This book presents a self-contained introduction to the analytic foundation of a level set approach for various surface evolution equations including curvature flow equations. These equations are important in many applications, such as material sciences, image processing and differential geometry. The goal is to introduce a generalized notion of solutions allowing singularities, and to solve the initial-value problem globally-in-time in a generalized sense. Various equivalent definitions of solutions are studied. Several new results on equivalence are also presented. Moreover, structures of level set equations are studied in detail. Further, a rather complete introduction to the theory of viscosity solutions is contained, which is a key tool for the level set approach.
Although most of the results in this book are more or less known, they are scattered in several references, sometimes without proofs. This book presents these results in a synthetic way with full proofs.
The intended audience are graduate students and researchers in various disciplines who would like to know the applicability and detail of the theory as well as its flavour. No familiarity with differential geometry or the theory of viscosity solutions is required. Only prerequisites are calculus, linear algebra and some basic knowledge about semicontinuous functions.

Presents results in a synthetic way with full proofs The audience is introduced to details of the theory as well as to its flavour No familiarity with differential geometry or the theory of viscosity solutions is required. Only prerequisites are calculus, linear algebra and some knowledge about semicontinuous functions Includes supplementary material: sn.pub/extras

Autorentext

Dr. Yoshikazu Giga is professor at the Graduate School of Mathematical Sciences of the University of Tokyo, Japan. He is a fellow of the American Mathematical Society as well as of the Japan Society for Industrial and Applied Mathematics. Through his more than 250 papers and 2 monographs, he has substantially contributed to the theory for parabolic partial differential equations including geometric evolution equations, semilinear heat equations, as well as the incompressible Navier-Stokes equations. He has received several prizes including the Medal of Honor with Purple Ribbon from the Government of Japan and the second Kodaira Kunihiko Prize from the Mathematical Society of Japan. Dr. Mi-Ho Giga is a project researcher at the Graduate School of Mathematical Sciences of the University of Tokyo, Japan. She is a mathematical analyst working on parabolic partial differential equations related to crystal growth at low temperature. She is a member of both the Mathematical Society of Japan and the Japan Society for Industrial and Applied Mathematics. She is one of the authors of the book 'Nonlinear Partial Differential Equations: Asymptotic Behavior of Solutions and Self-Similar Solutions' published from Birkhauser in 2010 as well as its Japanese original version published from Kyoritu-Shuppan in 1999.

Inhalt
Surface evolution equations.- Viscosity solutions.- Comparison principle.- Classical level set method.- Set-theoretic approach.