Some non-linear function theoretic properties of Riemannian manifolds

Abstract

We study the appropriate versions of parabolicity stochastic completeness and related Liouville properties for a general class of operators which include the p-Laplace operator, and the non linear singular operators in non-diagonal form considered by J. Serrin and collaborators. IntroductionThe starting point of the present note is the circle of ideas in classical potential theory, which relate the parabolicity and stochastic completeness of a manifold on one hand, and their function theoretic counterparts on the other, and in particular results due to Khas'minskii which provide sufficient conditions for a manifold to be parabolic, respectively stochastically complete. These ideas allow also to establish comparison theorems with model manifolds in the sense of Greene and Wu under curvature conditions. It was recently shown by the authors that the parabolicity/stochastic completeness of a manifold may be described in terms of suitable versions of a global weak maximum principle. Explicitly, M is parabolic (resp. stochastically complete) if and only if for every u ∈ C 2 (M ), u bounded above, and for every η > 0 we have inf {u>sup u−η} ∆u < 0 (resp. ≤ 0) (see [17] and [18]).

Aspects of Potential Theory on Manifolds, Linear and Non-linear

“…The following theorem ( [22]) extends to the non-linear situation at hand the equivalences already seen in the linear setting, and substantiates the point of view that these properties may serve as definitions of parabolicity and stochastic completeness in the present situation. Note that the equivalences (a) ⇔ (b) and (c) ⇔ (d) allow us to define the L ϕ -parabolicity (resp.…”

“…The Khas'minskii test may also be used to prove comparison results with models, as in the following theorem (see [8] or [22]). As a consequence we have the following result, due to Varopoulos, [34] (see also [11], and [16]), which detects the "maximum amount" of negative curvature that one can allow without destroying stochastic completeness.…”

“…This is guaranteed by imposing suitable conditions on ϕ (see [13] and [22]). Analogous conditions can be formulated in case of the non-diagonal operators mentioned above.…”

“…To see the role of the structural conditions (A) and (B) imposed on ϕ we outline the argument that shows that (c) ⇒ (d) (see [22]). …”

“…Motivated by the linear case one is lead to the following we have the following proposition (see [22]). Examples show that if p = 2 neither of the reverse implications holds.…”