Viscosity solutions: A primer.

*(English)*Zbl 0901.49026
Capuzzo Dolcetta, I. (ed.) et al., Viscosity solutions and applications. Lectures given at the 2nd session of the Centro Internazionale Matematico Estivo (CIME), Montecatini Terme, Italy, June 12-20, 1995. Berlin: Springer. Lect. Notes Math. 1660, 1-43 (1997).

Since the seminal papers by M. G. Crandall and P. L. Lions [Trans. Am. Soc. 277, 1-42 (1983; Zbl 0599.35024)] and M. G. Crandall, L. C. Evans and P. L. Lions [Trans. Am. Math. Soc. 282, 487-502 (1984; Zbl 0543.35011)], the viscosity solution approach has grown into a well established theory, essential to understand several aspects of fully nonlinear scalar partial differential equations of first and second order
\[
F(x,u,Du,D^2u)=0.\tag{1}
\]
To have an idea of the extension of this theory and of the amazing range of its applications, the reader is referred to the recent books by M. Bardi and I. Capuzzo-Dolcetta [“Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations” (1996; Zbl 0890.49011)] for first order equations, and by W. H. Fleming and H. M. Soner [“Controlled Markov processes and viscosity solutions” (1993; Zbl 0773.60070)] for second order ones.

Nevertheless, if one is interested in a first insight of viscosity solutions, having to go through the above detailed monographies may seem discouraging. For this purpose, the reader will profit from reading the paper under review. In fact, these CIME lectures address the most basic issues of this theory, such as existence, comparison, and stability of solutions to degenerate elliptic equations of type (1). No attempt is made to describe applications of viscosity solutions: connections with control, game theory and mathematical finance can be found in the above mentioned books, as well as in the other contributions to the lecture notes containing the present paper.

The exposition is plain and enriched by a number of remarks, examples and exercises that help the reader to understand the ideas that suggested definitions and proofs. Generality is often neglected to persue clarity, which also makes this paper a perfect introduction to the more technical work of M. G. Crandall, H. Ishii and P. L. Lions [Bull. Am. Math. Soc., New Ser. 27, No. 1, 1-67 (1992; Zbl 0755.35015)].

For the entire collection see [Zbl 0868.00050].

Nevertheless, if one is interested in a first insight of viscosity solutions, having to go through the above detailed monographies may seem discouraging. For this purpose, the reader will profit from reading the paper under review. In fact, these CIME lectures address the most basic issues of this theory, such as existence, comparison, and stability of solutions to degenerate elliptic equations of type (1). No attempt is made to describe applications of viscosity solutions: connections with control, game theory and mathematical finance can be found in the above mentioned books, as well as in the other contributions to the lecture notes containing the present paper.

The exposition is plain and enriched by a number of remarks, examples and exercises that help the reader to understand the ideas that suggested definitions and proofs. Generality is often neglected to persue clarity, which also makes this paper a perfect introduction to the more technical work of M. G. Crandall, H. Ishii and P. L. Lions [Bull. Am. Math. Soc., New Ser. 27, No. 1, 1-67 (1992; Zbl 0755.35015)].

For the entire collection see [Zbl 0868.00050].

Reviewer: P.Cannarsa (Roma)

##### MSC:

49L25 | Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games |

35J70 | Degenerate elliptic equations |

35J60 | Nonlinear elliptic equations |

49L20 | Dynamic programming in optimal control and differential games |

35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |

49-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to calculus of variations and optimal control |