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In this paper, we present a version of the Omori-Yau maximum principle, a Liouville-type result, and a Phragmen-Lindelöff-type theorem for a class of singular elliptic operators on a Riemannian manifold, which include the p-Laplacian and the mean curvature operator. Some applications of the results obtained are discussed. 相似文献
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Stefano?PigolaEmail author Marco?Rigoli Alberto G.?Setti 《Geometric And Functional Analysis》2003,13(6):1302-1328
We obtain a maximum principle, and a priori upper estimates for
solutions of a class of non-linear singular elliptic differential inequalities
on Riemannian manifolds under the sole geometrical assumption
of volume growth conditions. Various applications of the results obtained
are presented. 相似文献
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We study the qualitative behavior of non-negative entire solutions of differential inequalities with gradient terms on the Heisenberg group. We focus on two classes of inequalities: Δφu?f(u)l(|∇u|) and Δφu?f(u)−h(u)g(|∇u|), where f, l, h, g are non-negative continuous functions satisfying certain monotonicity properties. The operator Δφ, called the φ-Laplacian, generalizes the p-Laplace operator considered by various authors in this setting. We prove some Liouville theorems introducing two new Keller-Osserman type conditions, both extending the classical one which appeared long ago in the study of the prototype differential inequality Δu?f(u) in Rm. We show sharpness of our conditions when we specialize to the p-Laplacian. While proving these results we obtain a strong maximum principle for Δφ which, to the best of our knowledge, seems to be new. Our results continue to hold, with the obvious minor modifications, also for Euclidean space. 相似文献
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This article deals with the study of some properties of immersed curves in the conformal sphere \({\mathbb{Q}_n}\), viewed as a homogeneous space under the action of the Möbius group. After an overview on general well-known facts, we briefly focus on the links between Euclidean and conformal curvatures, in the spirit of F. Klein’s Erlangen program. The core of this article is the study of conformal geodesics, defined as the critical points of the conformal arclength functional. After writing down their Euler–Lagrange equations for any n, we prove an interesting codimension reduction, namely that every conformal geodesic in \({\mathbb{Q}_n}\) lies, in fact, in a totally umbilical 4-sphere \({\mathbb{Q}_4}\). We then extend and complete the work in Musso (Math Nachr 165:107–131, 1994) by solving the Euler–Lagrange equations for the curvatures and by providing an explicit expression even for those conformal geodesics not included in any conformal 3-sphere. 相似文献