Some properties of the relative rearrangement

AbstractWe develop some new properties of the relative rearrangement. Some of these properties generalize well-known results known as the Hardy-Littlewood inequality

Linear diffusion with singular absorption potential and/or unbounded convective flow: the weighted space approach

In this paper we prove the existence and uniqueness of very weak solutions to linear diffusion equations involving a singular absorption potential and/or an unbounded convective flow on a bounded open set of $\mathbb R^N$. In most of the paper we consider homogeneous Dirichlet boundary conditions but we prove that when the potential function grows faster than the distance to the boundary to the power -2 then no boundary condition is required to get the uniqueness of very weak solutions. This result is new in the literature and must be distinguished from other previous results in which such uniqueness of solutions without any boundary condition was proved for degenerate diffusion operators (which is not our case). Our approach, based on the treatment on some distance to the boundary weighted spaces, uses a suitable regularity of the solution of the associated dual problem which is here established. We also consider the delicate question of the differentiability of the very weak solution and prove that some suitable additional hypothesis on the data is required since otherwise the gradient of the solution may not be integrable on the domain

Characterization of interpolation between Grand, small or classical Lebesgue spaces

In this paper, we show that the interpolation spaces between Grand, small or classical Lebesgue are so called Lorentz-Zygmund spaces or more generally $G\Gamma$-spaces. As a direct consequence of our results any Lorentz-Zygmund space $L^{a,r}({\rm Log}\, L)^\beta$, is an interpolation space in the sense of Peetre between either two Grand Lebesgue spaces or between two small spaces provided that $1<a<\infty, \beta \not= 0$. The method consists in computing the so called K-functional of the interpolation space and in identifying the associated norm

Analyse de quelques équations différentielles à retard et EDP modélisant les instabilités de surfaces

Cette thèse est divisée en deux parties principales : La première partie concerne la déformation plastique d'un matériau contraint. Nous commençons cette partie par une introduction physique sur la dislocation et son rôle dans l'étude de la déformation plastique. Nous exposons ensuite deux types de modélisation de la déformation plastique ce qui nous conduit à deux équations différentielles à retard de Mecking-Lüke-Grilhé. Nous présentons une analyse mathématique complète des deux modèles linéaire et non linéaire. Nous terminons cette partie par des tests numériques et une comparaison des deux modèles. La deuxième partie de la thèse traite l'instabilité de Rayleigh-Plateau. Cette étude porte sur les instabilités de surface d'un pore cylindrique sans contraintes. Nous nous intéressons à une EDP parabolique non linéaire d'ordre quatre, obtenue à partir d'une équation d'évolution des films minces. Le résultat principal est l'existence globale de la solution et la convergence vers la valeur moyenne de la donnée initiale en temps long. L'étude théorique est aussi appuyée comme dans la première partie par une validation numérique.This thesis is divided into two main parts: The first part relates to the plastic deformation of a constrained material. We begin this part by physical introduction on the dislocation and its role in the study of plastic deformation. We also present two types modelling for the plastic deformation, which leads to two delayed differential equations of Mecking-Lücke-Grilhé. We present a complete mathematical analysis of linear and nonlinear models. We conclude this part by numerical tests and a comparison of the two models. The second part of the thesis treats the Rayleigh-Plateau instability. This study focuses on the surface instabilities of a cylindrical pore without constraints. We are interested in a nonlinear parabolic PDE of fourth order, obtained from an evolution equation model of thin films. The main result is the global existence of the solution and the convergence to the average value of the initial data in long time. Numerical validation of the theoretical results is also presented in this part.POITIERS-SCD-Bib. électronique (861949901) / SudocSudocFranceF

Quasilinear P.D.Es, Interpolation spaces and H\"olderian mappings

As in the work of Tartar ( Tartar L. Interpolation non lin\'eaire et r\'egularit\'e, 9, Journal of Functional Analysis, (1972), 469-489) we developed here some new results on non linear interpolation of $\alpha$-H\"olderian mappings between normed spaces, namely, by studying the action of the mappings on $K$-functionals and between interpolation spaces with logarithm functors. We apply those results to obtain regularity results on the gradient of the solution to quasilinear equations of the form $$-div(\widehat a(\nabla u ))+V(u)=f,$$ whenever $V$ is a nonlinear potential, $f$ belongs to non standard spaces as Lorentz-Zygmund spaces. We show among other that the mapping $T: \ Tf=\nabla u$ is locally or globally $\alpha$-H\"olderian under suitable values of $\alpha$ and adequate hypothesis on $V$ and $\widehat a.

An elliptic-parabolic equation with a nonlocal term for the transient regime of a plasma in a Stellarator

We prove the existence and the regularity of weak solutions of a nonlocal elliptic–parabolic free-boundary problem involving the notions of relative rearrangement and monotone rearrangement. The problem arises in the study of the dynamics of a magnetically confined fusion plasma in a Stellarator device when the dimensional analysis on the characteristic times suggests to neglect the inertial acceleration in presence of a time dependent magnetic field

Profil De Résistance Des Mycobabcterium Tuberculosis Des Malades En Retraitement Dans La Région De Haute Matsiatra, Madagascar

Introduction: Relapse, failure and defaulted treatment are a risk factor for Mycobacterium tuberculosis resistance to anti-tuberculosis. The objective of our study is to determine the resistance profile of Mycobacterium tuberculosis of patients in retreatment at the Haute Matriatra area. Method: This is a retrospective, descriptive study carried out on the basis of the data contained in the register of the Haute Matsiatra Regional Tuberculosis Laboratory of the University Hospital of Fianarantsoa, from May 2014 to December 2016 (31 months). We included patients with retreatment in the study. Results: We found 138 patients in retreatment. The average age was 39.32 years with a sex ratio of 2.11. Resistance to Rifampicin and Isoniazid was respectively 2.17% and 2.82%, The prevalence of multidrug-resistant tuberculosis was 0.72%. Conclusion: Tuberculosi resistance, monoresistance or multidrug resistance is a reality in the region of Haute Matsiatra with a prevalence that is still low, reflecting the effectiveness of the tuberculosis control program. However, monoresistances require special attention and monitoring to avoid the emergence of multidrug resistant strains

Facteurs De Retard Diagnostique De La Tuberculose Pulmonaire Vus À l’Unité De Soins, De Formation Et De Recherche De Pneumologie Befelatanana

Delay in diagnosis is significant to tuberculosis prognosis. The aim of this study is to evaluate the diagnosis delay of tuberculosis and to identify determinants of “patient delay”, “health system delay”, and “total delay” in the diagnosis of tuberculosis. It is a prospective study of the patients who were hospitalized at the Unité de Soins de Formation et de Recherche (USFR) of Pneumology in Befelatanana, Antananarivo Madagascar, during the period of 1 st October 2014 to 30 April 2015 (7 months). We included all patients with diagnosis of pulmonary tuberculosis. Sixty six patients were also included. The mean time of patient delay, health system delay, and total delay were 26,30± 36,87, 69,56±64, and 96,35±72,65 days respectively. The different variables that affected diagnosis delay were: tobacco smoke (OR : 3,6723), asthenia (OR : 5,4815), anorexia (OR : 2,9524), and hemoptysis (OR : 0,2406) for the total delay. Knowledge about tuberculosis signs (OR : 0,164) and transmissions (OR : 0,243) was for the patient delay. Hemoptysis (OR : 8,1250), asthenia (OR : 0,1081), breathlessness (OR : 0,3556), infiltrative syndrome (OR : 0,2500), and alveolar syndrome (OR : 0,0687) in chest Xray was for the health system delay. Having an understanding of these factors of tuberculosis diagnosis delay will result to a decrease in the diagnosis delay