Дисертації з теми "Problèmes Elliptique"
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Benmlih, Khalid. "Étude qualitative de certains problèmes semi-linéaires elliptiques." Nancy 1, 1994. http://www.theses.fr/1994NAN10075.
Maris, Mihai. "Sur quelques problèmes elliptiques non-linéaires." Paris 11, 2001. http://www.theses.fr/2001PA112247.
In this thesis we study particular solutions for some nonlinear dispersive partial differential equations which appear in physics, such the nonlinear Schrödinger equation, the Benney-Luke equation or the Benjamin-Ono equation. We are particularly interested in the stationary waves and in the travelling waves of these equations. This gives nonlinear elliptic problems in the whole space. Solitary and travelling waves for the considered equations have been observed in experiments and in numerical simulations. In some cases, these solutions seem to play an important role in the general dynamics of the corresponding evolution equations. In the first chapter we prove the analyticity and we find the optimal algebraic decay rate at infinity of solitary waves to the Benney-Luke equation and to the generalized Benjamin-Ono equation. The second chapter is devoted to the proof of existence of stationary solutions for a nonlinear Schrödinger equation with potential in one dimension which describes the flow of a fluid past an obstacle. .
Radulescu, Vicentiu. "Analyse de quelques problèmes aux limites elliptiques non linéaires." Habilitation à diriger des recherches, Université Pierre et Marie Curie - Paris VI, 2003. http://tel.archives-ouvertes.fr/tel-00980823.
Haial, Abdelillah El. "Problèmes aux limites pour une équation différentielle abstraite complète du second ordre de type elliptique." Le Havre, 1999. http://www.theses.fr/1999LEHA0002.
Raimondi, Federica. "Problèmes elliptiques singuliers dans des domaines perforés et à deux composants." Thesis, Normandie, 2018. http://www.theses.fr/2018NORMR093/document.
This thesis is mainly devoted to the study of some singular elliptic problems posed in perforated domains. Denoting by Ωɛ* e domain perforated by ɛ-periodic holes of ɛ-size, we prove existence and uniqueness of the solution , for fixed ɛ, as well as homogenization and correctors results for the following singular problem :{█(-div (A (x/ɛ,uɛ)∇uɛ)=fζ(uɛ) dans Ωɛ*@uɛ=0 sur Γɛ0@@(A (x/ɛ,uɛ)∇uɛ)υ+ɛγρ (x/ɛ) h(uɛ)= ɛg (x/ɛ) sur Γɛ1@)┤Where homogeneous Dirichlet and nonlinear Robin conditions are prescribed on the exterior boundary Γɛ0 and on the boundary of the holles Γɛ1, respectively. The quasilinear matrix field A is elliptic, bounded, periodic in the first variable and Carathéodory. The nonlinear singular lower order ter mis the product of a continuous function ζ (singular in zero) and f whose summability depends on the growth of ζ near its singularity. The nonlinear boundary term h is a C1 increasing function, ρ and g are periodic nonnegative functions with prescribed summabilities. To investigate the asymptotic behaviour of the problem, as ɛ -> 0, we apply the Periodic Unfolding Method by D. Cioranescu-A. Damlamian-G. Griso, adapted to perforated domains by D. Cioranescu-A. Damlamian-P. Donato-G. Griso-R. Zaki. Finally, we show existence and uniqueness of a weak solution of the same equation in a two-component domain Ω = Ω1 υ Ω2 υ Γ, being Γ the interface between the connected component Ω1 and the inclusions Ω2. More precisely we consider{█(-div (A(x, u)∇u)+ λu=fζ(u) dans Ω\Γ,@u=0 sur δΩ@(A(x, u1)∇u1)υ1= (A(x, u2)∇u2)υ1 sur Γ,@(A(x, u1)∇u1)υ1= -h(u1-u2) sur Γ@)┤Where ν1 is the unit external vector to Ω1 and λ a nonnegative real number. Here h represents the proportionality coefficient between the continuous heat flux and the jump of the solution and it is assumed to be bounded and nonnegative on Γ
Moutazaim, Fathallah. "EEtude de quelques problèmes inverses : parabolique et elliptique, à partir de données sur le bord d'un domaine borné." Compiègne, 1999. http://www.theses.fr/1999COMP1207.
Radulescu, Vicentiu. "Analyse de quelques problèmes liés à l'équation de Ginzburg-Landau." Phd thesis, Université Pierre et Marie Curie - Paris VI, 1995. http://tel.archives-ouvertes.fr/tel-00980811.
Vivier, Laurent. "Deux problèmes d'analyse non linéaire : comportement au bord des solutions d'une équation elliptique et approximation de mouvements de front." Tours, 1998. http://www.theses.fr/1998TOUR4015.
Mokrane, Abdelhafid. "Existence de solutions pour certains problèmes quasi linéaires elliptiques et paraboliques." Paris 6, 1986. http://www.theses.fr/1986PA066086.
Sha, Min. "Problèmes autour de courbes élliptiques et modulaires." Phd thesis, Université Sciences et Technologies - Bordeaux I, 2013. http://tel.archives-ouvertes.fr/tel-00879227.
Debussche, Arnaud. "Quelques problèmes concernant le comportement pour les grands temps des équations d'évolution dissipatives." Paris 11, 1989. http://www.theses.fr/1989PA112318.
In this work, we consider the long time behaviour of dissipative evolution equations. More precisely we study the existence of attracting sets such as attactors and inertial manifolds. In the first part, we describe a general method to construct inertial manifolds for a nonlinear parabolic equation. We obtain an existence theorem under the same type of assumptions as the methods that already exist. Our method is based on the resolution of a hyperbolic partial differentiai equation (the Sacker's equation) such that the graph of its solution is a positively invariant manifold. The second part is devoted to the existence of approximate inertial manifolds. These are substitute to inertial manifolds when their existence is not known. We prove in two cases (the reaction diffusion equation and the Cahn-Hilliard equation) the existence of an infinite family of approximate inertial manifolds with increasing order of approximation. Our method is general and can be applied to other equations. Finally, in the third part, we study a singular perturbation of the Cahn-Hilliard equation in space dimension one obtained by adding a second order derivative intime whose coefficient E is small. We prove the existence of attractors for the perturbed equation. Moreover, the Haussdorf semi distance from these attractors to the attractor of the unperturbed equation converges to zero when E goes to zero
Chouikha, Raouf. "Aspects des fonctions elliptiques. \\ Solutions périodiques d'équations différentielles.\\ Métriques pseudo-cylindriques. \\ Problèmes isopérimètriques plans." Habilitation à diriger des recherches, Université de Rouen, 2003. http://tel.archives-ouvertes.fr/tel-00003633.
Dardé, Jérémi. "Méthodes de quasi-réversibilité et de lignes de niveau appliquées aux problèmes inverses elliptiques." Phd thesis, Université Paris-Diderot - Paris VII, 2010. http://tel.archives-ouvertes.fr/tel-00551853.
Belhamiti, Omar. "Étude dans les espaces de Hölder de problèmes aux limites et de transmission dans un domaine avec couche mince." Phd thesis, Université du Havre, 2008. http://tel.archives-ouvertes.fr/tel-00330880.
Karami, Fahd. "Limite singulière de quelques problèmes de Réaction Diffusion: Analyse mathématique et numérique." Phd thesis, Université de Picardie Jules Verne, 2007. http://tel.archives-ouvertes.fr/tel-00180724.
Dousteyssier, Buvat Hélène. "Sur des techniques déterministes et stochastiques appliquées aux problèmes d'identification." Université Joseph Fourier (Grenoble), 1995. http://tel.archives-ouvertes.fr/tel-00346058.
Robbiano, Luc. "Unicité du problème de Cauchy : ensembles nodaux : régularité des problèmes elliptiques." Paris 11, 1990. http://www.theses.fr/1990PA112344.
Ponce, Augusto. "Quelques problèmes elliptiques avec singularités." Paris 6, 2004. https://tel.archives-ouvertes.fr/tel-00009043.
Meisner, Maëlis. "Étude unifiée d'équations aux dérivées partielles de type elliptique régies par des équations différentielles à coefficients opérateurs dans un cadre non commutatif : applications concrètes dans les espaces de Hölder et les espaces Lp." Phd thesis, Université du Havre, 2012. http://tel.archives-ouvertes.fr/tel-00712008.
Berdan, Nada El. "Régularité de problèmes à données dans les espaces pondérés par la distance au bord via l'inégalité uniforme de Hopf et le principe de dualité." Thesis, Poitiers, 2016. http://www.theses.fr/2016POIT2303/document.
We discuss the existence and non existence of the so called Hopf uniform Inequality (variant of a maximum principle) for the linear equation Lv = f with measurable coefficients and under the homogeneous Dirichlet Boundary condition. Then we apply such inequality to prove the W1;p 0 -regularity of a semi linear problem Lu = F(u), singular at u = 0, with the coefficients of the main operator of L in the space of vanishing mean oscillation. Moreover, when those coefficients are Lipschitz, we show that the gradient of the solution is at most in the space of bounded mean oscillation : bmor. In the last part of this thesis, we are concerned with the linear easticity system (Stationnary equation of the waves elasticity). But, here the second terms varies with respect to the distance function until the boundary.Using the duality method, we study the regularity of the solution of the elasticity system for the data belonging to various weighted spaces
Amrani, Mohamed. "Etude de la solution approchée de problèmes quasilinéaires et analyse d'un problème en théorie du signal." Metz, 1995. http://docnum.univ-lorraine.fr/public/UPV-M/Theses/1995/Amrani.Mohamed.SMZ9535.pdf.
In the first part of this work, we complete the results obtained by N. André and M. Chipot about the approximate solution of some class of elliptic problems, by using the simplest finite element method. In particular, in two dimensional case, we present a new technique to prove uniqueness for the approximate solution when the mesh size approaches zero, under optimal assumptions about the angles of the triangulation. Moreover, we studied the discrete maximum principle, the convergence, the regularity and the uniqueness in higher dimension. In the second part, we are interested in an industrial problem suggested by Landis & Gyr energy management. This problem consist of finding a numerical technique to measure the energy of an electrical signal
Winckler, Bruno. "Intersection arithmétique et problème de Lehmer elliptique." Thesis, Bordeaux, 2015. http://www.theses.fr/2015BORD0233/document.
In this thesis we consider the problem of lower bounds for the canonical height onelliptic curves, aiming for the conjecture of Lehmer. Our main diophantine result isan explicit version of a theorem of Laurent (who proved this conjecture for ellipticcurves with CM up to a " exponent) using arithmetic intersection, enlightening thedependence with parameters linked to the elliptic curve ; such a result can be motivatedby the conjecture of Lang, hoping for a lower bound proportional to, roughly,the Faltings height of the curve.Nevertheless, our dissertation begins with a part dedicated to a completely explicitversion of the density theorem of Chebotarev, along the lines of a previous workdue to Lagarias and Odlyzko, which will be crucial to investigate the elliptic Lehmerproblem. We also obtain upper bounds for Siegel zeros, and for the smallest primeideal whose Frobenius is in a fixed conjugacy class
LOHEAC, Jean-Pierre. "Problèmes elliptiques à données peu régulières, applications." Habilitation à diriger des recherches, Université Claude Bernard - Lyon I, 2002. http://tel.archives-ouvertes.fr/tel-00002062.
de recherche.
Le premier concerne la stabilisation-frontière de quelques systèmes
distribués, en présence de singularités. On s'intéresse principalement à l'équation des ondes et au système élastodynamique pour lesquels de nombreux auteurs ont obtenu des résultats de stabilisation en utilisant la méthode des multiplicateurs sous des conditions géométriques restrictives. Pour étendre ces résultats, on est amené à démontrer certaines propriétés de ``régularité cachée'' des solutions fortes, ce qui nécessite l'analyse des singularités d'un problème elliptique avec conditions aux limites mêlées. La connaissance de ces singularités permet de généraliser une relation de Rellich, cruciale dans l'obtentionédes estimations d'énergie conduisant aux résultats de stabilisation.
Le second thème a pour objet l'étude des écoulements de Hele-Shaw à
source ponctuelle. Le modèle de Stokes-Leibenson fait apparaître
une équation elliptique dont le second membre est la distribution de Dirac au point-source. Ce problème est de plus intrinsèquement non linéaire du fait que le domaine lui-même évolue d'une manière inconnue. On utilise la méthode de Helmholtz-Kirchhoff pour reformuler le problème. Ceci permet de démontrer un résultat d'existence et d'unicité locales d'une solution classique. On construit ensuite un modèle numérique, dit ``modèle quasi-contour'', destiné à étudier certaines propriétés qualitatives de ces écoulements.
Saoudi, Kamel. "Etude de quelques problèmes quasilinéaires elliptiques singuliers." Toulouse 1, 2009. https://tel.archives-ouvertes.fr/tel-00412365v2.
This thesis concerns the study od some singular elliptic problems. Precisely, in Chapter 2, we investigate the question of multiplicity of solutions for a singular problem with critical growth in dimension N = 3. In Chapter 3, we investigate the validity of C1 versus W0 1;p energy minimisers for a quasilinear elliptic singular problem. In Chapter 4, we present global bifurcation results for a semilinear elliptic singular problem with critical growth in dimension 2 with exponentiel growth
Smith, Graham. "Problèmes elliptiques pour des sous-variétés riemanniennes." Paris 11, 2004. http://www.theses.fr/2004PA112191.
The first part of the thesis treats special legendrian submanifolds which are positive in a certain sense. We obtain a compactness result and we study certain degenerate forms which appear. The second part treats the plateau problem for convex hypersurfaces of constant gaussian curvature immerged into three dimensional hyperbolic space. We show the existence of solutions and their continuous dependance on initial conditions for the case of conformally hyperbolic surfaces. The third part continues the work of the second part by treating the geometric structure of solutions which are conformally equivalent to pointed compact riemann surfaces. We show that such solutions are cylindrical near to critical points. The fourth part treats representations of compact fuchsian groups in compact kleinian groups. We show that if such a representation is non-elementary, and if its second stiefel-whitney class vanishes, then it has a realisation by a convex immersion of a compact surface into a three dimensional manifold
Stahlhut, Sebastian. "Problèmes aux limites pour les systèmes elliptiques." Thesis, Paris 11, 2014. http://www.theses.fr/2014PA112186.
In this this thesis we study boundary value problems for elliptic systems in divergence form with complex coefficients in L^{\infty}. We prove a priori estimates, discuss solvability and extrapolation of solvability. We use a transformation to generalized Cauchy-Riemann equations due to P. Auscher, A. Axelsson, and A. McIntosh. The generalized Cauchy-Riemann equations can be solved by the semi-group generated by a perturbed first order Dirac/differential operator. In relation to semi-group theory we setup the L^p theory by a discussion of bisectoriality, holomorphic functional calculus and off-diagonal estimates for operators in the functional calculus. In particular, we develop an L^p-L^q theory for operators in the functional calculus of the first order perturbed Dirac/differential operators. The formulation of Neumann, Regularity and Dirichlet problems involve square function estimates and nontangential maximal function estimates. This leads us to discuss square function estimates and nontangential maximal function estimates involving operators in the functional calculus of the perturbed first order Dirac/differential operator. We discuss the related Hardy spaces associated to operators and prove identifications by subspaces of classical Hardy and Lebesgue spaces. We obtain the a priori estimates by an extension of the square function estimates and nontangential maximal function estimates to Sobolev spaces associated to operators. We use the a priori estimates for a discussion of solvability and extrapolation of solvability
Sbihi, Karima. "Etude de quelques E.D.P. non linéaires dans L^1 avec des conditions générales sur le bord." Phd thesis, Université Louis Pasteur - Strasbourg I, 2006. http://tel.archives-ouvertes.fr/tel-00110417.
Moussaoui, Mimoun. "Questions d'existence dans les problèmes semi-linéaires elliptiques." Doctoral thesis, Universite Libre de Bruxelles, 1991. http://hdl.handle.net/2013/ULB-DIPOT:oai:dipot.ulb.ac.be:2013/213020.
Belahdji, Kheira. "Problèmes elliptiques dans des domaines à points cuspides." Ecully, Ecole centrale de Lyon, 1996. http://www.theses.fr/1996ECDL0005.
Megrez, Nasreddine. "Étude de certains problèmes elliptiques et sous elliptiques nonlinéaires sur des domaines non bornés." Toulouse 1, 2003. http://www.theses.fr/2003TOU10064.
This thesis is devoted to the study of some nonlinear elliptic and subelliptic problems on unbounded domains. Using variational methods, we investigate the existence of weak solutions for an elliptic problem involving the p-Laplacian operator defined on an unbounded domain of Rn. After this, and using also varational methods, we prove the existence of weak solutions for a subelliptic system involving the Heisenberg Laplacian on unbounded domains of the Heisenberg group Hn. Finally, using Rabinowitz's bifurcation theory, we prove the existence of bounded continuums of positive solutions for a semilinear elliptic problem defined on Rn with an indefinite nonlinearity
Ben-Kiran, Taoufiq. "Étude d'un problème de perturbation singulière elliptique non classique." Nancy 1, 1989. http://www.theses.fr/1989NAN10040.
Torne, Olaf. "Symétrie et brisure de symétrie dans quelques problèmes elliptiques." Doctoral thesis, Universite Libre de Bruxelles, 2004. http://hdl.handle.net/2013/ULB-DIPOT:oai:dipot.ulb.ac.be:2013/211121.
Wang, Chao. "Analyse de quelques problèmes elliptiques et paraboliques semi-linéaires." Phd thesis, Université de Cergy Pontoise, 2012. http://tel.archives-ouvertes.fr/tel-00809045.
Maghnouji, Abderrahman. "Problèmes elliptiques et paraboliques dans des domaines non-réguliers." Lille 1, 1992. http://www.theses.fr/1992LIL10161.
Bouhsiss, Fouzia. "Quelques résultats d'unicité pour des problèmes elliptiques et paraboliques." Besançon, 2001. http://www.theses.fr/2001BESA2048.
Redwane, Hicham. "Solutions normalisées de problèmes paraboliques et elliptiques non linéaires." Rouen, 1997. http://www.theses.fr/1997ROUES059.
Cohen, Laurent David. "Etude de quelques problèmes semi-linéaires paraboliques et elliptiques." Paris 6, 1986. http://www.theses.fr/1986PA066503.
Drogoul, Audric. "Méthode du gradient topologique pour la détection de contours et de structures fines en imagerie." Thesis, Nice, 2014. http://www.theses.fr/2014NICE4063/document.
This thesis deals with the topological gradient method applied in imaging. Particularly, we are interested in object detection. Objects can be assimilated either to edges if the intensity across the structure has a jump, or to fine structures (filaments and points in 2D) if there is no jump of intensity across the structure. We generalize the topological gradient method already used in edge detection for images contaminated by Gaussian noise, to quasi-linear models adapted to Poissonian or speckled images possibly blurred. As a by-product, a restoration model based on an anisotropic diffusion using the topological gradient is presented. We also present a model based on an elliptical linear PDE using an anisotropic differential operator preserving edges. After that, we study a variational model based on the topological gradient to detect fine structures. It consists in the study of the topological sensitivity of a cost function involving second order derivatives of a regularized version of the image solution of a PDE of Kirchhoff type. We compute the topological gradients associated to perforated and cracked 2D domains and to cracked 3D domains. Many applications performed on 2D and 3D blurred and Gaussian noisy images, show the robustness and the fastness of the method. An anisotropic restoration model preserving filaments in 2D is also given. Finally, we generalize our approach by the study of the topological sensitivity of a cost function involving the m − th derivatives of a regularization of the image solution of a 2m order PDE
Bruyère, Nicolas. "Comportement asymptotique de problèmes posés dans les cylindres. Problèmes d’unicité pour les systèmes Boussinesq." Rouen, 2007. http://www.theses.fr/2007ROUES032.
The thesis is divided in two independent parts. In the first part, we investigate the asymptotic behaviour of elliptic and parabolic problems with L1 + W 1,p’ data (respectively with L1+ Lp (0, T ; W-1,p’) data in the parabolic case), in domaine becoming unbounded. Using the framework of renormalized solutions and the regularity results of the solutions for such data, we prove, under structural conditions on space variables, convergence results in spaces containing the solutions. In the second part, in the 2-dimensional case, we study Boussinesq type systems. These systems derive from fluid mechanics models and couple incompressible Navier-Stokes equations and heat equation. We focus our attention on studying the uniqueness of the solution, which is intricate because of the very nonlinear coupling of the equations. We consider weak solutions for the Navier-Stokes equations and renormalized solutions are used for the heat equation. We state regularity results for these equations and then we prove few existence and uniuqueness results of the solution of the system for small data
Obeid-El, Hamidi Amira. "Sur une équation elliptique non linéaire dégénérée." Phd thesis, Université de Pau et des Pays de l'Adour, 2002. http://tel.archives-ouvertes.fr/tel-00002263.
Brada, Alain. "Comportement asymptotique de solutions d'équations elliptiques semi-linéaires dans un cylindre." Tours, 1987. http://www.theses.fr/1987TOUR4010.
Benkler, Yochai. "Potentiel de Riesz et problèmes elliptiques dans les espaces d'Orlicz." Doctoral thesis, Universite Libre de Bruxelles, 1988. http://hdl.handle.net/2013/ULB-DIPOT:oai:dipot.ulb.ac.be:2013/213349.
Ghergu, Marius. "Problèmes avec singularités sur la frontière pour des équations elliptiques." Chambéry, 2006. http://www.theses.fr/2006CHAMS016.
This work concerns the study of elliptic problems with singularities at the boundary. The first part deals with blow-op solutions for semilinear elliptic problems with gradient term. In this sense we establish some existence and nonexistence results for this kind of problems and for the associated elliptic systems. These results are obtained in the absence of the Keller-Osserman condition and assuming that the nonlinearities have a sublinear growth at infinity. We also point out the role played by the gradient term in the existence of a blow-up solution. The second part of the thesis concerns semilinear eliptic problems with singular nonlinearities. We are interested in existence, uniqueness and bifurcation with respect to the parameters. In the presence of asymptotically linear terms we establish a blow-up result for the solution around the bifurcation parameter. In the last chapter of this part we analyse the influence of the subquadratic gradient term. The proofs relies on the sub and super-solution method, combined with different techniques for singular elliptic equations. In the third part of this work we emphasize the collective behavior of a multi-building system subjected to time dependent impacts representing earthquakes, collisions and explosions. The approach is based on the spectral analysis of the problem combined with integral methods. The difficulty consists in the presence of the singularities at the end of the buildings foundations. An asymptotic study on the first frequency with respect to the number of the buildings is also presented
Vasseur, Baptiste. "Étude de problèmes différentiels elliptiques et paraboliques sur un graphe." Thesis, Littoral, 2014. http://www.theses.fr/2014DUNK0400/document.
After a quick presentation of usual notations for the graph theory, we study the set of harmonic functions on graphs, that is, the functions whose laplacian is zero. These functions form a vectorial space. On a uniformly locally finite tree, we shaw that this space has dimension one or infinity. When the graph has an infinite number of cycles, this result change and we describe some examples showing that there exists a graph on which the harmonic functions form a vectorial space of dimension n, for all n. We also treat the case of a particular periodic graph. Then, we study more precisely the eigenvalues of infinite dimension. In this case, the eigenspace contains a subspace isomorphic to the set of bounded sequences. An inequality concerning the spectral is given when edges length is equal to one. Examples show that these inclusions are optimal. We also study the asymptotic behavior of eigenvalues for elliptic operators under dynamical Kirchhoff node conditions. We write the problem as a Sturm-Liouville operator and we transform it in a matrix problem. Then we find a characteristic equation whose zeroes correspond to eigenvalues. We deduce a formula for the asymptotic behavior. In the last chapter, we study the stability of stationary solutions for some reaction-diffusion problem whose the non-linear term is polynomial
Cisternino, Marco. "A parallel second order Cartesian method for elliptic interface problems and its application to tumor growth model." Phd thesis, Université Sciences et Technologies - Bordeaux I, 2012. http://tel.archives-ouvertes.fr/tel-00690743.
Devillanova, Giuseppe. "Structures singulières de quelques problèmes variationnels." Phd thesis, École normale supérieure de Cachan - ENS Cachan, 2005. http://tel.archives-ouvertes.fr/tel-00132680.
El, Hachimi Abderrahmane. "Etude de quelques problèmes elliptiques et paraboliques liés au p-Laplacien." Doctoral thesis, Universite Libre de Bruxelles, 1993. http://hdl.handle.net/2013/ULB-DIPOT:oai:dipot.ulb.ac.be:2013/212842.
Bensedik, Ahmed. "Sur quelques problèmes elliptiques de type Kirchhoff et dynamique des fluides." Phd thesis, Université Jean Monnet - Saint-Etienne, 2012. http://tel.archives-ouvertes.fr/tel-00971279.
Sauvy, Paul. "Étude de quelques problèmes elliptiques et paraboliques quasi-linéaires avec singularités." Thesis, Pau, 2012. http://www.theses.fr/2012PAUU3020/document.
This thesis deals with the mathematical field of nonlinear partial differential equations analysis. More precisely, we focus on quasilinear and singular problems. By singularity, we mean that the problems that we have considered involve a nonlinearity in the equation which blows-up near the boundary. This singular pattern gives rise to a lack of regularity and compactness that prevent the straightforward applications of classical methods in nonlinear analysis used for proving existence of solutions and for establishing the regularity properties and the asymptotic behavior of the solutions. To overcome this difficulty, we establish estimations on the precise behavior of the solutions near the boundary combining several techniques : monotonicity method (related to the maximum principle), variational method, convexity arguments, fixed point methods and semi-discretization in time. Throughout the study of three problems involving the p-Laplacian operator, we show how to apply this different methods. The three chapters of this dissertation the describes results we get :– In Chapter I, we study a singular elliptic absorption problem. By using sub- and super-solutions and variational methods, we prove the existence of the solutions. In the case of a strong singularity, by using local comparison techniques, we also prove that the compact support of the solution. In Chapter II, we study a singular elliptic system. By using fixed point and monotonicity arguments, we establish two general theorems on the existence of solution. In a second time, we more precisely analyse the Gierer-Meinhardt systems which model some biological phenomena. We prove some results about the uniqueness and the precise behavior of the solutions. In Chapter III, we study a singular parabolic absorption problem. By using a semi-discretization in time method, we establish the existence of a solution. Moreover, by using differential energy inequalities, we prove that the solution vanishes in finite time. This phenomenon is called "quenching"
Djellit, Ali. "Valeurs propres de problèmes elliptiques indéfinis sur des ouverts non bornés." Toulouse 3, 1992. http://www.theses.fr/1992TOU30072.