Analysis of Heat Equations on Domains. (LMS-31)

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Princeton University Press, Jan 10, 2009 - Mathematics - 296 pages
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This is the first comprehensive reference published on heat equations associated with non self-adjoint uniformly elliptic operators. The author provides introductory materials for those unfamiliar with the underlying mathematics and background needed to understand the properties of heat equations. He then treats Lp properties of solutions to a wide class of heat equations that have been developed over the last fifteen years. These primarily concern the interplay of heat equations in functional analysis, spectral theory and mathematical physics.

This book addresses new developments and applications of Gaussian upper bounds to spectral theory. In particular, it shows how such bounds can be used in order to prove Lp estimates for heat, Schrödinger, and wave type equations. A significant part of the results have been proved during the last decade.

The book will appeal to researchers in applied mathematics and functional analysis, and to graduate students who require an introductory text to sesquilinear form techniques, semigroups generated by second order elliptic operators in divergence form, heat kernel bounds, and their applications. It will also be of value to mathematical physicists. The author supplies readers with several references for the few standard results that are stated without proofs.

 

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Contents

Chapter 1 SESQUILINEAR FORMS ASSOCIATED OPERATORS AND SEMIGROUPS
1
Chapter 2 CONTRACTIVITY PROPERTIES
43
Chapter 3 INEQUALITIES FOR SUBMARKOVIAN SEMIGROUPS
79
Chapter 4 UNIFORMLY ELLIPTIC OPERATORS ON DOMAINS
99
Chapter 5 DEGENERATEELLIPTIC OPERATORS
143
Chapter 6 GAUSSIAN UPPER BOUNDS FOR HEAT KERNELS
155
Chapter 7 GAUSSIAN UPPER BOUNDS AND LsuppSPECTRAL THEORY
193
Chapter 8 A REVIEW OF THE KATO SQUARE ROOT PROBLEM
253
Bibliography
265
Index
283
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