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The obstacle problem is a classic motivati … The obstacle problem is a classic motivating example in the mathematical study of variational inequalities and free boundary problems. The problem is to find the equilibrium position of an elastic membrane whose boundary is held fixed, and which is constrained to lie above a given obstacle. It is deeply related to the study of minimal surfaces and the capacity of a set in potential theory as well. Applications include the study of fluid filtration in porous media, constrained heating, elasto-plasticity, optimal control, and financial mathematics. The mathematical formulation of the problem is to seek minimizers of the Dirichlet energy functional, in some domain where the functions represent the vertical displacement of the membrane. In addition to satisfying Dirichlet boundary conditions corresponding to the fixed boundary of the membrane, the functions are in addition constrained to be greater than some given obstacle function . The solution breaks down into a region where the solution is equal to the obstacle function, known as the contact set, and a region where the solution is above the obstacle. The interface between the two regions is the free boundary. In general, the solution is continuous and possesses Lipschitz continuous first derivatives, but that the solution is generally discontinuous in the second derivatives across the free boundary. The free boundary is characterized as a Hölder continuous surface except at certain singular points, which reside on a smooth manifold.points, which reside on a smooth manifold.
, Le problème de l'obstacle est un exemple c … Le problème de l'obstacle est un exemple classique de motivation de l'étude mathématique des inégalités variationnelles et des problèmes à frontière libre. Il consiste à trouver la position d'équilibre d'une membrane élastique dont les bords sont fixes et devant passer au-dessus d'un obstacle donné. Il est profondément lié à l'étude des surfaces minimales et de la capacité d'un ensemble en théorie du potentiel. Les applications s'étendent à l'étude de la filtration d'un fluide en milieu poreux, au chauffage contraint, l'élastoplasticité, le contrôle optimal et les mathématiques financières. Mathématiquement, le problème se voit comme la minimisation de l'énergie de Dirichlet, sur un domaine D où les fonctions u décrivent le déplacement vertical de la membrane. Les solutions doivent satisfaire des conditions au bord de Dirichlet et être supérieures à une fonction obstacle χ(x). La solution se distingue sur deux parties : une où elle est égale à l'obstacle (ensemble de contact) et une où elle y est strictement supérieure. L'interface entre les régions est appelée frontière libre. En général, la solution est continue et sa dérivée est lipschitzienne, mais sa dérivée seconde est généralement discontinue à la frontière libre. La frontière libre est caractérisée comme une surface continue au sens de Hölder sauf en des points singuliers, situés sur une variété lisse. singuliers, situés sur une variété lisse.
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Qualche tempo dopo Stampacchia, partendo sempre dalla sua disequazione variazionale, aperse un nuovo campo di ricerche che si rivelò importante e fecondo. Si tratta di quello che oggi è chiamato il problema dell'ostacolo.
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Le problème de l'obstacle est un exemple c … Le problème de l'obstacle est un exemple classique de motivation de l'étude mathématique des inégalités variationnelles et des problèmes à frontière libre. Il consiste à trouver la position d'équilibre d'une membrane élastique dont les bords sont fixes et devant passer au-dessus d'un obstacle donné. Il est profondément lié à l'étude des surfaces minimales et de la capacité d'un ensemble en théorie du potentiel. Les applications s'étendent à l'étude de la filtration d'un fluide en milieu poreux, au chauffage contraint, l'élastoplasticité, le contrôle optimal et les mathématiques financières. optimal et les mathématiques financières.
, The obstacle problem is a classic motivati … The obstacle problem is a classic motivating example in the mathematical study of variational inequalities and free boundary problems. The problem is to find the equilibrium position of an elastic membrane whose boundary is held fixed, and which is constrained to lie above a given obstacle. It is deeply related to the study of minimal surfaces and the capacity of a set in potential theory as well. Applications include the study of fluid filtration in porous media, constrained heating, elasto-plasticity, optimal control, and financial mathematics.ptimal control, and financial mathematics.
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, Problème de l'obstacle
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