Green operators on manifolds with edges are known to be an ingredient of parametrices of elliptic (edge-degenerate) operators. They play a similar role as corresponding operators in boundary value problems. Close to edge singularities the Green operators have a very complex asymptotic behaviour. We give a new characterisation of Green edge symbols in terms of kernels with discrete and continuous asymptotics in the axial variable of local model cones.

The tension-compression vibration of an elastic cusped plate is studied under all the reasonable boundary conditions at the cusped edge, while at the noncusped edge displacements and at the upper and lower faces of the plate stresses are given.

We consider a (generally, non-coercive) mixed boundary value problem in a bounded domain for a second order elliptic differential operator A. The differential operator is assumed to be of divergent form and the boundary operator B is of Robin type. The boundary is assumed to be a Lipschitz surface. Besides, we distinguish a closed subset of the boundary and control the growth of solutions near this set. We prove that the pair (A,B) induces a Fredholm operator L in suitable weighted spaces of Sobolev type, the weight function being a power of the distance to the singular set. Moreover, we prove the completeness of root functions related to L.

In the first part of this thesis, suitable function spaces for analysis of partial differ- ential equations in unbounded domains are introduced and studied. The results are then applied in the second part on semilinear wave equation in Rd with non- linear source term and nonlinear damping. The source term is supposed to be bounded by a polynomial function with a subcritical growth. The damping term is strictly monotone and satisfying a polynomial-like growth condition. Global existence is proved using finite speed of propagation. Dissipativity in locally uni- form spaces and the existence of a locally compact attractor are then obtained after additional conditions imposed on the damping term.

The paper is devoted to the foundation of approximation methods for integral equations of the form (aI+SbI+K)f=g, where S is the Cauchy singular integral operator on (-1,1) and K is a weakly singular integral operator. Here a,b,g are given functions on (-1,1) and the unknown function f on (-1,1) is looked for. It is assumed that a and b are real-valued and Hölder continuous functions on [-1,1] without common zeros and that g belongs to some weighted space of Hölder continuous functions. In particular, g may have a finite number of singularities. Based on known spectral properties of Cauchy singular integral operators approximation methods for the numerical solution of the above equation are constructed, where both aspects the theoretical convergence and the numerical practicability are taken into account. The weighted uniform convergence of these methods is studied using a general approach based on the theory of approximation spaces. With the help of this approach it is possible to prove simultaneously the stability, the convergence and results on the order of convergence of the approximation methods under consideration.

The paper is devoted to the foundation of approximation methods for integral equations of the form (aI+SbI+K)f=g, where S is the Cauchy singular integral operator on (-1,1) and K is a weakly singular integral operator. Here a,b,g are given functions on (-1,1) and the unknown function f on (-1,1) is looked for. It is assumed that a and b are real-valued and Hölder continuous functions on [-1,1] without common zeros and that g belongs to some weighted space of Hölder continuous functions. In particular, g may have a finite number of singularities. Based on known spectral properties of Cauchy singular integral operators approximation methods for the numerical solution of the above equation are constructed, where both aspects the theoretical convergence and the numerical practicability are taken into account. The weighted uniform convergence of these methods is studied using a general approach based on the theory of approximation spaces. With the help of this approach it is possible to prove simultaneously the stability, the convergence and results on the order of convergence of the approximation methods under consideration.

La méthode des éléments finis inversés est une méthode sans troncature qui a été introduite pour résoudre des équations aux dérivées partielles en domaines non bornés. L’objective de cette thèse est d’analyser, d’adapter puis d’implémenter cette méthode pour résoudre quelques problèmes issus de la physique, notamment lorsque le domaine géométrique est l’espace R3 tout entier. Dans un premier temps, nous présentons de manière détaillée les aspects et les principes fondamentaux de la méthode. Ensuite, nous adapterons la méthode à des problèmes de type div-rot et de potentiels vecteurs posés dans R3. Après avoir analysé la convergence de la méthode, on montrera quelques résultats numériques obtenus avec un code tridimensionnel. On s’intéresse ensuite au problème de calcul de l’énergie magnétostatique dans des problèmes de micromagnétisme, où on développe avec succès une approche numérique utilisant les éléments finis inversés. Dans la dernière partie, on adapte la méthode à un problème provenant de la chimie quantique (modèle de continuum polarisable) pour lequel on prouve qu’elle donne des résultats numériques très prometteurs. La thèse comporte beaucoup de résultats numériques issus de codes tridimensionnels écrits ou co-écrits, notamment lorsque le domaine est l’espace tout entier. Elle comporte aussi des résultats théoriques liés à l’utilisation des espaces de Sobolev à poids comme cadre fonctionnel. On apporte en particulier une preuve constructive de quelques inégalités de type div-rot dans des domaines non bornés. / Inverted finite element method (IFEM) is a non runcature method which was introduced for solving partial differential equations in unbounded domains. The objective of this thesis is to analyze, to adapt and to implement IFEM for solving several problems arising in physics, especially when the domain is the whole space R3. We first give a presentation in which we detail the principles and the main features of the method. Then, we adapt IFEM for solving some div-curl systems and vector potential problems in the whole space. In a second part, we successfully develop an IFEM based approach for computing the stray-field energy in micromagnetism. In the last part, we are interested in the study of the polarizable continuum model arising in quantum chemistry. The manuscript contains a large number of numerical results obtained with some 3D codes, especially when the domain is the whole space R3. It also contains some theoretical results in relation with weighted Sobolev spaces. We give in particular a constructive proof of some div-curl inequalities in unbounded domains.

The Ph.D. Thesis ¿Operators on weighted spaces of holomorphic functions¿ presented here treats different areas of functional analysis such as spaces of holomorphic functions, infinite dimensional holomorphy and dynamics of operators. After a first chapter that introduces the notation, definitions and the basic results we will use throughout the thesis, the text is divided into two parts. A first one, consisting of Chapters 1 and 2, focused on a study of weighted (LB)-spaces of entire functions on Banach spaces, and a second one, corresponding to Chapters 3 and 4, where we consider differentiation and integration operators acting on different classes of weighted spaces of entire functions to study its dynamical behaviour. In what follows, we give a brief description of the different chapters: In Chapter 1, given a decreasing sequence of continuous radial weights on a Banach space X, we consider the weighted inductive limits of spaces of entire functions VH(X) and VH0(X). Weighted spaces of holomorphic functions appear naturally in the study of growth conditions of holomorphic functions and have been investigated by many authors since the work of Williams in 1967, Rubel and Shields in 1970 and Shields and Williams in 1971. We determine conditions on the family of weights to ensure that the corresponding weighted space is an algebra or has polynomial Schauder decompositions. We study Hörmander algebras of entire functions defined on a Banach space and we give a description of them in terms of sequence spaces. We also focus on algebra homomorphisms between these spaces and obtain a Banach-Stone type theorem for a particular decreasing family of weights. Finally, we study the spectra of these weighted algebras, endowing them with an analytic structure, and we prove that each function f ¿ VH(X) extends naturally to an analytic function defined on the spectrum. Given an algebra homomorphism, we also investigate how the mapping induced between the spectra acts on the corresponding analytic structures and we show how in this setting composition operators have a different behavior from that for holomorphic functions of bounded type. This research is related to recent work by Carando, García, Maestre and Sevilla-Peris. The results included in this chapter are published by Beltrán in [14]. Chapter 2 is devoted to study the predual of VH(X) in order to linearize this space of entire functions. We apply Mujica¿s completeness theorem for (LB)-spaces to find a predual and to prove that VH(X) is regular and complete. We also study conditions to ensure that the equality VH0(X) = VH(X) holds. At this point, we will see some differences between the finite and the infinite dimensional cases. Finally, we give conditions which ensure that a function f defined in a subset A of X, with values in another Banach space E, and admitting certain weak extensions in a space of holomorphic functions can be holomorphically extended in the corresponding space of vector-valued functions. Most of the results obtained have been published by the author in [13]. The rest of the thesis is devoted to study the dynamical behaviour of the following three operators on weighted spaces of entire functions: the differentiation operator Df(z) = f (z), the integration operator Jf(z) = z 0 f(¿)d¿ and the Hardy operator Hf(z) = 1 z z 0 f(¿)d¿, z ¿ C. In Chapter 3 we focus on the dynamics of these operators on a wide class of weighted Banach spaces of entire functions defined by means of integrals and supremum norms: the weighted spaces of entire functions Bp,q(v), 1 ¿ p ¿ ¿, and 1 ¿ q ¿ ¿. For q = ¿ they are known as generalized weighted Bergman spaces of entire functions, denoted by Hv(C) and H0 v (C) if, in addition, p = ¿. We analyze when they are hypercyclic, chaotic, power bounded, mean ergodic or uniformly mean ergodic; thus complementing also work by Bonet and Ricker about mean ergodic multiplication operators. Moreover, for weights satisfying some conditions, we estimate the norm of the operators and study their spectrum. Special emphasis is made on exponential weights. The content of this chapter is published in [17] and [15]. For differential operators ¿(D) : Bp,q(v) ¿ Bp,q(v), whenever D : Bp,q(v) ¿ Bp,q(v) is continuous and ¿ is an entire function, we study hypercyclicity and chaos. The chapter ends with an example provided by A. Peris of a hypercyclic and uniformly mean ergodic operator. To our knowledge, this is the first example of an operator with these two properties. We thank him for giving us permission to include it in our thesis. The last chapter is devoted to the study of the dynamics of the differentiation and the integration operators on weighted inductive and projective limits of spaces of entire functions. We give sufficient conditions so that D and J are continuous on these spaces and we characterize when the differentiation operator is hypercyclic, topologically mixing or chaotic on projective limits. Finally, the dynamics of these operators is investigated in the Hörmander algebras Ap(C) and A0 p(C). The results concerning this topic are included by Bonet, Fernández and the author in [16]. / Beltrán Meneu, MJ. (2014). Operators on wighted spaces of holomorphic functions [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/36578 / Premios Extraordinarios de tesis doctorales

Les problèmes de transmission avec couche mince sont délicats à approcher numériquement, en raison de la nécessité de construire des maillages à l’échelle de la couche mince. Il est courant d’éviter ces difficultés en usant de problèmes avec conditions aux limites approchées — dites d’impédance. Si l’approximation des problèmes de transmission par des problèmes d’impédance s’avère performante dans le cas de domaines réguliers, elle l’est beaucoup moins lorsque ceux-ci comportent des coins ou arêtes. L’objet de cette thèse est de proposer de nouvelles conditions d’impédance, plus performantes, afin de corriger cette perte de performance. Pour cela, les développements asymptotiques des différents problèmes-modèles sont construits et étudiés afin de localiser avec précision l’origine de la perte, en lien avec les profils singuliers associés aux coins et arêtes. De nouvelles conditions d’impédance sont construites, de type Robin multi-échelle ou Venctel. D’abord étudiées en dimension 2, elles sont ensuite généralisées à certaines situations en dimension 3. Des simulations viennent confirmer l’efficience des méthodes théoriques. / Transmission problems with thin layer are delicate to approximate numerically, because of the necessity to build meshes on the scale of the thin layer. It is common to avoid these difficulties by using problems with approximate boundary conditions — also called impedance conditions. Whereas the approximation of transmission problems by impedance problems turns out to be successful in the case of smooth domains, the situation is less satisfactory in the presence of corners and edges. The goal of this thesis is to propose new impedance conditions, more efficient, to correct this lack of performance. For that purpose, the asymptotic expansions of the various models -problems are built and studied to locate exactly the origin of the loss, in connection with the singular profiles associated to corners and edges. New impedance conditions are built, of multi-scale Robin or Venctel types. At first studied in dimension 2, they are then generalized in certain situations in dimension 3. Simulations have been carried out to confirm the efficiency of the theoretical methods to some.