This book demonstrates the influence of geometry on the qualitative behaviour of solutions of quasilinear PDEs on Riemannian manifolds. Motivated by examples arising, among others, from the theory of submanifolds, the authors study classes of coercive elliptic differential inequalities on domains of a manifold M with very general nonlinearities depending on the variable x, on the solution u and on its gradient. The book highlights the mean curvature operator and its variants, and investigates the validity of strong maximum principles, compact support principles and Liouville type theorems. In particular, it identifies sharp thresholds involving curvatures or volume growth of geodesic balls in M to guarantee the above properties under appropriate Keller-Osserman type conditions, which are investigated in detail throughout the book, and discusses the geometric reasons behind the existence of such thresholds. Further, the book also provides a unified review of recent results in the literature, and creates a bridge with geometry by studying the validity of weak and strong maximum principles at infinity, in the spirit of Omori-Yau’s Hessian and Laplacian principles and subsequent improvements.
Reviews
“The presentation of the book is very well ordered and Keller-Osserman type conditions are investigated in detail. … This is a very good book in this area of research.” (Shu-Yu Hsu, zbMATH 1470.58002, 2021)
Authors and Affiliations
Dipartimento di Matematica Pura e Applicata, Università degli Studi di Padova, Padova, Italy
Bruno Bianchini
Dipartimento di Matematica, Università degli Studi di Torino, Torino, Italy
Luciano Mari
Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, Perugia, Italy
Patrizia Pucci
Dipartimento di Matematica, Università degli Studi di Milano, Milano, Italy
Marco Rigoli
Bibliographic Information
Book Title: Geometric Analysis of Quasilinear Inequalities on Complete Manifolds
Book Subtitle: Maximum and Compact Support Principles and Detours on Manifolds
Authors: Bruno Bianchini, Luciano Mari, Patrizia Pucci, Marco Rigoli
