Free boundary problems arising in industry.- Convex free boundaries and the operator method.- The space SBV(?) and free discontinuity problems.- Wiener criterion for the obstacle problem relative to square Hörmander's operators.- Asymptotic behavior of solidification solutions of Stefan problems.- Blow-up and regularization for the Hele-Shaw problem.- A multidomain decomposition for the transport equation.- Axisymmetric MHD equilibria from Kruskal-Kulsrud to Grad.- A two-sided game for non local competitive systems with control on source terms.- The Stefan problem with surface tension.- The Rayleigh instability for a cylindrical crystal-melt interface.- Towards a unified approach for the adaptive solution of evolution phase changes.- Blowup and global existence for a non-equilibrium phase change process.
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.
This book presents recent research in free boundary problems. The articles are based on a variety of models and present mathematical methods which are meant to be further applied and developed for other models. This book will be useful to both mathematicians and engineers working with mathematical models.