This book deals with a basic notion in the theory of second order elliptic equations, i.e. the maximum principle. Combined with other notions, this principle provides a large amount of information on properties of solutions to, for example, elliptic problems. The authors study, chapter by chapter, the tangency and comparison theorems (starting from the results of Eberhard Hopf), the maximum principles for divergence structure of elliptic differential equations (for more general operators) and the two-point boundary value problems for nonlinear ordinary differential equations. The latter results are a preliminary to strong maximum principles, which are studied in chapter 5. The last chapters deal with maximum principles for the complete quasilinear divergence inequality and with local boundedness and Harnack’s inequalities. In the applications chapter we learn about the symmetry for overdetermined boundary value problems, the phenomenon of dead cores and the strong maximum principle for Riemannian manifolds. The book is well written, with a high mathematical standard. The book is meant for a wide audience of all those interested in the theory of partial differential equations, although some experience in the field is necessary (basic knowledge on existence and uniqueness results and an overview of the theory of elliptic partial differential equations), according to the opinion of the reviewer. The book will be appreciated by specialists in the field as well as by PhD students searching for an advanced book on the topic.

Reviewer:

mr