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Book
Positive harmonic functions and diffusion
Author:
ISBN: 0511526245 Year: 1995 Publisher: Cambridge : Cambridge University Press,

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In this book, Professor Pinsky gives a self-contained account of the theory of positive harmonic functions for second order elliptic operators, using an integrated probabilistic and analytic approach. The book begins with a treatment of the construction and basic properties of diffusion processes. This theory then serves as a vehicle for studying positive harmonic funtions. Starting with a rigorous treatment of the spectral theory of elliptic operators with nice coefficients on smooth, bounded domains, the author then develops the theory of the generalized principal eigenvalue, and the related criticality theory for elliptic operators on arbitrary domains. Martin boundary theory is considered, and the Martin boundary is explicitly calculated for several classes of operators. The book provides an array of criteria for determining whether a diffusion process is transient or recurrent. Also introduced are the theory of bounded harmonic functions, and Brownian motion on manifolds of negative curvature. Many results that form the folklore of the subject are here given a rigorous exposition, making this book a useful reference for the specialist, and an excellent guide for the graduate student.


Book
Spectral theory and differential operators
Author:
ISBN: 0511623720 Year: 1995 Publisher: Cambridge : Cambridge University Press,

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This book is an introduction to the theory of partial differential operators. It assumes that the reader has a knowledge of introductory functional analysis, up to the spectral theorem for bounded linear operators on Banach spaces. However, it describes the theory of Fourier transforms and distributions as far as is needed to analyse the spectrum of any constant coefficient partial differential operator. A completely new proof of the spectral theorem for unbounded self-adjoint operators is followed by its application to a variety of second-order elliptic differential operators, from those with discrete spectrum to Schrödinger operators acting on L2(RN). The book contains a detailed account of the application of variational methods to estimate the eigenvalues of operators with measurable coefficients defined by the use of quadratic form techniques. This book could be used either for self-study or as a course text, and aims to lead the reader to the more advanced literature on the subject.

Advances in pseudo-differential operators
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ISBN: 3764371404 Year: 2004 Publisher: Basel Birkhäuser

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Book
Heat kernels and spectral theory
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ISBN: 0511566158 Year: 1989 Publisher: Cambridge : Cambridge University Press,

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An advanced monograph on a central topic in the theory of differential equations, Heat Kernels and Spectral Theory investigates the theory of second-order elliptic operators. While the study of the heat equation is a classical subject, this book analyses the improvements in our quantitative understanding of heat kernels. The author considers variable coefficient operators on regions in Euclidean space and Laplace-Beltrami operators on complete Riemannian manifolds. He also includes results pertaining to the heat kernels of Schrödinger operators; such results will be of particular interest to mathematical physicists, and relevant too to those concerned with properties of Brownian motion and other diffusion processes.

Non-linear elliptic equations in conformal geometry
Author:
ISBN: 9783037190067 303719006X Year: 2004 Publisher: Zürich: European mathematical society,

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On spectral theory of elliptic operators
Authors: ---
ISBN: 3764353902 9783764353902 0817653902 Year: 1996 Publisher: Basel : Birkhäuser,


Book
Layer potentials and boundary-value problems for second order elliptic operators with data in Besov spaces
Authors: ---
ISBN: 9781470419899 Year: 2016 Publisher: Providence : American Mathematical Society,

Spectral theory and differential operators
Author:
ISBN: 0521587107 0521472504 9780521587105 9780521472500 9780511623721 Year: 2003 Volume: 42 Publisher: Cambridge Cambridge University Press

A variational inequality approach to free boundary problems with applications in mould filling
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ISBN: 9783764365820 376436582X Year: 2002 Publisher: Basel: Birkhäuser,

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Book
Traces and determinants of pseudodifferential operators
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ISBN: 0198568363 9780198568360 Year: 2010 Publisher: Oxford : Oxford University Press,

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