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Surface evolution equations. A level set approach. (English) Zbl 1096.53039
Monographs in Mathematics 99. Basel: Birkhäuser (ISBN 3-7643-2430-9/hbk). xii, 264 p. (2006).
This book gives an introduction to the level set approach for geometric evolution equations of the form \[ V=f(x,t,n,\nabla n)\quad\text{on }\Gamma_t.\tag{1} \] Here, \(\Gamma_t\) denotes the evolving surface, \(V\) is its normal velocity and \(n\) a unit normal field to \(\Gamma_t\). A well-known example of such an evolution law is motion by the mean curvature. In general, (1) will only have a smooth solution locally in time. One way to construct a global solution that allows the flow passing through singularities is to describe the surface \(\Gamma_t\) via a suitable level set of an auxiliary function \(u\). The law (1) then translates into a nonlinear PDE for \(u\) which is in many cases degenerate and singular, but which can be successfully treated within the framework of viscosity solutions. A careful description of this approach forms the major part of the book.
Chapter 1 introduces useful material from differential geometry and discusses relevant examples for the evolution law (1), such as motion by (anisotropic) mean curvature and Hamilton-Jacobi equations. Furthermore, the level set equation associated with (1) is derived and its properties are discussed. Chapter 2 introduces the concept of a viscosity solution and presents various stability results. Further topics include boundary conditions and existence proofs via Perron’s method. In Chapter 3, several versions of the comparison principle are proved, both for bounded and unbounded domains. Among other things, these are applied in order to obtain convexity and Lipschitz preserving properties for spatially homogeneous equations.
Chapter 4 contains a detailed description of the above mentioned level set approach to geometric evolution equations. The main steps in order to obtain a global well-defined evolution are: a) unique global solvability of the corresponding level set PDE within the framework of viscosity solutions; denote its solution by \(u\) and define \(\Gamma_t = \{x\in\mathbb R^n\mid u(x,t) = 0\}\); b) proof that \(\Gamma_t\) only depends on \(\Gamma_0\) but is independent of the function \(u_0\) whose zero level is \(\Gamma_0\). Chapter 4 also describes various properties of the evolving surfaces and addresses the fattening phenomenon. Finally, in Chapter 5 set-theoretic and barrier solutions are introduced and discussed.
In view of its detailed and thorough presentation this book will be a valuable source for everyone interested in the level set approach to surface evolution equations.

53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
35K10 Second-order parabolic equations
35K55 Nonlinear parabolic equations
74N20 Dynamics of phase boundaries in solids