Book•
Geometric Measure Theory
01 Jun 1969-
TL;DR: In this article, Grassmann algebras of a vectorspace have been studied in the context of the calculus of variations, and a glossary of some standard notations has been provided.
Abstract: Introduction Chapter 1 Grassmann algebra 1.1 Tensor products 1.2 Graded algebras 1.3 Teh exterior algebra of a vectorspace 1.4 Alternating forms and duality 1.5 Interior multiplications 1.6 Simple m-vectors 1.8 Mass and comass 1.9 The symmetric algebra of a vectorspace 1.10 Symmetric forms and polynomial functions Chapter 2 General measure theory 2.1 Measures and measurable sets 2.2 Borrel and Suslin sets 2.3 Measurable functions 2.4 Lebesgue integrations 2.5 Linear functionals 2.6 Product measures 2.7 Invariant measures 2.8 Covering theorems 2.9 Derivates 2.10 Caratheodory's construction Chapter 3 Rectifiability 3.1 Differentials and tangents 3.2 Area and coarea of Lipschitzian maps 3.3 Structure theory 3.4 Some properties of highly differentiable functions Chapter 4 Homological integration theory 4.1 Differential forms and currents 4.2 Deformations and compactness 4.3 Slicing 4.4 Homology groups 4.5 Normal currents of dimension n in R(-63) superscript n Chapter 5 Applications to the calculus of variations 5.1 Integrands and minimizing currents 5.2 Regularity of solutions of certain differential equations 5.3 Excess and smoothness 5.4 Further results on area minimizing currents Bibliography Glossary of some standard notations List of basic notations defined in the text Index
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Book•
01 Jan 1982
TL;DR: This book is a blend of erudition, popularization, and exposition, and the illustrations include many superb examples of computer graphics that are works of art in their own right.
Abstract: "...a blend of erudition (fascinating and sometimes obscure historical minutiae abound), popularization (mathematical rigor is relegated to appendices) and exposition (the reader need have little knowledge of the fields involved) ...and the illustrations include many superb examples of computer graphics that are works of art in their own right." Nature
24,199 citations
Book•
02 Jan 2013
TL;DR: In this paper, the authors provide a detailed description of the basic properties of optimal transport, including cyclical monotonicity and Kantorovich duality, and three examples of coupling techniques.
Abstract: Couplings and changes of variables.- Three examples of coupling techniques.- The founding fathers of optimal transport.- Qualitative description of optimal transport.- Basic properties.- Cyclical monotonicity and Kantorovich duality.- The Wasserstein distances.- Displacement interpolation.- The Monge-Mather shortening principle.- Solution of the Monge problem I: global approach.- Solution of the Monge problem II: Local approach.- The Jacobian equation.- Smoothness.- Qualitative picture.- Optimal transport and Riemannian geometry.- Ricci curvature.- Otto calculus.- Displacement convexity I.- Displacement convexity II.- Volume control.- Density control and local regularity.- Infinitesimal displacement convexity.- Isoperimetric-type inequalities.- Concentration inequalities.- Gradient flows I.- Gradient flows II: Qualitative properties.- Gradient flows III: Functional inequalities.- Synthetic treatment of Ricci curvature.- Analytic and synthetic points of view.- Convergence of metric-measure spaces.- Stability of optimal transport.- Weak Ricci curvature bounds I: Definition and Stability.- Weak Ricci curvature bounds II: Geometric and analytic properties.
5,524 citations
TL;DR: The notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems, existence theorem, and continuous dependence may now be proved by very efficient and striking arguments as discussed by the authors.
Abstract: The notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems, existence theorems, and theorems about continuous dependence may now be proved by very efficient and striking arguments. The range of important applications of these results is enormous. This article is a self-contained exposition of the basic theory of viscosity solutions
5,267 citations
Book•
01 Feb 1993TL;DR: Inequalities for mixed volumes 7. Selected applications Appendix as discussed by the authors ] is a survey of mixed volumes with bounding boxes and quermass integrals, as well as a discussion of their applications.
Abstract: 1. Basic convexity 2. Boundary structure 3. Minkowski addition 4. Curvature measure and quermass integrals 5. Mixed volumes 6. Inequalities for mixed volumes 7. Selected applications Appendix.
3,954 citations