## Abstract

Preface This book is intended to be a self-contained introduction to analytitc foundation of a level set method for various surface evolution equations including curvature flow equa-tions. These equations are important in various fields including material sciences, image processing and differential geometry. The goal of this book is to introduce a generalized notion of solutions allowing singularities and solve the initial-value problem globally-in-time in generalized sense. Various equivalent definitions of solutions are studied. Several new results on equivalence are also presented. This book contains rather complete introduction to the theory of viscosity solutions which is a key tool for the level set method. Also a self-contained explanation is given for general surface evolution equations of the second order. Although most of results in this book is more or less known, they are scattered in several references sometimes without proof. This book presents these results in a synthetic way with full proofs. However, the references are not exhaustive at all. This book is suitable for applied researchers who would like to know the detail of the theory as well as its flavour. No familiarity of differential geometry and the theory of viscosity solutions is required. Only prerequisites are calculus, linear algebra and some familiarity of semicontinuous functions. This book is also suitable for upper level of under graduate students who are interested in the field. This book is based on my Lipschitz lectures in Bonn 1997. The author is grateful to its audience for their interest. The author is also grateful to Professor Naoyuki Ishimura, Professor Reiner Sch�tzle, Professor Katsuyuki Ishii and Professor Masaki Ohnuma for their critical remarks on earlier versions of this book. The financial support of the Japan Society for the Promotion of Science (no. 10304010, 11894003, 12874024, 13894003) is gratefully acknowledged. Finally, the author is grateful to Ms. Hisako Iwai for careful typing of the manuscripts in latex style.