On a paper of Berestycki–Hamel–Rossi and its relations to the weak maximum principle at infinity, with applications

Abstract

The aim of this paper is to study a new equivalent form of the weak maximum principle for a large class of differential operators on Riemannian manifolds. This new form has been inspired by the work of Berestycki, Hamel and Rossi, [5], for trace operators and allows us to shed new light on it and to introduce a new sufficient bounded Khas'minskii type condition for its validity. We show its effectiveness by applying it to obtain some uniqueness results in a geometric setting.

Duality between Ahlfors-Liouville and Khas'minskii properties for nonlinear equations

“…Remark 1.4. A related type of Ahlfors property also appeared in the very recent [10], and we refer to [58] for its relationship with results in [3].…”

“…= −i − 1 on X\D j . As g j,k ≡ −i − 1 outside of D j−1 , the extension is smooth across ∂D j and (1) of Proposition 2.5 gives (58) u j,k ∈ F(X\V) and…”

“…The weak Khas'minskii property was first considered, for quasilinear operators, in [69,72,60] (when C = +∞) while the idea of considering potentials with finite C first appeared in the recent [58].…”

Maximum Principles at Infinity and the Ahlfors-Khas’minskii Duality: An Overview

“…Their work fits perfectly to the kind of problems considered in the present paper, and will be introduced later. Our major concern in the recent [54,50,52] is to put the above principles, as well as other properties to be discussed in a moment, into a unified framework where new relations, in particular an underlying duality, could emerge between them.…”