New Forms of the Maximum Principle

Abstract

In the previous chapter we described the Omori-Yau maximum principle for the Laplace-Beltrami operator Δ, giving some analytical motivations, and later we introduced the weak maximum principle, illustrating its deep equivalence with stochastic completeness. Furthermore, to show the power and effectiveness of these tools when applied to some specific problem, we gave a few applications to geometry. The aim of the present chapter is to extend the investigation to a much more general class of differential operators containing those that naturally appear when dealing with the geometry of submanifolds or, more generally, in tackling some analytical problems on complete manifolds: for instance, the p-Laplacian, the (generalized) mean curvature operator, trace operators, and so on. In doing so we give sufficient conditions for the validity of two types of maximum principles corresponding, respectively, to the Omori-Yau and to the weak maximum principle. In this chapter we focus our attention on conditions that basically require the existence of a function, indicated throughout with γ, whose existence is, in many instances, guaranteed by the geometry of the problem. First we deal with the linear case, that presents less analytical difficulties, and we conclude our discussion by providing a first a priori estimate; again by way of example, we show its use in a geometric problem. Note that in the next chapter we will provide a second type of sufficient condition for the validity of the weak maximum principle when the operator is in divergence form, basically in terms of the volume growth of geodesic balls with a fixed center on M. Clearly, this kind of condition is very mild and immediately implied by suitable curvature assumptions. We then move to the nonlinear case, where the analytical difficulties that we have to face are definitely deeper; for this reason and for an intrinsic interest, we devote an entire subsection to a careful proof of a general form of some auxiliary analytical results that we shall need for our purposes. We finally prove our general nonlinear results in Theorems 3.11 and 3.13, concluding the chapter.