Elliptic partial differential equations from an elementary viewpoint. A fresh glance at the classical theory (Q6601709)

The monograph under review is certainly interesting and reading it fully gives reason to the appropriate subtitle ``A fresh glance at the classical theory''. Elliptic equations are a very broad topic and therefore the authors have rightly decided to address many of the research topics (those closest to their research themes) but not all of them. Basic topics are covered such as the Laplace operator, the harmonic functions, the upper semicontinuity and subharmonicity, the regularity theory for equations in nondivergence form, the local existence, the Dirichlet problem, the capacity theory, the Poisson equation, and even the recent and celebrated moving plane method. Instead, the variational approach to regularity by De Giorgi and Campanato is not substantially covered even though it represents the greatest contribution of the Italian school in the topic. This choice is very functional as it allows the reader to perceive the authors' profound mastery of the topic. It is clear that this is a book born by actually teaching those topics to an audience of interested students: there is a lightness, a clarity and, almost, an irony capable of fascinating both the student who is tackling these topics for the first time and the expert reader. This monograph is very suitable for undergraduate courses (or courses in the first years of the doctoral program) under the guidance of an expert teacher capable of making the reader grasp the many nuances with which it is enriched or for a teacher who wants to read known results but stated and expressed with a new perspective and with considerable verve. A book definitely to be recommended to scholars interested in studying the subject in depth.