Ebook: Variational and Free Boundary Problems

This IMA Volume in Mathematics and its Applications VARIATIONAL AND FREE BOUNDARY PROBLEMS is based on the proceedings of a workshop which was an integral part of the 1990- 91 IMA program on "Phase Transitions and Free Boundaries. " The aim of the workshop was to highlight new methods, directions and problems in variational and free boundary theory, with a concentration on novel applications of variational methods to applied problems. We thank R. Fosdick, M. E. Gurtin, W. -M. Ni and L. A. Peletier for organizing the year-long program and, especially, J. Sprock for co-organizing the meeting and co-editing these proceedings. We also take this opportunity to thank the National Science Foundation whose financial support made the workshop possible. Avner Friedman Willard Miller, Jr. PREFACE In a free boundary one seeks to find a solution u to a partial differential equation in a domain, a part r of its boundary of which is unknown. Thus both u and r must be determined. In addition to the standard boundary conditions on the un­ known domain, an additional condition must be prescribed on the free boundary. A classical example is the Stefan problem of melting of ice; here the temperature sat­ isfies the heat equation in the water region, and yet this region itself (or rather the ice-water interface) is unknown and must be determined together with the tempera­ ture within the water. Some free boundary problems lend themselves to variational formulation.

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This volume contains articles based on recent research in Variational and Free Boundary Problems collected by the Institute for Mathematics and its Applications. The collection as a whole concentrates on novel applications of variational methods to applied problems. The book provides a wide cross section of current research in far growing areas. The articles are based on models which arise in phase transitions, in elastic/ plastic contact problems, Hele-Shaw cells, crystal growth, variational formulation of computer vision models, magneto-hydrodynamics, bubble growth, hydrodynamics (jets and cavities), and in stochastic control and economics. They present mathematical methods which can be further extended and developed for other models. The book should be of interest both to mathematicians and to engineers who are working with mathematical models.

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This volume contains articles based on recent research in Variational and Free Boundary Problems collected by the Institute for Mathematics and its Applications. The collection as a whole concentrates on novel applications of variational methods to applied problems. The book provides a wide cross section of current research in far growing areas. The articles are based on models which arise in phase transitions, in elastic/ plastic contact problems, Hele-Shaw cells, crystal growth, variational formulation of computer vision models, magneto-hydrodynamics, bubble growth, hydrodynamics (jets and cavities), and in stochastic control and economics. They present mathematical methods which can be further extended and developed for other models. The book should be of interest both to mathematicians and to engineers who are working with mathematical models.
Content:
Front Matter....Pages i-xv
Free Boundary Problems Arising in Industry....Pages 1-10
Convex Free Boundaries and the Operator Method....Pages 11-27
The Space SBV(?) and Free Discontinuity Problems....Pages 29-45
Wiener Criterion for the Obstacle Problem Relative to Square H?rmander’s Operators....Pages 47-62
Asymptotic Behavior of Solidification Solutions of Stefan Problems....Pages 63-72
Blow-Up and Regularization for the Hele-Shaw Problem....Pages 73-85
A Multidomain Decomposition for the Transport Equation....Pages 87-109
Axisymmetric MHD Equilibria from Kruskal-Kulsrud to Grad....Pages 111-134
A Two-Sided Game for Non Local Competitive Systems with Control on Source Terms....Pages 135-152
The Stefan Problem with Surface Tension....Pages 153-157
The Rayleigh Instability for a Cylindrical Crystal-Melt Interface....Pages 159-169
Towards a Unified Approach for the Adaptive Solution of Evolution Phase Changes....Pages 171-193
Blowup and Global Existence for a Non-Equilibrium Phase Change Process....Pages 195-204

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This volume contains articles based on recent research in Variational and Free Boundary Problems collected by the Institute for Mathematics and its Applications. The collection as a whole concentrates on novel applications of variational methods to applied problems. The book provides a wide cross section of current research in far growing areas. The articles are based on models which arise in phase transitions, in elastic/ plastic contact problems, Hele-Shaw cells, crystal growth, variational formulation of computer vision models, magneto-hydrodynamics, bubble growth, hydrodynamics (jets and cavities), and in stochastic control and economics. They present mathematical methods which can be further extended and developed for other models. The book should be of interest both to mathematicians and to engineers who are working with mathematical models.
Content:
Front Matter....Pages i-xv
Free Boundary Problems Arising in Industry....Pages 1-10
Convex Free Boundaries and the Operator Method....Pages 11-27
The Space SBV(?) and Free Discontinuity Problems....Pages 29-45
Wiener Criterion for the Obstacle Problem Relative to Square H?rmander’s Operators....Pages 47-62
Asymptotic Behavior of Solidification Solutions of Stefan Problems....Pages 63-72
Blow-Up and Regularization for the Hele-Shaw Problem....Pages 73-85
A Multidomain Decomposition for the Transport Equation....Pages 87-109
Axisymmetric MHD Equilibria from Kruskal-Kulsrud to Grad....Pages 111-134
A Two-Sided Game for Non Local Competitive Systems with Control on Source Terms....Pages 135-152
The Stefan Problem with Surface Tension....Pages 153-157
The Rayleigh Instability for a Cylindrical Crystal-Melt Interface....Pages 159-169
Towards a Unified Approach for the Adaptive Solution of Evolution Phase Changes....Pages 171-193
Blowup and Global Existence for a Non-Equilibrium Phase Change Process....Pages 195-204
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