Hardcover ISBN:  9783037190999 
Product Code:  EMSTM/17 
List Price:  $84.00 
AMS Member Price:  $67.20 
Hardcover ISBN:  9783037190999 
Product Code:  EMSTM/17 
List Price:  $84.00 
AMS Member Price:  $67.20 

Book DetailsEMS Tracts in MathematicsVolume: 17; 2011; 415 ppMSC: Primary 31
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 pharmonic 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 pharmonic 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, socalled Newtonian spaces, based on upper gradients on general metric spaces. In the second half, these spaces are used to study pharmonic 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.
A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.
ReadershipGraduate students and researchers interested in measure theory and functional analysis.

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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 pharmonic 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 pharmonic 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, socalled Newtonian spaces, based on upper gradients on general metric spaces. In the second half, these spaces are used to study pharmonic 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.
A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.
Graduate students and researchers interested in measure theory and functional analysis.