o de apresentar os Códigos Geométricos de Goppa via métodos elementares que foram introduzidos por J. H. van Lint, R. Pellikaan e T. Hfhold por volta de 1998. Numa primeira parte da dissertação são apresentados os conceitos fundamentais sobre corpos de funções racionais de uma curva algébrica na direção de se definir os códigos de Goppa de maneira clássica, neste estudo nos baseamos principalmente no livro ¿Algebraic Function Fields and Codes¿ de H. Stichtenoth. A segunda parte inicia-se com a introdução dos conceitos de funções peso, grau e ordem que são fundamentais para o estudo dos Códigos de Goppa via métodos elementares de álgebra linear e de semigrupos, tal estudo foi baseado em ¿Algebraic geometry codes¿ de J. H. van Lint, R. Pellikaan e T. Hfhold.A dissertação termina com a apresentação de exemplos que ilustram os métodos elementares que nos referimos acima / Abstract: The central objective of this dissertation was to present the Goppa Geometry Codes via elementary methods which were introduced by J. H. van Lint, R. Pellikaan and T. Hfhold about 1998. On the first past of such dissertation are presented the fundamental concepts about fields of rational functions of an algebraic curve in the direction as to define the Goppa Codes on a classical manner. In this study we based ourselves mainly on the book ¿Algebraic Function Fields and Codes¿ of H. Stichtenoth. The second part is initiated with an introduction about the functions weight, degree and order which are fundamental for the study of the Goppa Codes throught elementary methods of linear algebra and of semigroups and such study was based on ¿Algebraic Geometry Codes¿ of J. h. van Lint, R. Pellikaan and T. Hfhold. The dissertation ends up with a presentation of examples which illustrate the elementary methods that we have referred to above / Mestrado / Algebra / Mestre em Matemática

Sistemas de reação-difusão têm sido largamente estudados em diferentes contextos e através de diferentes métodos, motivados pela sua constante aparição em modelos de interação em contextos químicos, biológicos e ainda em fenômenos ecológicos. Neste trabalho nos propomos a estudar existência e unicidade - tanto do ponto de vista local como global - de soluções para uma classe de sistemas de reação-difusão acoplados, denidos em R^2, utilizando como ferramenta a teoria de semigrupos de operadores lineares. Apresentamos dois importantes exemplos: o modelo de Rosenzweig-MacArthur e um particular caso da classe de equações lambda-omega. Para o primeiro obtemos um resultado de existência e unicidade global utilizando um método de comparação envolvendo sub e super-soluções. Investigamos ainda a existência de soluções de ondas viajantes periódicas através do teorema de Bifurcação de Hopf. Já para o caso da equação lambda-omega obtemos a existência e unicidade de solucões, entretanto, a partir da aplicação da teoria de semigrupos de operadores lineares. / Reaction-diffusion systems have been widely studied in a broad variety of contexts in a large amount of disctinct approaches. It is due firstly by their constant appearance in interaction models in disciplines such as chemistry, biology and, more specific, ecology. The aim of this thesis is to provide an existence-uniqueness result - both from the local as well as from the global point of view - for solutions of a particular class of coupled reaction-diffusion systems defined over R^2. It is done applying the well established theory of semigroups of linear operators. Two remarkable examples of such systems are discussed: the Rosenzweig-MacArthur predator-prey model and a special case of lambda-omega class of equations. For the former one, an existence and uniqueness result is obtained through a comparison method - based on the notions of lower and upper solutions. Moreover, we investigate the existence of periodic travelling wave solutions via a Hopf bifurcation theorem. For the lambda-omega model another existence and uniqueness for solutions is obtained, on its turn, through the machinery obtained previously from the theory of semigroups for linear operators.

Orientador: Luiz Antonio Barrera San Martin / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica. / Made available in DSpace on 2018-08-19T06:18:00Z (GMT). No. of bitstreams: 1 Santos_ArianeLuziados_D.pdf: 829222 bytes, checksum: 870721241f42ea4a1c1748427ae28d99 (MD5) Previous issue date: 2011 / Resumo: Seja S um semigrupo com interior não vazio de um grupo de Lie simples G, conexo, complexo ou real. No caso em que o grupo G é real também considere-o não compacto, com centro finito e cuja álgebra de Lie é uma forma real, normal de uma álgebra clássica... ... Observação: O resumo, na íntegra, poderá ser visualizado no texto completo da tese digital / Abstract: Let S be a semigroup with nonempty interior of a complex or real connected simple Lie group G. In the case the group G is real also assume that G is non-compact, with finite center, whose algebra is a normal real form of a classic algebra. ... Note: The complete abstract is available with the full electronic digital thesis or dissertations / Doutorado / Matematica / Doutor em Matemática

[EN] The dynamics of linear operators, namely linear dynamics, is mainly concerned with the behaviour of iterates of linear transformations. Hypercyclicity is the study of linear operators that possess a dense orbit. Although the first examples of hypercyclic operators are due to G. D. Birkhoff (in 1929), G. R. MacLane (in 1952) and S. Rolewicz (in 1969), we can date the birth of the linear dynamics in 1982 with the unpublished PhD thesis of C. Kitai. Since then, many mathematicians have contributed to the development of this flourishing new area of the analysis. Linear dynamics connects functional analysis and dynamics. As for the classical dynamical systems, one can study the dynamics of linear operators from a topological point of view. In this context, we state that an operator has the specification property (SP). Precisely, the aim of this PhD thesis is to study the specification property on linear dynamical systems. A continuous map on a compact metric space satisfies the specification property if one can approximate pieces of orbits by a single periodic orbits with a certain uniformity. This Doctoral dissertation is a compendium of articles on the specification property. It is structured in four parts preceded by a chapter which introduces the notation, definitions and the basic results that will be needed throughout the thesis. The shift operators on sequence spaces constitute one of the most important test ground for discrete linear dynamical systems. Due to its simple structure, every time you introduce a new property in linear dynamics it is common to check it on weighted shifts operators. It is for this reason that the first part of this research work is devoted to study the specification property for unilateral and bilateral backward shift operators on weighted l^p-spaces and the relationship with other dynamical properties. In Chapter 3 we extend the results on the SP to shift operators on separable sequence F-spaces. An F-space is a vector space that is endowed with an F-norm and that is complete under the induced metric. The notion of an F-norm has the advantage that one can largely argue as if one was working in a Banach space. One need to be aware of the fact that the positive homogeneity of a norm is no longer available. The spaces l^p with 0 < p < 1 are F-spaces. Chaotic dynamical systems have received a great deal of attention in recent years. An operator is chaotic if it has a dense set of periodic points. The specification property is an interesting and rather strong notion of chaos (in the topological sense). We also consider a qualitative strengthening of hypercyclicity namely frequent hypercyclicity. It was introduced by Bayart and Grivaux, motivated by Birkhoff's ergodic theorem. An operator is frequently hypercyclic if there is some element whose orbit meets every non-empty open set very often. In Chapter 4 the specification property is deeply studied for linear and continuous operators on separable F-spaces. In addition, we are interested in finding out its relation with other dynamical properties such as mixing, Devaney chaos and frequent hypercyclicity. The results that we have achieved have been accepted to be publish in Journal of Mathematical Analysis and Applications. Finally, in the last chapter of this dissertation, we examine the specification property for strongly continuous semigroups on Banach spaces, that is, for C_0-semigroups. They can viewed as the continuous-time analogue of the discrete-time case of iterates of a single operator; in other words, the parameter in the continuous case plays the role of the iterations in the discrete case. Now the translation semigroups substitute the shift operators as test classes. Once again, we study the relationship between the specification property and mixing, chaos and frequent hypercyclicity properties of a C_0-semigroup. / [ES] La dinámica de operadores lineales, o simplemente dinámica lineal, estudia las órbitas generadas por las iteraciones de una transformación lineal. La hiperciclicidad es el estudio de los operadores lineales que poseen una órbita densa. Si bien G. D. Birkhoff (en 1929), G. R. MacLane (en 1952) y S. Rolewicz (en 1969) obtuvieron ejemplos de operadores lineales hipercíclicos, podemos fijar el nacimiento de la dinámica lineal en 1982 con la tesis de C. Kitai. Desde entonces muchos matemáticos han contribuido al desarrollo de esta floreciente área del análisis. La dinámica lineal conecta el análisis funcional y la dinámica. Al igual que en sistemas dinámicos clásicos, podemos estudiar la dinámica de operadores lineales desde un punto de vista topológico. En este contexto, hablamos de que un operador tiene la propiedad de especificación (SP). Precisamente, al estudio de la propiedad de especificación en sistemas dinámicos lineales está dedicada la presente tesis doctoral. Una aplicación continua en un espacio métrico satisface la propiedad de especificación si para cualquier familia de puntos podemos aproximar, con una cierta uniformidad, partes de sus órbitas por una sola órbita de un punto periódico. La tesis es un compendio de artículos sobre la propiedad de especificación. Se estructura en cuatro partes precedidas de un capítulo dedicado a introducir la notación, definir los conceptos y enunciar los resultados de ámbito general que van a ser utilizados en el resto de la memoria. Los operadores "shift" (desplazamiento) constituyen una de las clases más importantes, como campo de pruebas, en sistemas dinámicos lineales discretos. Debido a su estructura simple, siempre que se introduce un nuevo concepto en dinámica lineal es habitual comprobarlo sobre shifts ponderados. Por este motivo, en la primera parte de esta memoria, se estudia la propiedad de especificación para operadores desplazamiento unilaterales y bilaterales en espacios l^p ponderados y la relación con otras propiedades dinámicas. En el capítulo 3 se generalizan los resultados sobre la propiedad SP a operadores desplazamiento en F-espacios separables de sucesiones. Un F-espacio es un espacio vectorial, dotado de una F-norma, que es completo con la métrica inducida. La noción de F-norma tiene la ventaja de que permite trabajar como en un espacio de Banach llevando cuidado con la homogeneidad de la norma que ahora no se cumple Los sistemas dinámicos caóticos han recibido gran atención en los últimos años. Un operador lineal es caótico si admite un conjunto denso de puntos periódicos. La propiedad de especificación es una noción de caos (en el sentido topológico) más potente que la debida a Devaney. Otra variante más fuerte que la hiperciclicidad es la hiperciclicidad frequente. Este concepto fue introducido por Bayart y Grivaux motivados por el teorema ergódico de Birkhoff. Un operador es frecuentemente hipercíclico si algún elemento tiene una órbita que corta muy a menudo a cada conjunto abierto no vacío. En el capítulo 4 de esta tesis se estudia con profundidad la propiedad de especificación para operadores lineales y continuos definidos en F-espacios separables. Los resultados que presentamos han sido aceptados para su publicación en J. Math. Anal. Appl. Finalmente, en la cuarta parte de este trabajo, se extiende la propiedad de especificación a semigrupos de operadores fuertemente continuos en espacios de Banach, esto es, C_0-semigrupos. Estos operadores pueden verse como la versión continua del caso discreto correspondiente a las iteraciones de un único operador. Ahora, la labor de los operadores desplazamiento en espacios de sucesiones como clases de prueba la desempeñan los semigrupos de traslación. Al igual que en capítulos anteriores, se estudia la relación de la propiedad SP para C_0-semigrupos con otras propiedades dinámicas. / [CAT] La dinàmica d'operadors lineals, o simplement dinàmica lineal, estudie les òrbites generades per les iteracions d'una transformació lineal. La hiperciclicitat es el estudi dels operadors lineal que posseeixen una òrbita densa. Si bé G. D. Birkhoff (en 1929), G. R. MacLane (en 1952) y S. Rolewicz (en 1969) van obtenir exemples d'operadors lineals hipercíclics, podem fixar el naixement de la dinàmica lineal en 1982 amb la tesi de C. Kitai [68]. Des de llavors molts matemàtics han contribuït al desenvolupament d'esta florent area de l'anàlisi. La dinàmica lineal connecta el anàlisi funcional y la dinàmica. Igual que en sistemes dinàmics clàssics, podem estudiar la dinàmica d'operadors lineals des d'un punt de vista topològic. En eixe context, parlem que un operador té la propietat d'especificació (SP). Precisament, al estudi de la propietat d'especificació en sistemes dinàmics lineals està dedicada la present tesi doctoral. Una aplicació continua en un espai mètric compleix la propietat d'especificació si per a qualsevol família de punts podem aproximar, amb certa uniformitat, parts de les seues òrbites per una sola òrbita d'un punt periòdic. La tesi es un compendi de articles sobre la propietat d'especificació. S'estructura en quatre parts precedides d'un capítol dedicat a introduir la notació, definir els conceptes i enunciar els resultats d'àmbit general que seran utilitzats en la resta de la memòria. Els operadors "shifts" (desplaçaments) constitueixen una de les classes més importants, com a camp de proves, en sistemes dinàmics lineals discrets. Degut a la seua estructura simple, sempre que es introdueix un nou concepte en dinàmica lineal es habitual comprovar-ho sobre shifts ponderats. Per esta raó, en la primera part d'esta memòria, s'estudia la propietat d'especificació per a operadors desplaçament unilaterals i bilaterals en espais l^p ponderats i la relació amb altres propietats dinàmiques. En el capítol 3 es generalitzen els resultats sobre la propietat SP a operadors desplaçament en F-espais separables de successions. Un F-espai es un espai vectorial, dotat d'una F-norma, que és complet amb la mètrica induida. La noció de F-norma té l'avantatge que permet treballar com en un espai de Banach anant en compte amb l'homogeneitat de la norma que ara no es compleix. Els espais l^p amb 0 < p < 1 són exemples de F-espais. Els sistemes dinàmics caòtics han rebut gran atenció en els últims anys. Un operador lineal és caòtic si admet un conjunt dens de punts periòdics. La propietat d'especificació és una noció de caos (en el sentit topològic) més potent que la deguda a Devaney. Una altra variant més forta que la hiperciclicitat és la hiperciclicitat freqüent. Aquest concepte va ser introduït per Bayart i Grivaux motivats per el teorema ergòdic de Birkhoff. Un operador és freqüentment hipercíclic si algun element té una òrbita que talle molt sovint a cada conjunt obert no vuit. En el capítol 4 d'esta tesi se estudie amb profunditat la propietat d'especificació per a operadors lineals i continus definits en F-espais separables. També s'incideix en la connexió de dita propietat amb altres propietats dinàmiques. Els resultats que presentem han estat acceptats per a la seva publicació en J. Math. Anal. Appl. Finalment, en la quarta part d'aquest treball, s'estén la propietat d'especificació a semigrups d'operadors fortament continus en espais de Banach, això és, C_0-semigrups. Aquests operadors poden veure's com la versió continua del cas discret corresponen a les iteracions d'un únic operador; en altres paraules, el paper de les iteracions en el cas discret ho assumeix el paràmetre en el cas continu. Ara, la labor del operadors desplaçament en espais de successions com classes de prova l'exerceixen els semigrups de translació. Igual que en capítols anteriors, s'estudia la relació de la propietat SP per a C0-semigrups amb altres propie / Bartoll Arnau, S. (2016). The specification property in linear dynamics [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/61633 / TESIS

In this thesis we analyze lumpability of infinite dimensional dynamical systems. Lumping is a method to project a dynamics by a linear reduction operator onto a smaller state space on which a self-contained dynamical description exists. We consider a well-posed dynamical system defined on a Banach space X and generated by an operator F, together with a linear and bounded map M : X → Y, where Y is another Banach space. The operator M is surjective but not an isomorphism and it represents a reduction of the state space. We investigate whether the variable y = M x also satisfies a well-posed and self-contained dynamics on Y . We work in the context of strongly continuous semigroup theory. We first discuss lumpability of linear systems in Banach spaces. We give conditions for a reduced operator to exist on Y and to describe the evolution of the new variable y . We also study lumpability of nonlinear evolution equations, focusing on dissipative operators, for which some interesting results exist, concerning the existence and uniqueness of solutions, both in the classical sense of smooth solutions and in the weaker sense of strong solutions. We also investigate the regularity properties inherited by the reduced operator from the original operator F . Finally, we describe a particular kind of lumping in the context of C*-algebras. This lumping represents a different interpretation of a restriction operator. We apply this lumping to Feller semigroups, which are important because they can be associated in a unique way to Markov processes. We show that the fundamental properties of Feller semigroups are preserved by this lumping. Using these ideas, we give a short proof of the classical Tietze extension theorem based on C*-algebras and Gelfand theory.

No description available.

The study of integral operators on spaces of analytic functions has been considered for the past few decades. However, most of the studies in this line are based on spaces of analytic functions of the unit disc. For the analytic spaces of the upper half-plane, the literature is still scanty. Most notable is the recent work of Siskakis and Arvanitidis concerning the classical Ces`aro operator on Hardy spaces of the upper half-plane. In this dissertation, we characterize all continuous one-parameter groups of automorphisms of the upper halfplane according to the nature and location of their fixed points into three distinct classes, namely, the scaling, the translation, and the rotation groups. We then introduce the associated groups of weighted composition operators on both Hardy and weighted Bergman spaces of the half-plane. Interestingly, it turns out that these groups of composition operators form three strongly continuous groups of isometries. A detailed analysis of each of these groups of isometries is carried out. Specifically, we determine the spectral properties of the generators of every group, and using both spectral and semigroup theory of Banach spaces, we obtain concrete representations of the resolvents as integral operators on both Hardy and Bergman spaces of the half-plane. For the scaling group, the resulting resolvent operators are exactly the Ces`aro-like operators. The spectral properties of the obtained integral operators is also determined. Finally, we detail the theory of both Szeg¨o and Bergman projections of the half-plane, and use it to determine the duality properties of these spaces. Consequently, we obtain the adjoints of the resolvent operators on the reflexive Hardy and Bergman spaces of the half-plane.

This thesis is dedicated to the so-called Dirichlet-to-Neumann map associated with the weighted 𝑝-Laplace operator. In Chapter 1, we begin by deriving the Dirichlet-to-Neumann map by using classical modelling and outline why it is interesting to study this boundary operator. In the remaining part of Chapter 1, we dedicate each section an overview about the content of one chapter and summarize the main results. Chapter 2 is dedicated to the Poisson problem and the inverse of the Dirichlet-to-Neumann map. Chapter 3 provides the first main application of the Dirichlet-to-Neumann map, namely, it generates a strongly continuous semigroup of contractions on the Lebesgue space 𝐿2 and this contraction can be extrapolated to a contraction on 𝐿q for all 1 ≤ 𝑞 ≤ ∞. In Chapter 4, we develop an abstract theory to establish global 𝐿𝑞-𝐿∞ regularization estimates satisfied by the semigroup generated by the negative Dirichlet-to-Neumannmap. Chapter 5 is concerned with 𝐿1 and point-wise estimates on the time-derivative of the semigroup generated by the neagtive Dirichlet-to- Neumann map, which are known in the literatur as Aronson-Bénilan type estimates. In Chapter 6, we outline the theory of 𝑗-functional and its application to evolution problems. This theory allows us to study the Dirichlet problem on general open sets Ω, and to realize the Dirichlet-to-Neumann map as an operator in 𝐿2 (𝜕Ω). In Chapter 7, we consider the limit case 𝑝 = 1, which corresponds to the Dirichlet-to-Neumann map associated with the (unweighted) 1-Laplace operator. Each chapter covers parts of the authors papers mentioned in the references.:Chapter 1 Introduction................................................... 1 1.1 Motivation-physical background ............................. 2 1.2 The Dirichlet-to-Neumann map - an analyst’s perspective . . . . . . . . . 5 1.2.1 Step1. The Dirichlet problem.......................... 5 1.2.2 Step2. The Neumann boundary operator ................ 8 2 1.3 The Dirichlet-to-Neumann map on 𝐿2 ......................... 9 1.4 The Dirichlet-to-Neumann map and Leray-Lions operators . . . . . . . . 11 1.5 The Dirichlet-to-Neumann map is a nonlocal operator . . . . . . . . . . . . 12 1.6 The Dirichlet-to-Neumann map on open sets.................... 13 1.6.1 𝑗-elliptic functionals and their 𝑗-subgradient . . . . . . . . . . . . . 13 1.6.2 The construction of a weak trace on open sets ............ 15 1.6.3 Construction of the Dirichlet-to-Neumann map . . . . . . . . . . . 17 1.7 The Poisson problem and the Neumann-to-Dirichlet map . . . . . . . . . 19 1.8 Evolution problems governed by the Dirichlet-to-Neumann map . . . 21 1.9 𝐿𝑞-𝐿∞ regularization and decay estimates...................... 27 1.10 Aronson-Bénilantypeestimates .............................. 30 1.11 The Dirichlet-to-Neumann map associated with the 1-Laplacian . . . 33 Chapter 2 The Poisson problem and the Neumann-to-Dirichlet map . . . . . . . . . . . 45 2.1 The Poisson problem........................................ 45 2.2 Preliminaries .............................................. 46 2.3 The Dirichlet problem....................................... 48 2.4 The Dirichlet-to-Neumann map............................... 51 2.5 Proof of Theorem 2.1 ....................................... 56 2.5.1 Proof of claim (1) of Theorem 2.1 ...................... 56 2.5.2 Preliminaries for the proof of claim (2) of Theorem 2.1 . . . . 58 2.5.3 Proof of claim( 2) of Theorem 2.1 ...................... 60 Chapter 3 Nonlinear elliptic-parabolic evolution problems.................... 61 3.1 Main result................................................ 61 3.2 Preliminaries .............................................. 64 3.2.1 Some function spaces................................. 64 3.2.2 Nonlinear semigroupt heory - Part I..................... 65 3.2.3 Homogeneous operators - Part I ........................ 75 2 3.3 The Dirichlet-to-Neumann map on 𝐿2 ...................... 77 3.4 The Dirichlet-to-Neumann map on 𝐿1, 𝐿𝜓 and C................ 82 3.5 Proof of Theorem 3.1 ....................................... 84 Chapter 4 𝑳𝒒-𝑳∞ regularization and decay estimates ........................ 89 4.1 Main results............................................... 89 4.2 Preliminaries .............................................. 91 4.3 Sobolev implies 𝐿𝑞 -𝐿𝑟 regularization estimates ................. 92 4.4 Extrapolation towards 𝐿1 .................................... 98 4.5 A nonlinear interpolation theorem.............................100 4.6 Extrapolation towards 𝐿∞ via interpolation of the semigroup . . . . . . 107 4.7 Proof of Theorem 4.1 .......................................115 Chapter 5 Aronson-Bénilan type estimates..................................117 5.1 Main results ...............................................117 5.2 Preliminaries ..............................................119 5.2.1 Nonlinearsemigrouptheory-PartII ....................119 5.2.2 Homogeneousaccretiveoperators ......................130 5.2.3 Homogeneous completely accretive operators . . . . . . . . . . . . 138 5.3 Proof of Theorem 5.1 .......................................141 Chapter 6 The Dirichlet-to-Neumann map on open sets ......................143 6.1 Main results ...............................................143 6.2 The 𝑗-subgradient and basic properties ........................146 6.2.1 Definition and characterisation as a classical gradient . . . . . . 146 6.2.2 Ellipticextensions ...................................151 H 6.2.3 Identification of 𝜑 ..................................152 6.2.4 The case when 𝑗 is a weakly closed operator .............155 6.3 Semigroups and invariance of convex sets ......................156 6.3.1 Positive semigroups ..................................160 6.3.2 Comparison and domination of semigroups ..............161 6.3.3 𝐿∞-contractivity and extrapolation of semigroups . . . . . . . . . 163 6.4 Application:The Dirichlet-to-Neumann map....................168 Chapter 7 The Dirichlet-to-Neumann map associated with the 1-Laplacian . . . . . 171 7.1 Preliminaries ..............................................171 7.1.1 Functions of bounded variation.........................171 7.1.2 Nonlinear semigroup theory - Part III ...................178 7.2 The Dirichlet problem for the 1-Laplace operator................180 7.3 A Robin-type problem for the 1-Laplace operator................187 7.4 Proofs of the main results....................................189 7.4.1 The Dirichlet-to-Neumann operator in 𝐿1 ................189 7.4.2 The Dirichlet-to-Neumann operator in 𝐿2 ................200 7.4.3 The Dirichlet-to-Neumann operator in 𝐿1 (continued)...........204 7.4.4 Long-timestability...................................206 Appendix A Weighted Sobolev Spaces........................................213 A.1 p-admissible weights........................................213 B Mean spaces by Lions and Peetre ................................215 B.1 The connection between mean spaces and 𝐿p spaces.............215 References . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . 219 Index .............................................................227

Neste trabalho, estudamos uma generalização dos semigrupos gradientes, os semigrupos gradiente-like, algumas de suas propriedades e a sua invariância por pequenas perturbações; isto é, pequenas perturbações de sistemas gradiente-like continuam sendo gradiente-like. Como consequência da caracterização dos atratores para este tipo de sistema, estudamos a atração exponencial de atratores. Por fim, estudamos o concetio de dimensão de Hausdorff e dimensão fractal de atratores e apresentamos alguns resultados sobre este assunto, e estudamos a construção de uma nova classe de atratores, os atratores exponenciais fractais / In this work, we study a generalization of gradient discrete semigroups, the gradientlike semigroups, some of its properties and its invariance under small perturbations; that is, small perturbations of gradient-like semigroups are still gradient-like semigroups. As a consequence of the characterization of the attractors for this sort of semigroups, we study the exponential attraction of attractors. Finally, we study some concepts of Hausdorff dimension and fractal dimension and present some results about this subject, and we studied the construction of a new class of attractors, the exponential fractal attractors