A Note on the Strong Maximum Principle for Fully Nonlinear Equations on Riemannian Manifolds

Alessandro Goffi,Francesco Pediconi

doi:10.1007/s12220-021-00607-2

Alessandro Goffi, Francesco Pediconi

Open Access

https://doi.org/10.1007/s12220-021-00607-2

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Abstract

We investigate strong maximum (and minimum) principles for fully nonlinear second-order equations on Riemannian manifolds that are non-totally degenerate and satisfy appropriate scaling conditions. Our results apply to a large class of nonlinear operators, among which Pucci’s extremal operators, some singular operators such as those modeled on the p- and infty -Laplacian, and mean curvature-type problems. As a byproduct, we establish new strong comparison principles for some second-order uniformly elliptic problems when the manifold has nonnegative sectional curvature.

Highlights

This paper is devoted to analyze strong maximum and comparison principles for viscosity solutions to fully nonlinear second order equations on Riemannian manifolds (M, g) of the general formF(x, u, du, D2u) = 0 in ⊂ M, (1.1)being a connected open subset and F : J 2 M → R proper, namely non-decreasing in the second entry and non-increasing in the last entry
Our second motivation is to lay the groundwork for investigating Liouville-type results for fully nonlinear elliptic problems as (1.1) on general Riemannian structures. This would be mainly inspired by the recent nonlinear studies in [6], which make use of (SMP) and (SmP), and the linear Liouville properties given in [32], which are intimately connected with the stochastic completeness of the manifold, see [44,54]
Before stating our main results, we begin with a glimpse on the literature on maximum principles, starting with (SMP)-(SmP) and concluding with (SCP)

Summary

Introduction

This paper is devoted to analyze strong maximum and comparison principles for viscosity solutions to fully nonlinear second order equations on (finite dimensional) Riemannian manifolds (M, g) of the general form. In the fully nonlinear case, strong comparison principles have been addressed in [7, Rem 3] (when one of the functions is C2 via the arguments in [39]), by Ishii-Yoshimura in [36, Thm 5.3] for second order uniformly elliptic equations with Lipschitz growth in (u, du), N.S. Trudinger [56] for Lipschitz continuous viscosity solutions, Y. 5, we first set up a simple proof for Pucci’s extremal operators when one of the involved functions is smooth (see Lemma 5.1) and we prove (SCP) following a strategy implemented in [36] based on a combination of the (SMP) with the weak comparison principle in [4] that yields the result in the uniformly elliptic case.

Preliminaries of Riemannian Geometry

Notation

Universal Riemannian Operators

The Strong Maximum Principle

An Application