Singular behavior of the Dirichlet problem in Hölder spaces of the solutions to the Dirichlet problem in a cone

In the present study we consider the solution of the Dirichlet problem in conical domain. For general elliptic problems in non Hilbertian Sobolev spaces built on $L^{p},1<p<\infty$ the theory of sums of operators developed by Dore-Venni $\left[8\right]$ provides an optimal result. Holder spaces, as opposed to LP spaces, are not UMD. Using the results of Da Prato-Grisvard $\left[6\right]$ and Labbas $\left[14\right]$ we cope with the singular behaviour of the solution in the framework of H$\ddot{\textrm{o}}$lder and little H$\ddot{\textrm{o}}$lder spaces

Semantic Segmentation of Medical Images with Deep Learning: Overview

Semantic segmentation is one of the biggest challenging tasks in computer vision, especially in medical image analysis, it helps to locate and identify pathological structures automatically. It is an active research area. Continuously different techniques are proposed. Recently Deep Learning is the latest technique used intensively to improve the performance in medical image segmentation. For this reason, we present in this non-systematic review a preliminary description about semantic segmentation with deep learning and the most important steps to build a model that deal with this problem

Coins et arrondis en éléments finis - Une approche mathématique des coins et arrondis pour les solutions par éléments finis de l'équation de Laplace

La modélisation par éléments finis d'un objet technique conduit souvent à négliger certains détails de structure. C'est en particulier le cas des arêtes et des coins. Ces points particuliers jouent cependant parfois un rôle physique important : c'est le cas par exemple en électrostatique, en raison des effets de pointe. Il est donc important, après une résolution par éléments finis, de savoir estimer le champ au voisinage de ces singularités, en prenant en compte les rayons de courbure réels. Nous nous intéressons ici aux liens qui existent entre la solution singulière théorique, la solution numérique obtenue par éléments finis avec un angle vif, et celle qui est obtenue avec un maillage décrivant un arrondi. Un estimateur non local du champ sur l'arrondi est proposé. Some geometrical details like exact curvatures near edges and corners are often neglected in finite element meshes. Nevertheless they could greatly change the local solutions and the physical behavior (electric arc, ...). Therefore, it could be useful to be able to estimate the real field values near these singular points, using an adequate post-processing, which has to take into account the real curve radii. This paper presents the links between the theoretical singular solution, the numerical solution with a sharp angle, and the solutions with rounded angle. A non-local estimator for the field on the rounded edges and corners is proposed

PU-NET Deep Learning Architecture for Gliomas Brain Tumor Segmentation in Magnetic Resonance Images

Automatic medical image segmentation is one of the main tasks for many organs and pathology structures delineation. It is also a crucial technique in the posterior clinical examination of brain tumors, like applying radiotherapy or tumor restrictions. Various image segmentation techniques have been proposed and applied to different image types. Recently, it has been shown that the deep learning approach accurately segments images, and its implementation is usually straightforward. In this paper, we proposed a novel approach, called PU-NET, for automatic brain tumor segmentation in multi-modal magnetic resonance images (MRI). We introduced an input processing block to a customized fully convolutional network derived from the U-Net network to handle the multi-modal inputs. We performed experiments over the Brain Tumor Segmentation (BRATS) dataset collected in 2018 and achieved Dice scores of 90.5%, 82.7%, and 80.3% for the whole tumor, tumor core, and enhancing tumor classes, respectively. This study provides promising results compared to the deep learning methods used in this context

Singularités des solutions du problème mêlé, contrôlabilité exacte et stabilisation frontière

On considère dans ce travail l'influence des singularités des solutions du problème mêlé pour le Laplacien sur les questions de contrôlabilité exacte et de stabilisation frontière pour l'équation des ondes. On étudie plus précisément les conditions géométriques nécessaires pour la mise en oeuvre de la méthode d'unicité hilbertienne introduite par J.L. Lions en contrôlabilité exacte et de la méthode initiée par J. Lagnese et développée par exemple par V. Komornik et E. Zuazua pour la stabilisation frontière. On généralise, entre autres, des résultats de P. Grisvard en dimension deux et trois à une dimension N quelconque

The Dirichlet and Neumann problems in Lipschitz and in C1,1\mathscr{C}^{1, 1} domains. Abstract

The main purpose of this paper is to address some questions concerning boundary value problems related to the Laplacian and bi-Laplacian operators, set in the framework of classical $H^s$ Sobolev spaces on a bounded Lipschitz domain of R^N. These questions are not new and a lot of work has been done in this direction by many authors using various techniques since the 80's. If for regular domains almost every thing is elucidated, it is not the case for Lipschitz ones and for $s$ of the form $s = k + 1/2$, with $k$ integer. It is well known that this framework is delicate. Even in these cases many results are well established but sometimes not satisfactory. Several questions remain posed. Our main goal through this work is on one hand to give some improvements to the theory and on another one by using techniques which do not require too intricate calculations. We also tried to obtain maximal regularity for the solutions and as far as we can optimality of the results

Existence of radial positive solutions vanishing at infinity for asymptotically homogeneous systems

In this article we study elliptic systems called asymptotically homogeneous because their nonlinearities may not have polynomial growth. Using the Gidas-Spruck Blow-up method, we obtain a priori estimates, and then using Leray-Schauder topological degree theory, we obtain radial positive solutions vanishing at infinity

Existence and regularity of solutions for Stokes systems with non-smooth boundary data in a polyhedron

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Inégalités de Rellich et de Carleman (Applications à la stabilisation et au contrôle d'équations aux dérivées partielles)

Dans cette thèse, nous présentons quelques méthodes dans le but d étudier la stabilisation et le contrôle d équations aux dérivées partielles. Nous nous concentrons tout d abord sur les inégalités de Rellich propres à l étude de l équation des ondes avec singularités. Nous obtenons ainsi des résultats de décroissance exponentielle ou polynomiale dans le cas où le bord présente une interface entre la partie Dirichlet et la partie Neumann avec éventuellement présence de terme mémoire sur la partie Neumann. Nous nous intéressons ensuite aux inégalités de Carleman afin d étudier la contrôlabilité d un problème parabolique dégénérescent d une part et dans le but d obtenir des estimations spectrales sur le système des ondes avec interface Dirichlet Neumann d autre part. Ceci nous permet d obtenir un résultat intuitif sur le comportement du contrôle du problème parabolique et d espérer pouvoir obtenir une condition suffisante faible de décroissance logarithmique des solutions régulières à l équation des ondes avec interface.In this thesis, we present some methods in order to study the stabilization and control of partial differential equations. We focus first on the Rellich inequality to study the wave equation with singularities. We obtain some results of polynomial or exponential decay in the case where the boundary presents an interface between an homogeneous Dirichlet part and a Neumann part where the feedback is concentrated. We also deal with the presence of an extra term of memory type on the Neumann part. We then focus on Carleman inequalities to study the controllability of a parabolic problem with vanising viscosity on the one hand and in order to obtain some spectral estimates for wave equation with Dirichlet Neumann interface on the other hand. This allows us to obtain an intuitive result on the behavior of our parabolic control problem and to conjecture a weak sufficient condition of logarithmic decay for regular solutions to the wave equation with interface.LYON-Ecole Centrale (690812301) / SudocSudocFranceF