Geometric Analysis of Quasilinear Inequalities on Complete Manifolds

Maximum and Compact Support Principles and Detours on Manifolds

Bruno Bianchini, Luciano Mari, Patrizia Pucci, Marco Rigoli (Autoren)

Buch | Softcover

X, 286 Seiten

2021 | 1st ed. 2021
Springer International Publishing (Verlag)
978-3-030-62703-4 (ISBN)

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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.

- 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. - StrongMaximum Principle and Khas'minskii Potentials. - The Compact Support Principle. - Keller-Osserman, A Priori Estimates and the (SL) Property.

"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)

“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)

Erscheinungsdatum	20.01.2021
Reihe/Serie	Frontiers in Mathematics
Zusatzinfo	X, 286 p. 1 illus.
Verlagsort	Cham
Sprache	englisch
Maße	168 x 240 mm
Gewicht	505 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Analysis
Schlagworte	Coercive Differential Inequalities • Compact Support Principle • Liouville Properties • Maximum principle • Mean Curvature Flow Solitons
ISBN-10	3-030-62703-9 / 3030627039
ISBN-13	978-3-030-62703-4 / 9783030627034
Zustand	Neuware