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Award-winning monograph of the Ferran Sunyer i Balaguer Prize 2004. This book studies regularity properties of Mumford-Shah minimizers. The Mumford-Shah functional was introduced in the 1980s as a tool for automatic image segmentation, but its study gave rise to many interesting questions of analysis and geometric measure theory. The main object under scrutiny is a free boundary K where the minimizer may have jumps. The book presents an extensive description of the known regularity properties of the singular sets K, and the techniques to get them. Some time is spent on the C^1 regularity theorem (with an essentially unpublished proof in dimension 2), but a good part of the book is devoted to applications of A. Bonnet's monotonicity and blow-up techniques. In particular, global minimizers in the plane are studied in full detail. The book is largely self-contained and should be accessible to graduate students in analysis.The core of the book is composed of regularity results that were proved in the last ten years and which are presented in a more detailed and unified way.
Calculus of variations. --- Geometric measure theory. --- Manifolds (Mathematics) --- Boundary value problems. --- Differential equations, Partial. --- Isoperimetrical problems --- Variations, Calculus of --- Maxima and minima --- Partial differential equations --- Boundary conditions (Differential equations) --- Differential equations --- Functions of complex variables --- Mathematical physics --- Initial value problems --- Geometry, Differential --- Topology --- Measure theory --- Mathematical optimization. --- Functional analysis. --- Differential equations, partial. --- Calculus of Variations and Optimal Control; Optimization. --- Functional Analysis. --- Partial Differential Equations. --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Mathematical analysis --- Operations research --- Simulation methods --- System analysis --- Partial differential equations.
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