Magíster en Ciencias de la Ingeniería, Mención Matemáticas Aplicadas. Ingeniero Civil Matemático / En el presente trabajo se muestran algunos resultados obtenidos recientemente en ciertos espacios de Banach, los llamados espacios Lipschitz-libres. Junto con las definiciones básicas y resultados que principalmente se encuentran en \cite{GK} y \cite{K}, se añaden resultados presentes en diversos artículos y trabajos publicados. Así mismo, se incluye una introducción a los conceptos de integración de funciones vector-valuadas, más precisamente, la noción de Bochner-integrabilidad, la cual resulta ser un punto clave en el desarrollo del resultado principal. Se muestra dentro de estos resultados una identificación que puede ser hallada, por ejemplo, en \cite{W} para el espacio Lipschitz-libre $\mathcal{F}(\R)$. En virtud de esto, se propone una generalización para el caso finito-dimensional, con el fin de entregar una nueva herramienta para el estudio de los espacios Lipschitz-libres en el caso mencionado. En el transcurso de la identificación de este espacio, se hace uso de herramientas clásicas de espacios de Banach y de teoría de la medida. Además, se define el espacio de funciones esencialmente Lipschitz, así como un subespacio de éste que refleja la estructura de las funciones Lipschitz nulas en $0$. Haciendo uso del espacio obtenido, se propone una vía de estudio para los espacios $\mathcal{F}(\ell^{p})$, para $1\leq p < +\infty$, usando para ello la densidad de $c_{00}$ en $\ell^{p}$ y la estructura de los espacios que identifican a $\mathcal{F}(\R^{n})$. Se incluye por completitud además en el anexo una demostración de un resultado clásico asociado a las funciones Lipchitz definidas y a valores en espacios de dimensión finita, el Teorema de Rademacher. Éste último es la pieza clave en la identificación de $\mathcal{F}(\R)$ y así mismo se proponen posibles generalizaciones en la identificación de $\mathcal{F}(\R^{n})$ para espacios de dimensión infinita en los cuales existan resultados similares a dicho teorema.

The investigation of the multifractal spectrum of the equilibrium measure for a parabolic rational map with a Lipschitz continuous potential, φ, which satisfies sup φ < P(φ) x∈J(T) is conducted. More specifically, the multifractal spectrum or spectrum of singularities, f(α) is studied.

Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-08-08T20:21:22Z No. of bitstreams: 1 ARTHUR GILZEPH FARIAS ALMEIDA - DISSERTAÇÃO PPGMAT 2013..pdf: 508810 bytes, checksum: 02ca89b269a1cb82e4ba0a5d102acff9 (MD5) / Made available in DSpace on 2018-08-08T20:21:22Z (GMT). No. of bitstreams: 1 ARTHUR GILZEPH FARIAS ALMEIDA - DISSERTAÇÃO PPGMAT 2013..pdf: 508810 bytes, checksum: 02ca89b269a1cb82e4ba0a5d102acff9 (MD5) Previous issue date: 2013-10 / CNPq / Neste trabalho estudamos a existência de, pelo menos, três soluções distintas para dois problemas de inclusão diferencial. Para isto, faremos uso da teoria da análise convexa para funcionais localmente Lipschitz, bem como métodos variacionais. / In this work we study the existence of, at least, three distinct solutions to two problems of differential inclusion. For this, we use the theory of convex functional analysis Lipschitz locally, and variational methods.

Quelques aspects de la géométrie des espaces LipschitzEn premier lieu, nous donnons les propriétés fondamentales des espaces Lipschitz libres. Puis, nous démontrons que l'image canonique d'un espace métrique M est faiblement fermée dans l'espace libre associé F(M). Nous prouvons un résultat similaire pour l'ensemble des molécules.Dans le second chapitre, nous étudions les conditions sous lesquelles F(M) est isométriquement un dual. En particulier, nous généralisons un résultat de Kalton sur ce sujet. Par la suite, nous nous focalisons sur les espaces métriques uniformément discrets et sur les espaces métriques provenant des p-Banach.Au chapitre suivant, nous explorons le comportement de type l1 des espaces libres. Entre autres, nous démontrons que F(M) a la propriété de Schur dès que l'espace des fonctions petit-Lipschitz est 1-normant pour F(M). Sous des hypothèses supplémentaires, nous parvenons à plonger F(M) dans une somme l_1 d'espaces de dimension finie.Dans le quatrième chapitre, nous nous intéressons à la structure extrémale de $F(M)$. Notamment, nous montrons que tout point extrémal préservé de la boule unité d'un espace libre est un point de dentabilité. Si F(M) admet un prédual, nous obtenons une description précise de sa structure extrémale.Le cinquième chapitre s'intéresse aux fonctions Lipschitziennes à valeurs vectorielles. Nous généralisons certains résultats obtenus dans les trois premiers chapitres. Nous obtenons également un résultat sur la densité des fonctions Lipschitziennes qui atteignent leur norme. / Some aspects of the geometry of Lipschitz free spaces.First and foremost, we give the fundamental properties of Lipschitz free spaces. Then, we prove that the canonical image of a metric space M is weakly closed in the associated free space F(M). We prove a similar result for the set of molecules.In the second chapter, we study the circumstances in which F(M) is isometric to a dual space. In particular, we generalize a result due to Kalton on this topic. Subsequently, we focus on uniformly discrete metric spaces and on metric spaces originating from p-Banach spaces.In the next chapter, we focus on l1-like properties. Among other things, we prove that F(M) has the Schur property provided the space of little Lipschitz functions is 1-norming for F(M). Under additional assumptions, we manage to embed F(M) into an l1-sum of finite dimensional spaces.In the fourth chapter, we study the extremal structure of F(M). In particular, we show that any preserved extreme point in the unit ball of a free space is a denting point. Moreover, if F(M) admits a predual, we obtain a precise description of its extremal structure.The fifth chapter deals with vector-valued Lipschitz functions.We generalize some results obtained in the first three chapters.We finish with some considerations of norm attainment. For instance, we obtain a density result for vector-valued Lipschitz maps which attain their norm.

Made available in DSpace on 2015-03-03T15:28:32Z (GMT). No. of bitstreams: 1 FrancinarioOA_DISSERT.pdf: 2024986 bytes, checksum: 9f49537413fc37dd20232d05db6223d5 (MD5) Previous issue date: 2011-12-15 / Universidade Federal do Rio Grande do Norte / The problem treated in this dissertation is to establish boundedness for the iterates of an iterative algorithm in <d which applies in each step an orthogonal projection on a straight line in <d, indexed in a (possibly infinite) family of lines, allowing arbitrary order in applying the projections. This problem was analyzed in a paper by Barany et al. in 1994, which found a necessary and suficient condition in the case d = 2, and analyzed further the case d > 2, under some technical conditions. However, this paper uses non-trivial intuitive arguments and its proofs lack suficient rigor. In this dissertation we discuss and strengthen the results of this paper, in order to complete and simplify its proofs / O problema abordado nesta disserta??o e a prova da propriedade de limita??o para os iterados de um algoritmo iterativo em Rd que aplica em cada passo uma proje??o ortogonal sobre uma reta em Rd, indexada em uma fam?lia de retas dada (possivelmente infinita) e permitindo ordem arbitr?ria na aplica??o das v?rias proje??es. Este problema foi abordado em um artigo de Barany et al. em 1994, que encontrou uma condi??o necess?ria e suficiente para o caso d = 2 e analisou tamb?m o caso d > 2 sob algumas condi??es t?cnicas. Por?m, este artigo usa argumentos intuitivos n?o triviais e nas suas demonstra??es nos parece faltar rigor. Nesta disserta??o detalhamos e completamos as demonstra??es do artigo de Barany, fortalecendo e clareando algumas de suas proposi??es, bem como propiciando pontos de vista complementares em alguns aspectos do artigo em tela

The thesis consists of two papers and one preprint. The two papers are de- voted to the approximation properties of Lipschitz-free spaces. In the first pa- per we prove that the Lipschitz-free space over a doubling metric space has the bounded approximation property. In particular, the Lipschitz-free space over a closed subset of Rn has the bounded approximation property. We also show that the Lipschitz-free spaces over ℓ1 and over ℓn 1 admit a monotone finite-dimensional Schauder decomposition. In the second paper we improve this work and obtain even a Schauder basis in the Lipschitz-free spaces over ℓ1 and ℓn 1 . The topic of the preprint is rigidity of ℓ∞ and ℓn ∞ with respect to uniformly differentiable map- pings. Our main result is a non-linear analogy of the classical result on rigidity of ℓ∞ with respect to non-weakly compact linear operators by Rosenthal, and it generalises the theorem on non-complementability of c0 in ℓ∞ due to Phillips. 1

A multiple-valued function is a "function" that assumes two or more distinct values in its range for at least one point in its domain. While these "functions" are not functions in the normal sense of being single-valued, the usage is so common that there is no way to dislodge it. This thesis is devoted to a particular class of multiple-valued functions: Q-valued functions. A Q-valued function is essentially a rule assigning Q unordered and not necessarily distinct points of R^n to each element of R^m. This object is one of the key ingredients of Almgren's 1700 pages proof that the singular set of an m-dimensional mass minimizing integral current in R^n has dimension at most m-2. We start by developing a decomposition theory and show for instance when a continuous Q-valued function can or cannot be seen as Q "glued" continuous classical functions. Then, the decomposition theory is used to prove intrinsically a Rademacher type theorem for Lipschitz Q-valued functions. A couple of Lipschitz extension theorems are also obtained for partially defined Lipschitz Q-valued functions. The second part is devoted to a Peano type result for a particular class of nonconvex-valued differential inclusions. To the best of the author's knowledge this is the first theorem, in the nonconvex case, where the existence of a continuously differentiable solution is proved under a mere continuity assumption on the corresponding multifunction. An application to a particular class of nonlinear differential equations is included. The third part is devoted to the calculus of variations in the multiple-valued framework. We define two different notions of Dirichlet nearly minimizing Q-valued functions, generalizing Dirichlet energy minimizers studied by Almgren. Hölder regularity is obtained for these nearly minimizers and we give some examples showing that the branching phenomena can be much worse in this context.

Univerzita Karlova Abstract of the diploma thesis Analysis in Banach spaces Matěj Novotný, Praha 2013 In the thesis, connection between two certain types of equivalence on Ba- nach spaces is studied: Between Lipschitz and linear one. In general, linear equivalence of two Banach spaces implies their Lipschitz equivalence, but the converse need not be true, which is shown by some nonseparable examples. There are summarized several examples to this question in the thesis, both positive and negative ones. Moreover, it is shown that James' quasi-reflexive space and its dual space have unique Lipschitz structure. To prove this, theory of linearization of Lipschitz mappings and at the same time linear structure of the two mentioned spaces is used. 1

We show that solutions to the mixed problem on a Lipschitz domain Ω can be approximated in the Sobolev space H1(Ω) by solutions to a family of related mixed Dirichlet-Robin boundary value problems which converge in H1(Ω), and we give a rate of convergence. Further, we propose a method of solving the related problem using layer potentials.

This thesis is motivated by some of the recent results of the solvability of elliptic PDE in Lipschitz domains and the relationships between the solvability of different boundary value problems. The parabolic setting has received less attention, in part due to the time irreversibility of the equation and difficulties in defining the appropriate analogous time-varying domain. Here we study the solvability of boundary value problems for second order linear parabolic PDE in time-varying domains, prove two main results and clarify the literature on time-varying domains. The first result shows a relationship between the regularity and Dirichlet boundary value problems for parabolic equations of the form Lu = div(A∇u)−ut = 0 in Lip(1, 1/2) time-varying cylinders, where the coefficient matrix A = [aij(X, t)] is uniformly elliptic and bounded. We show that if the Regularity problem (R)p for the equation Lu = 0 is solvable for some 1 < p < then the Dirichlet problem (D*) 1 p, for the adjoint equation L*v = 0 is also solvable, where p' = p/(p − 1). This result is analogous to the one established in the elliptic case. In the second result we prove the solvability of the parabolic Lp Dirichlet boundary value problem for 1 < p ≤ ∞ for a PDE of the form ut = div(A∇u)+B ·∇u on time-varying domains where the coefficients A = [aij(X, t)] and B = [bi(X, t)] satisfy a small Carleson condition. This result brings the state of affairs in the parabolic setting up to the current elliptic standard. Furthermore, we establish that if the coefficients of the operator A and B satisfy a vanishing Carleson condition, and the time-varying domain is of VMO-type then the parabolic Lp Dirichlet boundary value problem is solvable for all 1 < p ≤ ∞. This is related to elliptic results where the normal of the boundary of the domain is in VMO or near VMO implies the invertibility of certain boundary operators in Lp for all 1 < p < ∞. This then (using the method of layer potentials) implies solvability of the Lp boundary value problem in the same range for certain elliptic PDE. We do not use the method of layer potentials, since the coefficients we consider are too rough to use this technique but remarkably we recover Lp solvability in the full range of p's as the elliptic case. Moreover, to achieve this result we give new equivalent and localisable definitions of the appropriate time-varying domains.