A priori estimates and reduction principles for quasilinear elliptic problems and applications. (English) Zbl 1273.35138

The paper under review is written by leading experts in quasilinear degenerate elliptic inequalities who have made extensive and deep contributions to the subject for many years. Indeed, a substantial part of the bibliography contains the authors’ research. The paper is an in-depth, up-to-date, modern, clear exposition of several variants of the Kato inequality and applications to positivity results, as in the celebrated papers of H. Brezis from the 1980s, and to Liouville theorems. The paper under review treats this classical subject for very general and different frameworks. On the other hand, the proofs of the main theorems are entirely self-contained. Therefore, the paper should be accessible to a large audience of researchers in the field of elliptic partial differential inequalities. The paper is also full of comments that provide a short historical background, and of examples, some of which are of independent interest. The connections among different topics are clearly exhibited, and much attention is given to bibliographical and historical notes. I strongly recommend this excellent paper to every researcher somehow interested in the field of elliptic equations.


35J62 Quasilinear elliptic equations
35B45 A priori estimates in context of PDEs
35B51 Comparison principles in context of PDEs
35B53 Liouville theorems and Phragmén-Lindelöf theorems in context of PDEs
35J70 Degenerate elliptic equations
35R03 PDEs on Heisenberg groups, Lie groups, Carnot groups, etc.
35R45 Partial differential inequalities and systems of partial differential inequalities
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations