A beginner's guide to the theory of viscosity solutions
As stated in the Preface, the aim of this textbook is ``to present a brief introduction to the theory of viscosity solutions'' for (undergraduate !) students who have knowledge on Advanced Calculus and, ``hopefully, a little on Lebesgue Integral and Functional Analysis''. To the more advanced readers, the author recommends the user's guide in [\textit{M. G. Crandall, H. Ishii} and \textit{P.-L. Lions}, Bull. Am. Math. Soc. 27, No. 1, 1--67 (1992; Zbl 0755.35015)]. Written indeed as a manual for (rather advanced) students in Mathematics, the book begins with a ``gentle introduction'' to the theory of Partial Differential Equations, illustrating on several classical examples (Laplace, Burger and Hamilton-Jacobi equations) the necessity of introducing the concepts of ``weak'' (generalized) solutions, in particular that of viscosity solutions for nonlinear PDEs. In Section 2, after a discussion of Kruzhkov's ``vanishing viscosity method'', one presents the basic definitions of the viscosity (semi-) solutions and also some equivalent definitions expressed in terms of the so-called ``semi-jets'' of a real function; the definitions look rather complicated as they are presented directly for second order elliptic equations. Section 3 is dedicated to the so-called ``comparison principles'' (implying uniqueness, in certain cases), separately for first-order and, respectively, for second-order PDEs while in Section 4 one proves the existence of viscosity solutions using Perron's method for second order equations and representation formulas in the theory of optimal control and differential games for Bellman's and, respectively, Isaacs', Hamilton-Jacobi equations. In Section 5 one proves the existence of viscosity solutions for Dirichlet and Neumann boundary value problems and in Section 6 is presented ``a short introduction to the \(L^p\)-viscosity solutions theory''. Finally, in the Appendix, the author presents the proofs of some of the more technical results used in the book. The book is very well written, in a clear and systematic way and therefore could be recommended as a good textbook for beginners in the theory of viscosity solutions.
scientific article; zbMATH DE number 2116094