Topics on Analysis in Metric Spaces (Oxford Lecture Series in Mathematics and Its Applications)

Ambrosio, Luigi/ Tilli, Paolo

Oxford University Press（2003/12発売）

• 製本 Hardcover:ハードカバー版／ページ数 142 p.
• 言語 ENG
• 商品コード 9780198529385
• DDC分類 514.32

Full Description

Based on lecture notes from the 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.
The book covers abstract measure theory, Hausdorff measures, Lipschitz functions, covering theorems, lower semicontinuity of the one-dimensional Hausdorff measure, Sobolev spaces of maps between metric spaces, and Gromov-Hausdorff theory, all developed in a general metric setting. The existence of geodesics (and more generally of minimal Steiner connections) is discussed in 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 Cavalieri's formula as the definition of the integral.

Contents

1. Some preliminaries in measure theory ; 2. Hausdorff measures and covering theorems in metric spaces ; 3. Lipschitz functions in metric spaces ; 4. Geodesic problem and Gromov-Hausdorff convergence ; 5. Sobolev spaces in a metric framework ; 6. A quick overview on the theory of integration