On the interplay among maximum principles, compact support principles and Keller-Osserman conditions on manifolds

Abstract

This paper is about 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, we study classes of coercive differential inequalities of the formon domains of a manifold M , for suitable ϕ, b, f, l, with emphasis on mean curvature type operators. We investigate the validity of strong maximum principles, compact support principles and Liouville type theorems; in particular, the goal is to identify sharp thresholds, involving curvatures or volume growth of geodesic balls in M , to guarantee the above properties under appropriate Keller-Osserman type conditions, and to discuss the geometric reasons behind the existence of such thresholds. The paper also aims to give a unified view of recent results in the literature. The bridge with Geometry is realized 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.

Bernstein and half-space properties for minimal graphs under Ricci lower bounds

“…Proof of Theorem 4. Because of volume comparison, B r ≤ Cr m for some C > 0 and thus, by Theorem 8.1 in [3], f (u(x)) = 0 for every x ∈ M . The function u is therefore a minimal graph, and since f ≡ 0 then u is bounded on one side.…”

“…3], and P. Pucci and J. Serrin [48,Thm. 8.1.3], still in Euclidean space, while the most up-to-date results in a manifold setting can be found in Subsections 5.2.1, 6.1.1 and 8.5.2 in [3], see in particular Theorems 5.8, 8.1, 8.9.…”

“…By [46], this is equivalent to the stochastic completeness of M . However, the picture is subtle and a more precise account goes beyond the scope of the present work: the reader who is interested in the relations between Keller-Osserman conditions and maximum principles at infinity, is referred to [3] for a detailed discussion.…”

Detecting the completeness of a Finsler manifold via potential theory for its infinity Laplacian

“…for large . By Theorem 2.24 and Proposition 7.4 in [14], property (6) for solutions of (7) holds on any complete manifold satisfying…”

“…For instance, the first operator appears when studying entire graphs with prescribed mean curvature, and the validity of maximum principles at infinity are therefore instrumental to prove Bernstein type theorems [14,20], while the -Laplacian, in the limit → 1, gives an efficient way to construct solutions of the inverse mean curvature flow on spaces with mild curvature requirements, see [39], and maximum principles at infinity serve to guarantee the global gradient estimates needed to perform the approximation procedure. For both operators, criteria in the spirit of ( 5) have been established in [48,14], still showing a substantial difference between the "parabolic" case ≡ 0 and the case > 0 on ℝ + . More precisely, the formal limit…”