Sobolev type inequalities on Riemannian manifolds

This paper studies Sobolev type inequalities on Riemannian manifolds. We show that on a complete non-compact Riemannian manifold the constant in the Gagliardo-Nirenberg inequality cannot be smaller than the optimal one on the Euclidean space of the same dimension. We also show that a complete non-compact manifold with asymptotically non-negative Ricci curvature admitting some Gagliardo-Nirenberg inequality is not very far from the Euclidean space.     

Keller-Osserman conditions for diffusion-type operators on Riemannian manifolds

In this paper we obtain essentially sharp generalized Keller-Osserman conditions for wide classes of differential inequalities of the form Lu?b(x)f(u)?(|∇u|) and Lu?b(x)f(u)?(|∇u|)−g(u)h(|∇u|) on weighted Riemannian manifolds, where L is a non-linear diffusion-type operator. Prototypical examples of these operators are the p-Laplacian and the mean curvature operator. The geometry of the underlying manifold is reflected, via bounds for the modified Bakry-Emery Ricci curvature, by growth conditions for the functions b and ?. A weak maximum principle which extends and improves previous results valid for the φ-Laplacian is also obtained. Geometric comparison results, valid even in the case of integral bounds for the modified Bakry-Emery Ricci tensor, are presented.     

On the elliptic equation on complete Riemannian manifolds and their geometric applications

We study the elliptic equation on complete noncompact Riemannian manifolds with nonnegative. Three fundamental theorems for this equation are proved in this paper. Complete analyses of this equation on the Euclidean space and the hyperbolic space are carried out when is a constant. Its application to the problem of conformal deformation of nonpositive scalar curvature will be done in the second part of this paper.

     

Hardy and Rellich type inequalities on complete manifolds

We prove some Hardy and Rellich type inequalities on complete noncompact Riemannian manifolds supporting a weight function which is not very far from the distance function in the Euclidean space.     

Uniqueness result for non-negative solutions of semi-linear inequalities on Riemannian manifolds

We consider certain semi-linear partial differential inequalities on complete connected Riemannian manifolds and provide a simple condition in terms of volume growth for the uniqueness of a non-negative solution. We also show the sharpness of this condition.     

Conditional extremals in complete Riemannian manifolds

Conditional extremal curves in a complete Riemannian manifold M are defined as the critical points of the squared L² distance between the tangent vector field of a curve and a so-called prior vector field. We prove that this L² distance satisfies the Palais-Smale condition on the space of absolutely continuous curves joining two submanifolds of M, and thus establish the existence of critical points. We also prove a Morse index theorem in the case where the two submanifolds are single points, and use the Morse inequalities to place lower bounds on the number of critical points of each index.     

Functional inequalities on arbitrary Riemannian manifolds

For any connected (not necessarily complete) Riemannian manifold, we construct a probability measure of type , where dx is the Riemannian volume measure and V is a function C^∞-smooth outside a closed set of zero volume, satisfying Poincaré–Sobolev type functional inequalities. In particular, V is C^∞-smooth on the whole manifold when the Poincaré and the super-Poincaré inequalities are considered. The Sobolev inequality for infinite measures are also studied.     

Hardy type inequalities on complete Riemannian manifolds

In this paper, we show that complete Riemannian manifolds with asymptotically non-negative Ricci curvature in which some Hardy type inequalities hold are not far from the Euclidean space.     

Existence and uniqueness for a semilinear elliptic problem on Lipschitz domains in Riemannian manifolds II

Extending our recent work for the semilinear elliptic equation on Lipschitz domains, we study a general second-order Dirichlet problem in . We improve our previous results by studying more general nonlinear terms with polynomial (and in some cases exponential) growth in the variable . We also study the case of nonnegative solutions.

     

Existence of solutions for variational inequalities on Riemannian manifolds

We establish the existence and uniqueness results for variational inequality problems on Riemannian manifolds and solve completely the open problem proposed in [S.Z. Németh, Variational inequalities on Hadamard manifolds, Nonlinear Anal. 52 (2003) 1491–1498]. Also the relationships between the constrained optimization problem and the variational inequality problems as well as the projections on Riemannian manifolds are studied.     

Second-order semilinear elliptic inequalities in exterior domains

We study the problem of existence and nonexistence of positive solutions of the semilinear elliptic inequalities in divergence form with measurable coefficients in exterior domains in . For W(x)?|x|^−σ at infinity we compute the critical line on the plane (p,σ), which separates the domains of existence and nonexistence, and reveal the class of potentials V that preserves the critical line. Example are provided showing that the class of potentials is maximal possible, in certain sense. The case of (p,σ) on the critical line has also been studied.     

Positive solutions of quasilinear elliptic inequalities on noncompact Riemannian manifolds

In this paper, we consider the generalized solutions of the inequality $$ - div(A(x,u,\nabla u)\nabla u) \geqslant F(x,u,\nabla u)u^q , q > 1,$$ on noncompact Riemannian manifolds. We obtain sufficient conditions for the validity of Liouville’s theorem on the triviality of the positive solutions of the inequality under consideration. We also obtain sharp conditions for the existence of a positive solution of the inequality ? Δu ≥ u ^q, q > 1, on spherically symmetric noncompact Riemannian manifolds.     

A class of quasilinear elliptic hemivariational inequalities

1. IntroductionIn this paper we shall deal with the quasilinear elliptic hemivaxiational inequalitieswhere the symbol OJ designates Clarke's generalized gradiellt of a locally Lipschitz functionalJ (see [1, 2]). The discontinuity is assumed to be in the lower order term OJ(u(x)). If Ais a linear operator and the disconiinuity is monotone, the problems have been studied,for example, in [3-6]. If A is a linear opertor and the discolltinuity is nonmonotone, thiskind of problems have been inve…     

Quasilinear elliptic equations with Neumann boundary and singularity

Let Ω be a bounded domain with a smooth C2 boundary in RN（N ≥ 3）, 0 ∈Ω, and n denote the unit outward normal to ЭΩ.We are concerned with the Neumann boundary problems： -div（｜x｜α｜△u｜p-2△u）=｜x｜βup（α,β）-1-λ｜x｜γup-1,u（x）〉0,x∈Ω,Эu/Эn=0 on ЭΩ,where 1〈p〈N and α〈0,β〈0 such that p（α,β）△=p（N＋β）/N-p＋α〉p,y〉α-p.For various parameters α,βorγ,we establish certain existence results of the solutions in the case 0∈Ω or 0∈ЭΩ.     

Weighted isoperimetric inequalities in warped product manifolds

We prove some sharp isoperimetric type inequalities in warped product manifolds, or more generally, multiply warped product manifolds. We also relate them to inequalities involving the higher order mean-curvature integrals. Applications include some sharp eigenvalue estimates, Pólya-Szegö inequality, Faber-Krahn inequality, Sobolev inequality and some sharp geometric inequalities in some warped product spaces.     

紧流形上椭圆微分算子预解式的一致L^p-L^q估计——献给陆善镇教授75华诞

假设n和m是两个正整数,P（x,D）是定义在维数为n的紧致无边流形M上的一般m阶椭圆自伴微分算子.在一定条件下,本文主要证明微分算子P（x,D）的预解式的一致L^p-L^q估计,其中n〉m≥2,（p,q）在Sobolev线上并满足1/p-1/q=m/n,p≤2（n＋1）/n＋3,q≥2（n＋1）/n-1.本文的一个核心引理是建立曲面Σx={ξ∈Tx^＊（M）：p（x,ξ）=1}上测度的Fourier变换衰减估计的具体表达式,并利用它来得到局部算子的一致L^p-L^q估计.     

Weighted Hardy and Rellich type inequalities on Riemannian manifolds

In this paper we present new results on two‐weight Hardy, Hardy–Poincaré and Rellich type inequalities with remainder terms on a complete noncompact Riemannian Manifold M. The method we use is flexible enough to obtain more weighted Hardy type inequalities. Our results improve and include many previously known results as special cases.     

Quasi‐subdifferential operator approach to elliptic variational and quasi‐variational inequalities

We prove an existence theorem for an abstract operator equation associated with a quasi‐subdifferential operator and then apply it to concrete elliptic variational and quasi‐variational inequalities. Copyright © 2016 John Wiley & Sons, Ltd.