Morse Theoretic Aspects of p-Laplacian Type Operators

Author(s): 
K. Perera, R.P. Agarwal, D. O'Regan
Publisher: 
AMS
Year: 
2011
ISBN: 
978-082184968
Price (tentative): 
$69
Short description: 

The purpose of this book is to present a Morse theoretic study of a very general class of homogeneous operators that includes the p-Laplacian as a special case. The p-Laplacian operator is a quasilinear differential operator that arises in many applications such as non-Newtonian fluid flows and turbulent filtration in porous media.

URL for publisher, author, or book: 
http://www.ams.org/bookstore-getitem/item=SURV-161
MSC main category: 
47 Operator theory
MSC category: 
47J05
Other MSC categories: 
58E05, 47J10, 35J60
Review: 

Standard Morse theory is a useful tool to compute homology groups of finite dimensional manifolds from the critical points of a sufficiently generic function. The tools of Morse theory can be developed in an infinite dimensional setting to some extent. This point of view can be applied to solve differential equations, by means of writing a Morse functional from a Banach vector space whose critical points are the solutions to the original equation. Each critical point has a degree and some cohomological information. This together with some standard tools of algebraic topology may allow finding restrictions to the number and properties of critical points.

The current book is devoted to apply infinite dimensional Morse theory to study the p-Laplacian and other quasilinear operators where the Euler functional is not defined on a Hilbert space or is not $C^2$ or where there are no eigenspaces to work with. The p-Laplacian operator arises in many applications such as non-Newtonian fluid flows and turbulent filtration in porous media. The p-Laplacian problem consists on finding the eigenfunctions of $\Delta_p u= \lambda |u|^{p-2} u$, where $\Delta_p u= div( |\nabla u|^{p-2} \nabla u)$.

The book is very technical, and it is only accessible to experts in the field. Even the introduction and overview is written in a technical language, assuming some knowledge with the notation. Chapters 2, 3 and 4 are more readable, containing background material. A large part of the rest of the book is of research level. Researchers in the area will find the book of interest, containing many results, and with a large bibliography.

Reviewer: 
Vicente Muñoz
Affiliation: 
UCM

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