On the first part, we consider nonlinear operators I depending on a family of nonlocal linear operators [mathematical equations]. We study the solutions of the Dirichlet initial and boundary value problems [mathematical equations]. We do not assume even symmetry for the kernels. The odd part bring some sort of nonlocal drift term, which in principle competes against the regularization of the solution. Existence and uniqueness is established for viscosity solutions. Several Hölder estimates are established for u and its derivatives under special assumptions. Moreover, the estimates remain uniform as the order of the equation approaches the second order case. This allows to consider our results as an extension of the classical theory of second order fully nonlinear equations. On the second part, we study two phase problems posed over a two dimensional cone generated by a smooth curve [mathematical symbol] on the unit sphere. We show that when [mathematical equation] the free boundary avoids the vertex of the cone. When [mathematical equation]we provide examples of minimizers such that the vertex belongs to the free boundary. / text

L’opérateur du p-Laplacien non local, l’équation d’évolution et la régularisation variationnelle associées régies par un noyau donné ont des applications dans divers domaines de la science et de l’ingénierie. En particulier, ils sont devenus des outils modernes pour le traitement massif des données (y compris les signaux, les images, la géométrie) et dans les tâches d’apprentissage automatique telles que la classification. En pratique, cependant, ces modèles sont implémentés sous forme discrète (en espace et en temps, ou en espace pour la régularisation variationnelle) comme approximation numérique d’un problème continu, où le noyau est remplacé par la matrice d’adjacence d’un graphe. Pourtant, peu de résultats sur la consistence de ces discrétisations sont disponibles. En particulier, il est largement ouvert de déterminer quand les solutions de l’équation d’évolution ou du problème variationnel des tâches basées sur des graphes convergent (dans un sens approprié) à mesure que le nombre de sommets augmente, vers un objet bien défini dans le domaine continu, et si oui, à quelle vitesse. Dans ce manuscrit, nous posons les bases pour aborder ces questions.En combinant des outils de la théorie des graphes, de l’analyse convexe, de la théorie des semi- groupes non linéaires et des équations d’évolution, nous interprétons rigoureusement la limite continue du problème d’évolution et du problème variationnel du p-Laplacien discrets sur graphes. Plus précisé- ment, nous considérons une suite de graphes (déterministes) convergeant vers un objet connu sous le nom de graphon. Si les problèmes d’évolution et variationnel associés au p-Laplacien continu non local sont discrétisés de manière appropriée sur cette suite de graphes, nous montrons que la suite des solutions des problèmes discrets converge vers la solution du problème continu régi par le graphon, lorsque le nombre de sommets tend vers l’infini. Ce faisant, nous fournissons des bornes d’erreur/consistance.Cela permet à son tour d’établir les taux de convergence pour différents modèles de graphes. En parti- culier, nous mettons en exergue le rôle de la géométrie/régularité des graphons. Pour les séquences de graphes aléatoires, en utilisant des inégalités de déviation (concentration), nous fournissons des taux de convergence nonasymptotiques en probabilité et présentons les différents régimes en fonction de p, de la régularité du graphon et des données initiales. / The non-local p-Laplacian operator, the associated evolution equation and variational regularization, governed by a given kernel, have applications in various areas of science and engineering. In particular, they are modern tools for massive data processing (including signals, images, geometry), and machine learning tasks such as classification. In practice, however, these models are implemented in discrete form (in space and time, or in space for variational regularization) as a numerical approximation to a continuous problem, where the kernel is replaced by an adjacency matrix of a graph. Yet, few results on the consistency of these discretization are available. In particular it is largely open to determine when do the solutions of either the evolution equation or the variational problem of graph-based tasks converge (in an appropriate sense), as the number of vertices increases, to a well-defined object in the continuum setting, and if yes, at which rate. In this manuscript, we lay the foundations to address these questions.Combining tools from graph theory, convex analysis, nonlinear semigroup theory and evolution equa- tions, we give a rigorous interpretation to the continuous limit of the discrete nonlocal p-Laplacian evolution and variational problems on graphs. More specifically, we consider a sequence of (determin- istic) graphs converging to a so-called limit object known as the graphon. If the continuous p-Laplacian evolution and variational problems are properly discretized on this graph sequence, we prove that the solutions of the sequence of discrete problems converge to the solution of the continuous problem governed by the graphon, as the number of graph vertices grows to infinity. Along the way, we provide a consistency/error bounds. In turn, this allows to establish the convergence rates for different graph models. In particular, we highlight the role of the graphon geometry/regularity. For random graph se- quences, using sharp deviation inequalities, we deliver nonasymptotic convergence rates in probability and exhibit the different regimes depending on p, the regularity of the graphon and the initial data.

In this thesis we present some results on cross-diffusion and nonlocal interaction. In the first part we study a PDE model for two diffusing species interacting by local size exclusion and global attraction. This leads to a nonlinear degenerate cross-diffusion system, for which we provide a global existence result. The analysis is motivated by the formulation of the system as a formal gradient flow for an appropriate energy functional consisting of entropic terms as well as quadratic nonlocal terms. Key ingredients are entropy dissipation methods as well as the recently developed boundedness by entropy principle. Moreover, we investigate phase separation effects inherent in the cross-diffusion model by an analytical and numerical study of minimizers of the energy functional and their asymptotics to a previously studied case as the diffusivity tends to zero. Finally we briefly discuss coarsening dynamics in the system, which can be observed in numerical results and is motivated by rewriting the PDEs as a system of nonlocal Cahn-Hilliard equations. Proving the uniqueness of solutions to multi-species cross-diffusion systems is a difficult task in the general case, and very few results exist in this direction. In the second part of this thesis, we study a particular system with zero-flux boundary conditions for which the existence of a weak solution has been proven in [60]. Under additional assumptions on the value of the cross-diffusion coefficients, we are able to show the existence and uniqueness of nonnegative strong solutions. The proof of the existence relies on the use of an appropriate linearized problem and a fixed-point argument. In addition, a weak-strong stability result is obtained for this system in dimension one which also implies uniqueness of weak solutions. In the third part we focus on a class of integral functionals known as nonlocal perimeters. Intuitively, these functionals express a weighted interaction between a set and its complement. The weight is provided by a positive kernel K which might be singular. We show that these functionals are indeed perimeters in a generalised sense and we establish existence of minimisers for the corresponding Plateau’s problem. Also, when K is radial and strictly decreasing, we prove that halfspaces are minimisers if we prescribe “flat” boundary conditions. Furthermore, a Γ-convergence result is discussed. We study the limiting behaviour of the nonlocal perimeters associated with certain rescalings of a given kernel which might be singular in the origin but that have faster-than-L 1 decay at infinity and we show that the Γ-limit is the classical perimeter, up to a multiplicative constant that we give explicitly.

Made available in DSpace on 2016-06-02T19:06:17Z (GMT). No. of bitstreams: 1 6277.pdf: 15816665 bytes, checksum: 105661656ee67fe816f34a96605797f9 (MD5) Previous issue date: 2014-01-23 / The development of new methods and noisy images filtering techniques still attract researchers, which seek to reduce the noise with the minimal loss of details, edges, resolution and removal of fine structures of the image. Moreover, it is extremely important to expand the capacity of the filters for the different noise models present in the Image and Signal Processing literature, like the multiplicative noise speckle, present in the synthetic aperture radar (SAR) images. This Master s degree thesis aims to use a recent denoising algorithm: the nonlocal means (NLM), developed for the additive white gaussian noise (AWGN), and expand, analyze and compare its capacity for intensity SAR images denoising (despeckling), which are contaminated with the speckle. This expansion of the NLM filter is based with the use of the stochastic distances and the comparison of the estimated parameters with de G0 and the inverse Gamma distributions. Finally, this work compares the synthetic and real results of the proposed filter with some filters of the literature. / A elaboração de novos métodos e técnicas de filtragem de imagens ruidosas ainda atraem pesquisadores, que buscam a redução de ruído com a mínima perda dos detalhes, bordas, resolução e remoção de estruturas finas da imagem. Além disto, é de extrema importância ampliar a capacidade dos filtros para diversos modelos de ruído existentes na literatura de Processamento de Imagens e Sinais, como o ruído multiplicativo speckle , presente em imagens de radar de abertura sintética (SAR). Esta dissertação de Mestrado tem o objetivo de utilizar um algoritmo de filtragem recente: o nonlocal means (NLM), desenvolvido para o ruído branco aditivo gaussiano (AWGN), e ampliar, analisar e comparar a sua capacidade para a filtragem de imagens SAR de intensidade ( despeckling ), as quais são contaminadas com o speckle . Esta ampliação do filtro NLM é baseada no uso das distâncias estocásticas e na comparação dos parâmetros estimados através das distribuições G0 e da inversa da Gama. Por fim, este trabalho compara os resultados sintéticos e reais do filtro proposto com alguns filtros da literatura.

Peridynamics is an emerging nonlocal continuum theory which allows governing field equations to be applicable at discontinuities. This applicability at discontinuities is achieved by replacing the spatial derivatives, which lose meaning at discontinuities, with integrals that are valid regardless of the existence of a discontinuity. Within the realm of solid mechanics, the peridynamic theory is one of the techniques that has been employed to model material fracture. In this work, the peridynamic theory is used to investigate different fracture problems in order to establish its fidelity for predicting crack growth. Various fracture experiments are modeled and analyzed. The peridynamic predictions are made and compared against experimental findings along with predictions from other commonly used numerical fracture techniques. Additionally, this work applies the peridynamic framework to model heat transfer. Generalized peridynamic heat transfer equation is formulated using the Lagrangian formalism. Peridynamic heat conduction quantites are related to quanties from the classical theory. A numerical procedure based on an explicit time stepping scheme is adopted to solve the peridynamic heat transfer equation and various benchmark problems are considered for verification of the model. This paves the way for the coupling of thermal and structural fields within the framework of peridynamics. The fully coupled peridynamic thermomechanical equations are derived based on thermodynamic considerations, and a nondimensional form of the coupled thermomechanical peridynamic equations is also presented. An explicit staggered algorithm is implemented in order to numerically approximate the solution to these coupled equations. The coupled thermal and structural responses of a thermoelastic semi-infinite bar and a thermoelastic vibrating bar are subsequently investigated.

Darbe nagrinėjamas parabolinių lygčių su nelokaliąja daugiataške sąlyga ir tikrinių reikšmių uždaviniai antrosios eilės paprastajam diferencialiniam operatoriui. Uždavinio specifika yra ta, kad vietoje vienos arba abiejų klasikinių kraštinių sąlygų duota nelokalioji sąlyga. Tokio tipo kraštiniai uždaviniai diferiancialinėms lygtims paskutiniaisiais metais gana intensyviai nagrinėjami matematinėje literatūroje. Darbe naudojamas M. Sapagovo ir A. Štikono 2005 straipsnio metodika tikrinių reikšmių savybėms tirti. / The parabolic equation with nonlocal multi-point condition and the eigenvalue problem for differential operation with nonlocal multi-point condition is investigated in the work. Nonlocal condition is given instead one or both classical boundary conditions. These problems are investigated in the mathematical literature in recent years. The method of analysis to eigenvalue of the article [5] by M. Sapagovas and A. Štikonas on 2005 are used in the work.

O sistema de FitzHugh-Nagumo possui um tipo especial de solução chamadas ondas viajantes, que são da forma &micro(x,t)=&oslash(x+ct) e w(x,t)=&#1137(x+ct) e além disso sabe-se que ela é estável. Tem-se o interesse de obter uma caracterização de seu perfil (&oslash,&#1137) e sua velocidade de propagação c. Fazendo uma mudança de variáveis, transformamos tal problema em encontrar equilíbrios de uma equação não local. Esta equação não local possui uma onda viajante de velocidade zero cujo perfil é o mesmo da equação original e, com esta equação, é possível aproximar, ao mesmo tempo, o perfil e a velocidade da onda viajante. Como a intenção é usar métodos numéricos para aproximar tais soluções, o problema não local foi analisado em um intervalo limitado verificando a existência e algumas propriedades espectrais em domínios limitados. / The FitzHugh-Nagumo systems have a special kind of solution named traveling wave, which has a form &micro(x,t)=&oslash(x+ct) and w(x,t)=&#1137(x+ct) and furthermore it is a stable solution. It is our interest to obtain a characterization of its profile (&oslash,&#1137) and speed of propagation c. Changing variables, we transform the problem of finding these solutions in the problem of finding an equilibria in a nonlocal equation. This nonlocal equation has a traveling wave with zero speed whose profile is the same of the original equation, and the nonlocal equation is used to approximate the profile and speed of the traveling wave at the same time. To use numerical methods for approximating such solutions, the nonlocal problem was analyzed in a finite interval to check that the existence and some spectral properties on bounded domains.

Adhesion, which includes cell-to-cell and cell-to-extracellular-matrix adhesion, plays an important role in cancer invasion and metastasis. After undergoing morphological changes malignant and invasive tumour cells, i.e., cancer cells, break away from the primary tumour by loss of cell-cell adhesion, degrade their basement membrane and migrate through the extracellular matrix by enhancement of cell-matrix adhesion. These processes require interactions and signalling cross-talks between proteins and cellular components facilitating the cell adhesion. Although such processes are very complex, the necessity to fully understand the mechanism of cell adhesion is crucial for cancer studies, which may contribute to improving cancer treatment strategies. We consider mathematical models in an attempt to understand better the roles of cell adhesion involved in cancer invasion. Using mathematical models and computational simulations, the underlying complex biological processes can be better understood and their properties can be predicted that might not be evident in laboratory experiments. Cancer cell migration and invasion of the extracellular matrix involving adhesive interactions between cells mediated by cadherins and between cell and matrix mediated by integrins, are modelled by employing two types of mathematical models: a continuum approach and an individual-based approach. In the continuum approach, we use Partial Differential Equations in which cell adhesion is treated as non-local and formulated by integral terms. In the individual-based approach, we first develop pathways for cell-cell and cell-matrix adhesion using Ordinary Differential Equations and later incorporate the pathways in a simulation environment for multiscale computational modelling. The computational simulation results from the two different mathematical models show that we can predict invasive behaviour of cancer cells from cell adhesion properties. Invasion occurs if we reduce cell-cell adhesion and increase cell-matrix adhesion and vice versa. Changing the cell adhesion properties can affect the spatio-temporal behaviour of cancer cell invasion. These results may lead to broadening our understanding of cancer cell invasion and in the long term, contributing to methods of patient treatment.

De nombreux noyaux atomiques et hypernoyaux se modélisent comme des structures à trois corps. C'est le cas, par exemple, de noyaux à halo, comme 6He, ou de noyaux stables, comme 12C et 9Be. En effet, 6He se caractérise comme un système à trois corps, formé d'un coeur (une particule alpha) et de deux neutrons de valence faiblement liés. Le noyau de 12C peut s'étudier comme un système lié formé de trois particules alphas, tandis que 9Be peut être décrit comme la liaison de deux particules alphas et d'un neutron. Dans les exemples précédents, les particules alphas sont des amas de nucléons. Elles possèdent donc une structure interne dont il faut tenir compte en raison du principe de Pauli. Les modèles les plus réalistes pour décrire les structures à trois corps sont les modèles "microscopiques". Ces modèles prennent en compte explicitement tous les nucléons et respectent exactement le principe d'antisymétrisation de Pauli. Cependant, l'application de ces modèles est fortement limitée en pratique, car ils exigent de trop nombreux et trop longs calculs. Par conséquent, pour simplifier considérablement les calculs et permettre l'étude des structures à trois corps, des modèles moins détaillés, de type "semi-microscopiques", sont également développés. Dans ces modèles, on représente les amas de nucléons comme de simples particules ponctuelles. Dans ce cas, la modélisation consiste à construire les potentiels effectifs entre les amas, puis à les employer dans les modèles à trois corps. Dans ce travail, nous avons développé les modèles "semi-microscopiques à trois corps". Les potentiels effectifs entre amas sont directement déduits des forces entre nucléons (selon la RGM à 2 corps). Ces potentiels sont "non-locaux", et dépendent des énergies des amas qui interagissent. Ils permettent de simuler le principe de Pauli et les échanges de nucléons entre les amas. La dépendance en l'énergie se révèle être un inconvénient dans les modèles à trois corps. Les potentiels effectifs sont par conséquent transformés en de nouveaux potentiels (non-locaux) indépendants de l'énergie, bien adaptés aux modèles à trois corps. Les modèles "semi-microscopiques" sont beaucoup plus simples et plus rapides que les modèles "microscopiques". Ils fournissent les fonctions d'onde des états liés à trois corps des noyaux légers et hypernoyaux. Cela permet d'une part de comprendre les propriétés spectroscopiques nucléaires, et d'autre part, cela ouvre la voie pour de futurs modèles de réactions nucléaires impliquant les structures à trois corps. / Several atomic nuclei and hypernuclei can be modelled as three-body structures: e.g., two-neutron halo nuclei, such as 6He, and other nuclei, such as 12C and 9Be. Indeed 6He can be represented as a three-body system, made up of a core (an alpha particle) and two weakly bound valence neutrons. The 12C nucleus can be studied as a bound system formed by three alpha particles, while the 9Be nucleus can be described as the binding of two alpha particles and one neutron. In these typical examples, the alpha particles are clusters of nucleons. They have an internal structure that must be taken into account because of the Pauli principle. The most realistic models are the "microscopic models". In these models, all the nucleons are taken into account, and the Pauli antisymmetrisation principle is fully respected. However, the application of the "microscopic models" is limited in practice, because they require too many laborious calculations. Therefore, in order to greatly simplify the calculations, "semi-microscopic models" are developed. In those models, the clusters of nucleons are treated as ("structureless") pointlike particles. The models then consist in determining the effective potentials between the clusters, and in using them in three-body models. In the present work, we have developed "semi-microscopic models". The effective potentials between the clusters are directly obtained from the interactions between nucleons (according to the two-cluster RGM). These potentials are "nonlocal", and depend on the energy of the interacting clusters. The non-locality is a direct consequence of the Pauli principle and the exchanges of nucleons between the clusters. The energy-dependence of the potentials turns out to be a drawback in three-body models. Therefore, the effective potentials are transformed into energy-independent potentials, which can be used in three-body models. The "semi-microscopic models" are much simpler and faster than the "microscopic models". They provide the three-body bound-state wave functions (i.e., the spectroscopic properties and the structure) of light nuclei and hypernuclei. Such wave functions are also the basic ingredient that will be used in future reactions models.

In this thesis the nonlinear effects and heating are studied in inductively coupled plasma (ICP) in a regime of anomalous skin effect (nonlocal regime). In this regime the thermal motion of plasma electrons plays an important role, significantly influencing the processes associated with the penetration of electromagnetic field into plasma, such as the ponderomotive effect and heating of plasma by the field. We have developed a linear kinetic theory that describes the electron dynamics in ICP taking into account the electron thermal motion and collisions of electrons. This theory yields relatively simple expressions for the electron current in plasma, the ponderomotive force, and plasma heating. It describes correctly the thermal reduction of ponderomotive force in the nonlocal regime, which has been previously observed experimentally. It also describes the collisionless heating of plasma due to resonant interaction between the electromagnetic wave and plasma electrons. There is a good overall agreement of the results of our theory with the experimental data on ponderomotive force and plasma heating. Using our theory, we predicted a new effect of reduction of plasma heating compared to the purely collisional value, occurring at low frequencies. This effect has not been previously reported. The nonlinear effects of the electromagnetic field on the electron distribution function and on plasma heating, that are not accounted for in the linear kinetic theory, have been studied using a quasilinear kinetic theory, also developed in this thesis. Within the quasilinear approximation we have formulated the system of equations describing the slow response of plasma electrons to the fast oscillating electromagnetic field. As an example, these equations have been solved in the simplest case of cold plasma with collisions, and the nonlinear perturbation of the electron distribution function and its effect on the plasma heating have been found. It has been shown that the nonlinear modification of plasma heating occurs mainly due to the nonlinear effect of the magnetic component of the electromagnetic field. It has also been shown that at high frequencies the nonlinear effects vanish, and the heating is well described by the linear theory. To verify the predicted new effect of plasma heating reduction at low frequencies, as well as to investigate the nonlinear effect of the magnetic field on plasma heating for arbitrary amplitudes of electromagnetic field in plasma, we have developed a 1d3v Particle-In-Cell (PIC) numerical simulation code with collisions. The collisions were implemented into the PIC code using two different techniques: the direct Monte-Carlo technique for the electron-atom collisions, and the stochastic technique based on the Langevin equation for the electron-electron collisions. The series of numerical simulations by this code confirmed the results of our linear theory, particularly the effect of heating reduction at low frequencies that we predicted theoretically. Also, the nonlinear effects of electromagnetic field on plasma heating were studied using the PIC code in the cases of weak and strong electromagnetic fields. It has been shown that in the case of weak electromagnetic fields (corresponding to weak nonlinearity) the nonlinear effects lead to some enhancement of heating (compared to the linear theory) at low frequencies, followed by a small reduction of heating at higher frequencies. This observed nonlinear perturbation of heating in warm plasma with collisions is similar to that predicted by the quasilinear theory for the case of cold plasma with collisions. In the case of strong electromagnetic fields (corresponding to strong nonlinearity) the nonlinear effects lead to a further reduction of heating (compared to the linear theory) at low frequencies, as shown by the simulation, thus adding to the effect of reduction of heating predicted by the linear theory. The nonlinear effects are shown to vanish at high frequencies, as expected.