Nonlinear Potential Theory on Metric Spaces
Title | Nonlinear Potential Theory on Metric Spaces PDF eBook |
Author | Anders Björn |
Publisher | European Mathematical Society |
Pages | 422 |
Release | 2011 |
Genre | Harmonic functions |
ISBN | 9783037190999 |
The $p$-Laplace equation is the main prototype for nonlinear elliptic problems and forms a basis for various applications, such as injection moulding of plastics, nonlinear elasticity theory, and image processing. Its solutions, called p-harmonic functions, have been studied in various contexts since the 1960s, first on Euclidean spaces and later on Riemannian manifolds, graphs, and Heisenberg groups. Nonlinear potential theory of p-harmonic functions on metric spaces has been developing since the 1990s and generalizes and unites these earlier theories. This monograph gives a unified treatment of the subject and covers most of the available results in the field, so far scattered over a large number of research papers. The aim is to serve both as an introduction to the area for interested readers and as a reference text for active researchers. The presentation is rather self contained, but it is assumed that readers know measure theory and functional analysis. The first half of the book deals with Sobolev type spaces, so-called Newtonian spaces, based on upper gradients on general metric spaces. In the second half, these spaces are used to study p-harmonic functions on metric spaces, and a nonlinear potential theory is developed under some additional, but natural, assumptions on the underlying metric space. Each chapter contains historical notes with relevant references, and an extensive index is provided at the end of the book.
Nonlinear Potential Theory on Metric Spaces
Title | Nonlinear Potential Theory on Metric Spaces PDF eBook |
Author | Tero Mäkäläinen |
Publisher | |
Pages | 98 |
Release | 2008 |
Genre | Embedding theorems |
ISBN | 9789513932695 |
Nonlinear Potential Theory on Metric Spaces
Title | Nonlinear Potential Theory on Metric Spaces PDF eBook |
Author | Juha Kinnunen |
Publisher | |
Pages | 23 |
Release | 2001 |
Genre | |
ISBN |
Lectures on Analysis on Metric Spaces
Title | Lectures on Analysis on Metric Spaces PDF eBook |
Author | Juha Heinonen |
Publisher | Springer |
Pages | 141 |
Release | 2000-12-21 |
Genre | Mathematics |
ISBN | 9780387951041 |
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.
Function Spaces and Potential Theory
Title | Function Spaces and Potential Theory PDF eBook |
Author | David R. Adams |
Publisher | Springer Science & Business Media |
Pages | 372 |
Release | 2012-12-06 |
Genre | Mathematics |
ISBN | 3662032821 |
"..carefully and thoughtfully written and prepared with, in my opinion, just the right amount of detail included...will certainly be a primary source that I shall turn to." Proceedings of the Edinburgh Mathematical Society
Sobolev Spaces on Metric Measure Spaces
Title | Sobolev Spaces on Metric Measure Spaces PDF eBook |
Author | Juha Heinonen |
Publisher | Cambridge University Press |
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.
Lectures on Analysis on Metric Spaces
Title | Lectures on Analysis on Metric Spaces PDF eBook |
Author | Juha Heinonen |
Publisher | |
Pages | 152 |
Release | 2011-04-26 |
Genre | |
ISBN | 9781461301325 |