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

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 |

265 | |

283 | |