Titlebook: Geometric Analysis of Quasilinear Inequalities on Complete Manifolds; Maximum and Compact Bruno Bianchini,Luciano Mari,Marco Rigoli Book 2

entreat

https://doi.org/10.1007/978-981-99-9569-1Consider the problem . We recall that an end Ω ⊂ . is a connected component with non-compact closure of .∖., for some compact set ..

放纵

Discourse, the Body, and IdentityIn this section, we relate the Keller–Osserman condition . to the strong Liouville property (SL) for solutions of (..). It is particularly interesting to see how geometry comes into play via the validity of the weak or the strong maximum principle for (.). Δ.. Hereafter, we require . and moreover

Meditate

Preliminaries from Riemannian Geometry,We briefly recall some facts from Riemannian Geometry, mostly to fix notation and conventions. Our main source for the present chapter is P. Petersen’s book. Let (.., 〈 , 〉) be a connected Riemannian manifold. We denote with ∇ the Levi–Civita connection induced by 〈 , 〉, and with . the (4, 0) curvature tensor of ∇, with the usual sign agreement

Hallowed

Gyrate

Boundary Value Problems for Nonlinear ODEs,At the beginning of Chap. ., we observed that to find radial solutions of (..) and (..) one is lead to solve the following ODE: . on an interval of ., where we have extended . to an odd function on all of .. The functions .. and . are bounds, respectively, for the volume of geodesic spheres of . and for ..

起来了

Comparison Results and the Finite Maximum Principle,In this section, we collect two comparison theorems and a “pasting lemma” for Lip. solutions that will be repeatedly used in the sequel. Throughout the section, we assume

nominal

KIN

,Strong Maximum Principle and Khas’minskii Potentials,The aim of this section is to prove Theorem . in the Introduction. We observe that the argument is based on the existence of what we call a “Khas’minskii potential”, according to the following.

integral

CLOT

,Keller–Osserman, A Priori Estimates and the (,) Property,In this section, we relate the Keller–Osserman condition . to the strong Liouville property (SL) for solutions of (..). It is particularly interesting to see how geometry comes into play via the validity of the weak or the strong maximum principle for (.). Δ.. Hereafter, we require . and moreover