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.
Cuprins
– Some Geometric Motivations. – An Overview of Our Results. – Preliminaries from Riemannian Geometry. – Radialization and Fake Distances. – Boundary Value Problems for Nonlinear ODEs. – Comparison Results and the Finite Maximum Principle. – Weak Maximum Principle and Liouville’s Property. – Strong Maximum Principle and Khas’minskii Potentials. – The Compact Support Principle. – Keller–Osserman, A Priori Estimates and the (SL) Property.
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