Geometric Analysis of Quasilinear Inequalities on Complete Manifolds: Maximum and Compact Support Principles and Detours on Manifolds (Frontiers in Mathematics)

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. Preface Contents List of Symbols 1 Some Geometric Motivations 1.1 Prescribing the Mean Curvature of a Graph The Seeming Lack of a Keller–Osserman Condition Geodesic Graphs in R h M Mean Curvature Flow Solitons Equidistant Graphs in M h R 1.2 A Problem from the Theory of Stochastic Control 2 An Overview of Our Results 2.1 Setting and Main Properties Under Investigation 2.2 Keller–Osserman Conditions 2.3 Notation and Conventions 2.4 The Finite Maximum Principle (FMP) 2.5 Strong and Weak Maximum Principles at Infinity 2.6 The Strong Liouville Property (SL) 2.7 The Compact Support Principle (CSP) 2.8 More General Inequalities 3 Preliminaries from Riemannian Geometry 3.1 Model Manifolds 3.2 Comparison Theory for the Distance Function 4 Radialization and Fake Distances 4.1 Basic Facts on Nonlinear Potential Theory Model Manifolds Uniqueness and Comparison Theory for G 4.2 The Fake Distance and Its Basic Properties Relations with Hardy Weights 4.3 Gradient Estimate 4.4 Properness of 5 Boundary Value Problems for Nonlinear ODEs 5.1 The Dirichlet Problem 5.2 The Mixed Dirichlet–Neumann Problem 6 Comparison Results and the Finite Maximum Principle 6.1 Basic Comparisons and a Pasting Lemma 6.2 The Finite Maximum Principle 7 Weak Maximum Principle and Liouville's Property 7.1 The Equivalence Between (WMP∞) and (L) 7.2 Volume Growth and (WMP∞) 7.3 Bernstein Theorems for Minimal and MCF Soliton Graphs 7.4 Counterexamples 8 Strong Maximum Principle and Khas'minskii Potentials 8.1 Ricci Curvature and (SMP∞) 8.2 Bernstein Theorems for Prescribed Mean Curvature Graphs 9 The Compact Support Principle 9.1 Necessity of (KO0) for the Compact Support Principle 9.2 Sufficiency of (KO0) for the Compact Support Principle General Operators and No Cut-Locus A Second ODE Lemma: Locating the Support Non-Empty Cut-Locus: The p-Laplacian Case A Further Fake Distance and the Feller Property Proof of Proposition 9.19 10 Keller–Osserman, A Priori Estimates and the (SL) Property 10.1 Necessity of (KO∞) for the (SL) Property 10.2 Sufficiency of (KO∞) for the (SL) Property 10.3 Ricci Curvature and (SL) 10.4 Sharpness 10.5 Volume Growth and (SL) 10.6 Applications: Yamabe and Capillarity Equations Yamabe Type Equations The Capillarity Equation 10.7 Other Ranges of Parameters Bibliography Index