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| Author | : Juha Heinonen |
| Publisher | : Cambridge University Press |
| Total Pages | : 447 |
| Release | : 2015-02-05 |
| Genre | : Mathematics |
| ISBN | : 1107092345 |
This coherent treatment from first principles is an ideal introduction for graduate students and a useful reference for experts.
| Author | : Juha Heinonen |
| Publisher | : Cambridge University Press |
| Total Pages | : 447 |
| Release | : 2015-02-05 |
| Genre | : Mathematics |
| ISBN | : 1316241033 |
Analysis on metric spaces emerged in the 1990s as an independent research field providing a unified treatment of first-order analysis in diverse and potentially nonsmooth settings. Based on the fundamental concept of upper gradient, the notion of a Sobolev function was formulated in the setting of metric measure spaces supporting a Poincaré inequality. This coherent treatment from first principles is an ideal introduction to the subject for graduate students and a useful reference for experts. It presents the foundations of the theory of such first-order Sobolev spaces, then explores geometric implications of the critical Poincaré inequality, and indicates numerous examples of spaces satisfying this axiom. A distinguishing feature of the book is its focus on vector-valued Sobolev spaces. The final chapters include proofs of several landmark theorems, including Cheeger's stability theorem for Poincaré inequalities under Gromov–Hausdorff convergence, and the Keith–Zhong self-improvement theorem for Poincaré inequalities.
| Author | : Nageswari Shanmugalingam |
| Publisher | : |
| Total Pages | : 186 |
| Release | : 1999 |
| Genre | : |
| ISBN | : |
| Author | : Juha Heinonen |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 149 |
| Release | : 2012-12-06 |
| Genre | : Mathematics |
| ISBN | : 1461301319 |
The purpose of this book is to communicate some of the recent advances in this field while preparing the reader for more advanced study. The material can be roughly divided into three different types: classical, standard but sometimes with a new twist, and recent. The author first studies basic covering theorems and their applications to analysis in metric measure spaces. This is followed by a discussion on Sobolev spaces emphasizing principles that are valid in larger contexts. The last few sections of the book present a basic theory of quasisymmetric maps between metric spaces. Much of the material is recent and appears for the first time in book format.
| Author | : Heli Tuominen |
| Publisher | : |
| Total Pages | : 96 |
| Release | : 2004 |
| Genre | : Functional equations |
| ISBN | : |
| Author | : |
| Publisher | : |
| Total Pages | : |
| Release | : 2016 |
| Genre | : |
| ISBN | : |
| Author | : Fabrice Baudoin |
| Publisher | : Springer Nature |
| Total Pages | : 312 |
| Release | : 2022-02-04 |
| Genre | : Mathematics |
| ISBN | : 3030841413 |
This book includes four courses on geometric measure theory, the calculus of variations, partial differential equations, and differential geometry. Authored by leading experts in their fields, the lectures present different approaches to research topics with the common background of a relevant underlying, usually non-Riemannian, geometric structure. In particular, the topics covered concern differentiation and functions of bounded variation in metric spaces, Sobolev spaces, and differential geometry in the so-called Carnot–Carathéodory spaces. The text is based on lectures presented at the 10th School on "Analysis and Geometry in Metric Spaces" held in Levico Terme (TN), Italy, in collaboration with the University of Trento, Fondazione Bruno Kessler and CIME, Italy. The book is addressed to both graduate students and researchers.
| Author | : Luigi Ambrosio |
| Publisher | : Oxford University Press, USA |
| Total Pages | : 148 |
| Release | : 2004 |
| Genre | : Mathematics |
| ISBN | : 9780198529385 |
This book presents the main mathematical prerequisites for analysis in metric spaces. It covers abstract measure theory, Hausdorff measures, Lipschitz functions, covering theorums, lower semicontinuity of the one-dimensional Hausdorff measure, Sobolev spaces of maps between metric spaces, and Gromov-Hausdorff theory, all developed ina general metric setting. The existence of geodesics (and more generally of minimal Steiner connections) is discussed on general metric spaces and as an application of the Gromov-Hausdorff theory, even in some cases when the ambient space is not locally compact. A brief and very general description of the theory of integration with respect to non-decreasing set functions is presented following the Di Giorgi method of using the 'cavalieri' formula as the definition of the integral. Based on lecture notes from Scuola Normale, this book presents the main mathematical prerequisites for analysis in metric spaces. Supplemented with exercises of varying difficulty it is ideal for a graduate-level short course for applied mathematicians and engineers.
| Author | : Lars Diening |
| Publisher | : Springer |
| Total Pages | : 516 |
| Release | : 2011-03-29 |
| Genre | : Mathematics |
| ISBN | : 3642183638 |
The field of variable exponent function spaces has witnessed an explosive growth in recent years. The standard reference article for basic properties is already 20 years old. Thus this self-contained monograph collecting all the basic properties of variable exponent Lebesgue and Sobolev spaces is timely and provides a much-needed accessible reference work utilizing consistent notation and terminology. Many results are also provided with new and improved proofs. The book also presents a number of applications to PDE and fluid dynamics.
| Author | : Petteri Harjulehto |
| Publisher | : |
| Total Pages | : 25 |
| Release | : 2004 |
| Genre | : |
| ISBN | : |
