Height estimates and half-space type theorems in weighted product spaces with nonnegative Bakry–Émery–Ricci curvature

We prove height estimates concerning compact hypersurfaces with nonzero constant weighted mean curvature and whose boundary is contained into a slice of a weighted product space of nonnegative Bakry–Émery–Ricci curvature. As applications of our estimates, we obtain half-space type results related to complete noncompact hypersurfaces properly immersed in such an ambient space.     

Potential function estimates for quasi-Einstein metrics

In this paper, we derive the growth pinching estimates for potential functions of τ-quasi-Einstein metrics on complete noncompact connected manifolds, based on the estimates for the scalar curvature and the using of the weighted measure comparison theorem. Our results show that the estimates for potential functions rely on the sign of constants λ and μ.     

Local gradient estimate for harmonic functions on Finsler manifolds

In this paper, we prove the local gradient estimate for harmonic functions on complete, noncompact Finsler measure spaces under the condition that the weighted Ricci curvature has a lower bound. As applications, we obtain Liouville type theorems on noncompact Finsler manifolds with nonnegative Ricci curvature.     

Nonlinear Riemann-Hilbert Problems with Unbounded Restriction Curves

Using a norm inequality for singular integral operators in pairs of weighted Lebesgue spaces we prove new existence and uniqueness results for solutions of nonlinear Riemann-Hilbert problems with noncompact restriction curves.     

Stability of complex hyperbolic space under curvature-normalized Ricci flow

Using the maximal regularity theory for quasilinear parabolic systems, we prove two stability results of complex hyperbolic space under the curvature-normalized Ricci flow in complex dimensions two and higher. The first result is on a closed manifold. The second result is on a complete noncompact manifold. To prove both results, we fully analyze the structure of the Lichnerowicz Laplacian on complex hyperbolic space. To prove the second result, we also define suitably weighted little Hölder spaces on a complete noncompact manifold and establish their interpolation properties.     

Weighted Strichartz estimates for the Schrödinger and wave equations on Damek–Ricci spaces

We prove Strichartz estimates for radial solutions of the Schrödinger and wave equations on Damek–Ricci spaces, and in particular on symmetric spaces of noncompact type and rank one, using the perturbative theory with potentials. The curvature of the noncompact manifold has an influence on the dispersive properties, and indeed we obtain Strichartz estimates with weights at spatial infinity, which are stronger than the standard ones in the flat case.     

Gradient Estimates for p-Laplacian Lichnerowicz Equation on Noncompact Metric Measure Space

Abstract In this paper,the authors obtain the gradient estimates for positive solutions to the weighted p-Laplacian Lichnerowicz equation■on noncompact smooth metric measure space,where c is a nonnegative constant,and p,σ(1p≤2,σ≤p-1)are real constants.Moreover,by the gradient estimate,they can get the corresponding Liouville theorem and Harnack inequality.     

Eigenvalue estimate on complete noncompact Riemannian manifolds and applications

We obtain some sharp estimates on the first eigenvalues of complete noncompact Riemannian manifolds under assumptions of volume growth. Using these estimates we study hypersurfaces with constant mean curvature and give some estimates on the mean curvatures.

     

The order of convergence of eigenfrequencies in finite element approximations of fluid-structure interaction problems

In this paper we prove a double order for the convergence of eigenfrequencies in fluid-structure vibration problems. We improve estimates given recently for compressible and incompressible fluids. To do this, we extend classical results on finite element spectral approximation to nonconforming methods for noncompact operators.

     

The Critical Exponent for Weighted Sobolev Imbeddings

We define the critical exponent of (compact and noncompact) imbeddings of certain special weighted Sobolev spaces into weighted Lebesgue spaces.     

Riesz transform and perturbation

We prove that, on a complete noncompact Riemannian manifold with bounded geometry, the L^p boundedness of the Riesz transform, for p>2, is stable under a quasi-isometric and integrable change of metric. As an intermediate step, we treat the case of weighted divergence form operators in the Euclidean space.     

Dirichlet型空间上Toeplitz算子的本性范数

本文研究单位圆盘上Dirichlet型空间非紧Toeplitz算子的本性范数, 它事实上等于到紧Toeplitz算子集的距离. 并且这个距离可由无限多个紧Toeplitz算子来刻画, 这个结果与加权Bergman空间情形的类似.     

广义凸空间内的拟平衡问题和约束多目标对策

在没有线性结构的广义凸空间内研究了一类拟平衡问题和一类约束多目标对策.首先在非紧广义凸空间内对拟平衡问题证明了两个解的存在性定理.然后作为拟平衡存在定理的应用,在广义凸空间内对约束多目标对策建立了几个加权Nash-平衡和帕雷多平衡存在定理.这些定理改进、统一和推广了最近文献中多目标对策的相应结果.     

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.     

Wavelet Frames,Sobolev Embeddings and Negative Spectrum of Schrödinger Operators on Manifolds with Bounded Geometry

We consider weighted function spaces of Sobolev-Besov type and Schrödinger type operators on noncompact Riemannian manifolds with bounded geometry. First we give characterization of the spaces in terms of wavelet frames. Then we describe the necessary and sufficient conditions for the compactness of Sobolev embeddings between the spaces. An asymptotic behavior of the corresponding entropy numbers is calculated. At the end we use the asymptotic behavior to estimate the number of negative eigenvalues of the Schrödinger type operators.     

Existence and Multiplicity of solutions for a quasilinear elliptic system on unbounded domains involving nonlinear boundary conditions

We prove two existence results for the nonlinear elliptic boundary value system involving $p$-Laplacian over an unbounded domain in $R^N$ with noncompact boundary. The proofs are based on variational methods applied to weighted spaces.     

Lower bounds of error estimates for singular perturbation problems with Jacobi-Spectral approximations

With weighted orthogonal Jacobi polynomials, we study spectral approximations for singular perturbation problems on an interval. The singular parameters of the model are included in the basis functions, and then its stiff matrix is diagonal. Considering the estimations for weighted orthogonal coefficients, a special technique is proposed to investigate the a posteriori error estimates. In view of the difficulty of a posteriori error estimates for spectral approximations, we employ a truncation projection to study lower bounds for the models. Specially, we present the lower bounds of a posteriori error estimates with two different weighted norms in details.     

Universal inequalities on complete noncompact smooth metric measure spaces

In this paper, we obtain universal inequalities for the eigenvalues of the Dirichlet problem and clamped plate problem of drifting Laplacian on (\(n+1\))-dimensional (\(n\ge 4\)) complete noncompact simply connected smooth metric measure spaces which meet some conditions of the sectional curvature and radial weighted Ricci curvature.     

Coincidence Theorems with Applications to Minimax Inequalities, Section Theorem, Best Approximation and Multiobjective Games in Topological Spaces

Some new coincidence theorems involving admissible set-valued mappings are proved in general noncompact topological spaces. As applications, some new minimax inequalities, section theorem, best approximation theorem, existence theorems of weighted Nash equilibria and Pareto equilibria for multiobjective games are given in general topological spaces.     

Persistence of noncompact normally hyperbolic invariant manifolds in bounded geometry

We prove a persistence result for noncompact normally hyperbolic invariant manifolds in Riemannian manifolds of bounded geometry. The bounded geometry of the ambient manifold is a crucial assumption in order to control the uniformity of all estimates throughout the proof.