Problèmes aux valeurs propres non linéaires dans les inéquations variaionnelles : Etude locale

ON S'INTERESSE A UNE CLASSE D'INEQUATIONS VARIATIONNELLES ELLIPTIQUES, ASSOCIEES A UN PROBLEME D'OBSTACLE ET DEPENDANT D'UN PARAMETRE LAMBDA : A(A,V-U) >OU= SOM::(OMEGA )LAMBDA F(U)(V-U)DX POUR TOUT V APPARTIENT A K, U APPARTIENT A K=(W APPARTIENT A H::(0)**(1)(OMEGA )/W OU= PSI P.P. SUR OMEGA ). UNE TELLE INEQUATION ADMET AU MOINS UNE BRANCHE DE SOLUTIONS EQUATION ET UNE BRANCHE DE SOLUTIONS INEQUATIONS. ON CHERCHE A CONNAITRE LA STRUCTURE LOCALE DES BRANCHES INEQUATIONS. GRACE A UN PROCESSUS DE LINEARISATION "CONIQUE" ON CLASSE LES POINTS DE LA BRANCHE INEQUATION EN POINTS REGULIERS OU SINGULIERS. AU VOISINAGE D'UN POINT REGULIER, ON MONTRE QUE LES SOLUTIONS ADMETTENT UN DEVELOPPEMENT SELON LAMBDA . PUIS ON ETUDIE LE COMPORTEMENT LOCAL DES SOLUTIONS AU VOISINAGE D'UN POINT SINGULIER VERIFIANT CERTAINES HYPOTHESES, QUI SONT A RAPPROCHER DE CELLES FAITES POUR LES EQUATIONS ET QUI ASSURENT QU'ON A DU POINT DE RETOURNEMENT. ON MONTRE QUE, DANS CERTAINS CAS, IL EXISTE UN TEL POINT SINGULIER, SUR LA BRANCHE DE SOLUTIONS MAXIMALES. LES BRANCHES EQUATIONS ET INEQUATIONS SONT RELIEES PAR UN POINT APPELE POINT DE TRANSITION. ON FAIT UNE ETUDE PLUS FINE AU VOISINAGE DE CE POINT. UNE ETUDE DES DIVERSES CONDITIONS INTRODUITES POUR L'ETUDE LOCALE, MONTRE QU'ELLES SONT FORTEMENT LIEES A UNE CONDITION TYPE STABILITE. ILLUSTRATION NUMERIQUE SUR QUELQUES PROBLEMES D'OBSTACLENo abstrac

Well-Posedness of a Dissolution-Growth Problem on an Unbounded Domain

: We consider a spherical grain which may be growing by accretion or dissolving in a dilute solution of the same substance where one also has reaction and diffusion. The resulting free boundary problem is shown to be well-posed and additional regularity is obtained. Key Words: Stefan Problem with surface tension, unbounded domain, Free Boundary Problem, radial geometry, system, parabolic, partial differential equation, reaction/diffusion, well-posedness, regularity. 1. Introduction In this paper we study a simple model of a solid/liquid interface which forms a free boundary for the problem. This model is composed of a solid phase of a single substance and an incompressible liquid phase which is a dilute solution of the same substance involving both diffusion and possible chemical reaction. One interpretation of this model is the corrosion of a material in water with both dissolution of the material into the water and accretion of dissolved material to the solid possible so the inter..

A non-smooth version of the Lojasiewicz–Simon theorem with applications to non-local phase-field systems

AbstractWe consider a simple system modelling phase transition phenomena with long term interactions. It is shown that any solution converges with growing time to a single stationary state. To this end, a non-smooth version of the celebrated Simon-Lojasiewicz theorem is proved

Long time convergence for a class of variational phase field models

In this paper we analyze a class of phase field models for the dynamics of phase transitions which extend the well-known Caginalp and Penrose-Fife models. Existence and uniqueness of the solution to the related initial boundary value problem are shown. Further regularity of the solution is deduced by exploiting the so-called regularizing effect. Then, the large time behavior of such a solution is studied and several convergence properties of the trajectory as time tends to infinity are discussed