Approximation Of Free Discontinuity Problems

Functionals involving both volume and surface energies have a number of applications ranging from Computer Vision to Fracture Mechanics. In order to tackle numerical and dynamical problems linked to such functionals many approximations by functionals defined on smooth functions have been proposed (using high-order singular perturbations, finite-difference or non-local energies, etc.) The purpose of this book is to present a global approach to these approximations using the theory of gamma-convergence and of special functions of bounded variation. The book is directed to PhD students and researchers in calculus of variations, interested in approximation problems with possible applications.

This book contains a collection of articles corresponding to some of the talks delivered at the Foundations of Computational Mathematics conference held at IMPA in Rio de Janeiro in January 1997. Some ofthe others are published in the December 1996 issue of the Journal of Complexity. Both of these publications were available and distributed at the meeting. Even in this aspect we hope to have achieved a synthesis of the mathematics and computer science cultures as well as of the disciplines. The reaction to the Park City meeting on Mathematics of Numerical Analy sis: Real Number Algorithms which was chaired by Steve Smale and had around 275 participants, was very enthusiastic. At the suggestion of Narendra Karmar mar a lunch time meeting of Felipe Cucker, Arieh Iserles, Narendra Karmarkar, Jim Renegar, Mike Shub and Steve Smale decided to try to hold a periodic meeting entitled "Foundations of Computational Mathematics" and to form an organization with the same name whose primary purpose will be to hold the meeting. This is then the first edition of FoCM as such. It has been organized around a small collection of workshops, namely - Systems of algebraic equations and computational algebraic geometry - Homotopy methods and real machines - Information-based complexity - Numerical linear algebra - Approximation and PDEs - Optimization - Differential equations and dynamical systems - Relations to computer science - Vision and related computational tools There were also twelve plenary speakers.

The theory of Gamma-convergence is commonly recognized as an ideal and flexible tool for the description of the asymptotic behaviour of variational problems. Its applications range from the mathematical analysis of composites to the theory of phase transitions, from Image Processing to Fracture Mechanics. This text, written by an expert in the field, provides a brief and simple introduction to this subject, based on the treatment of a series of fundamental problems that illustrate the main features and techniques of Gamma-convergence and at the same time provide a stimulating starting point for further studies. The main part is set in a one-dimensional framework that highlights the main issues of Gamma-convergence without the burden of higher-dimensional technicalities. The text deals in sequence with increasingly complex problems, first treating integral functionals, then homogenisation, segmentation problems, phase transitions, free-discontinuity problems and their discrete and continuous approximation, making stimulating connections among those problems and with applications. The final part is devoted to an introduction to higher-dimensional problems, where more technical tools are usually needed, but the main techniques of Gamma-convergence illustrated in the previous section may be applied unchanged. The book and its structure originate from the author's experience in teaching courses on this subject to students at PhD level in all fields of Applied Analysis, and from the interaction with many specialists in Mechanics and Computer Vision, which have helped in making the text addressed also to a non-mathematical audience. The material of the book is almost self-contained, requiring only some basic notion of Measure Theory and Functional Analysis.

This book presents a series of lectures on three of the best known examples of free discontinuity problems: the Mumford-Shah model for image segmentation, a variational model for the epitaxial growth of thin films, and the sharp interface limit of the Ohta-Kawasaki model for pattern formation in dyblock copolymers.

This is the first volume of the procedings of the second European Congress of Mathematics. Volume I presents the speeches delivered at the Congress, the list of lectures, and short summaries of the achievements of the prize winners. Together with volume II it contains a collection of contributions by the invited lecturers. Finally, volume II also presents reports on some of the Round Table discussions. This two-volume set thus gives an overview of the state of the art in many fields of mathematics and is therefore of interest to every professional mathematician. Contributors: Vol. I: N. Alon, L. Ambrosio, K. Astala, R. Benedetti, Ch. Bessenrodt, F. Bethuel, P. Bjørstad, E. Bolthausen, J. Bricmont, A. Kupiainen, D. Burago, L. Caporaso, U. Dierkes, I. Dynnikov, L.H. Eliasson, W.T. Gowers, H. Hedenmalm, A. Huber, J. Kaczorowski, J. Kollár, D.O. Kramkov, A.N. Shiryaev, C. Lescop, R. März. Vol. II: J. Matousek, D. McDuff, A.S. Merkurjev, V. Milman, St. Müller, T. Nowicki, E. Olivieri, E. Scoppola, V.P. Platonov, J. Pöschel, L. Polterovich , L. Pyber, N. Simányi, J.P. Solovej, A. Stipsicz, G. Tardos, J.-P. Tignol, A.P. Veselov, E. Zuazua

"This book attempts to bring together a selection of the latest results of state-of-the art research in image and video segmentation, one of the most critical tasks of image and video analysis that has the objective of extracting information (represented by data) from an image or a sequence of images (video)"--Provided by publisher.

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.

At the summer school in Pisa in September 1996, Luigi Ambrosio and Norman Dancer each gave a course on the geometric problem of evolution of a surface by mean curvature, and degree theory with applications to PDEs respectively. This self-contained presentation accessible to PhD students bridged the gap between standard courses and advanced research on these topics. The resulting book is divided accordingly into 2 parts, and neatly illustrates the 2-way interaction of problems and methods. Each of the courses is augmented and complemented by additional short chapters by other authors describing current research problems and results.