New gap results on the 4‐dimensional sphere

A result showed by M. Gursky in 4 ensures that any metric g on the 4‐dimensional sphere satisfying and is isometric to the round metric. In this note, we prove that there exists a universal number i ₀ such that any metric g on the 4‐dimensional sphere satisfying and is isometric to the round metric. Moreover, there exists a universal such that any metric g on the 4‐dimensional sphere with nonnegative sectional curvature, and is isometric to the round metric. This last result slightly improves a rigidity theorem also proved in 4 .     

On Lorentzian Ricci solitons on nilpotent Lie groups

The aim of this article is to exhibit the variety of different Ricci soliton structures that a nilpotent Lie group can support when one allows for the metric tensor to be Lorentzian. In stark contrast to the Riemannian case, we show that a nilpotent Lie group can support a number of non‐isometric Lorentzian Ricci soliton structures with decidedly different qualitative behaviors and that Lorentzian Ricci solitons need not be algebraic Ricci solitons. The analysis is carried out by classifying all left invariant Lorentzian metrics on the connected, simply‐connected five‐dimensional Lie group having a Lie algebra with basis vectors and and non‐trivial bracket relations and , investigating the various curvature properties of the resulting families of metrics, and classifying all Lorentzian Ricci soliton structures.     

Classification of surfaces in a pseudo‐sphere with 2‐type pseudo‐spherical Gauss map

In this article, we study submanifolds in a pseudo‐sphere with 2‐type pseudo‐spherical Gauss map. We give a characterization theorem for Lorentzian surfaces in the pseudo‐sphere with zero mean curvature vector in and 2‐type pseudo‐spherical Gauss map. We also prove that non‐totally umbilical proper pseudo‐Riemannian hypersurfaces in a pseudo‐sphere with non‐zero constant mean curvature has 2‐type pseudo‐spherical Gauss map if and only if it has constant scalar curvature. Then, for we obtain the classification of surfaces in with 2‐type pseudo‐spherical Gauss map. Finally, we give an example of surface with null 2‐type pseudo‐spherical Gauss map which does not appear in Riemannian case, and we give a characterization theorem for Lorentzian surfaces in with null 2‐type pseudo‐spherical Gauss map.     

Ridigity of Ricci solitons with weakly harmonic Weyl tensors

In this paper, we prove rigidity results on gradient shrinking or steady Ricci solitons with weakly harmonic Weyl curvature tensors. Let be a compact gradient shrinking Ricci soliton satisfying with constant. We show that if satisfies , then is Einstein. Here denotes the Weyl curvature tensor. In the case of noncompact, if M is complete and satisfies the same condition, then M is rigid in the sense that M is given by a quotient of product of an Einstein manifold with Euclidean space. These are generalizations of the previous known results in 10 , 14 and 19 . Finally, we prove that if is a complete noncompact gradient steady Ricci soliton satisfying , and if the scalar curvature attains its maximum at some point in the interior of M, then either is flat or isometric to a Bryant Ricci soliton. The final result can be considered as a generalization of main result in 3 .     

On Fejér's inequalities for the Legendre polynomials

We present various inequalities for the sum where denotes the Legendre polynomial of degree k . Among others we prove that the inequalities hold for all and . The constant factors 2/5 and are sharp. This refines a classical result of Fejér, who proved in 1908 that is nonnegative for all and .     

Four‐dimensional contact ‐submanifolds in

The analogue of ‐submanifolds in (almost) Kählerian manifolds is the concept of contact ‐submanifolds in Sasakian manifolds. These are submanifolds for which the structure vector field ξ is tangent to the submanifold and for which the tangent bundle of M can be decomposed as , where is invariant with respect to the endomorphism φ and is antiinvariant with respect to φ. The lowest possible dimension for M in which this decomposition is non trivial is the dimension 4. In this paper we obtain a complete classification of four‐dimensional contact ‐submanifolds in and for which the second fundamental form restricted to and vanishes identically.     

Existence of solutions to a non‐variational singular elliptic system with unbounded weights

In this paper we prove an existence result for the following singular elliptic system where Ω is a bounded open set in (), is the p‐laplacian operator, and are suitable Lebesgue functions and , , are positive parameters satisfying suitable assumptions.     

Well‐posedness,blow‐up phenomena and analyticity for a two‐component higher order Camassa–Holm system

In this paper, the local well‐posedness for the Cauchy problem of a two‐component higher‐order Camassa–Holm system (2HOCH) is established in Besov spaces with and (and also in Sobolev spaces with ), which improves the corresponding results for higher‐order Camassa–Holm in 7 , 24 , 25 , where the Sobolev index is required, respectively. Then the precise blow‐up mechanism and global existence for the strong solutions of 2HOCH are determined in the lowest Sobolev space with . Finally, the Gevrey regularity and analyticity of the 2HOCH are presented.     

Upper bound for the ratios of eigenvalues of Schrödinger operators with nonnegative single‐barrier potentials

In this paper we prove the optimal upper bound for one‐dimensional Schrödinger operators with a nonnegative differentiable and single‐barrier potential , such that , where . In particular, if satisfies the additional condition , then for . For this result, we develop a new approach to study the monotonicity of the modified Prüfer angle function.     

Approximation of Schrödinger operators with δ‐interactions supported on hypersurfaces

We show that a Schrödinger operator with a δ‐interaction of strength α supported on a bounded or unbounded C²‐hypersurface , can be approximated in the norm resolvent sense by a family of Hamiltonians with suitably scaled regular potentials. The differential operator with a singular interaction is regarded as a self‐adjoint realization of the formal differential expression , where is an arbitrary bounded measurable function. We discuss also some spectral consequences of this approximation result.     

Boundedness in a fully parabolic chemotaxis‐consumption system with nonlinear diffusion and sensitivity,and logistic source

In this paper we study the zero‐flux chemotaxis‐system Ω being a convex smooth and bounded domain of , , and where , and . For any the chemotactic sensitivity function is assumed to behave as the prototype , with and . We prove that for nonnegative and sufficiently regular initial data and , the corresponding initial‐boundary value problem admits a unique globally bounded classical solution provided μ is large enough.     

Well‐posedness of fractional degenerate differential equations with finite delay on vector‐valued functional spaces

We study the well‐posedness of the fractional degenerate differential equations with finite delay on Lebesgue–Bochner spaces , periodic Besov spaces and periodic Triebel–Lizorkin spaces , where A and M are closed linear operators on a Banach space X satisfying , F is a bounded linear operator from (resp. and ) into X, where is given by when and . Using known operator‐valued Fourier multiplier theorems, we give necessary or sufficient conditions for the well‐posedness of in the above three function spaces.     

Spectral gaps for the Dirichlet‐Laplacian in a 3‐D waveguide periodically perturbed by a family of concentrated masses

We consider a spectral problem for the Laplace operator in a periodic waveguide perturbed by a family of “heavy concentrated masses”; namely, Π contains small regions of high density, which are periodically distributed along the z axis. Each domain has a diameter and the density takes the value in and 1 outside; m and ε are positive parameters, , . Considering a Dirichlet boundary condition, we study the band‐gap structure of the essential spectrum of the corresponding operator as . We provide information on the width of the first bands and find asymptotic formulas for the localization of the possible gaps.     

CMC hypersurfaces with canonical principal direction in space forms

A hypersurface of the space form has a canonical principal direction (CPD) relative to the closed and conformal vector field Z of if the projection of Z to M is a principal direction of M . We show that CPD hypersurfaces with constant mean curvature are foliated by isoparametric hypersurfaces. In particular, we show that a CPD surface with constant mean curvature of space form is invariant by the flow of a Killing vector field whose action is polar on . As consequence we show that a compact CPD minimal surface of the sphere is a Clifford torus. Finally, we consider the case when a CPD Euclidean hypersurface has zero Gauss–Kronecker curvature.     

Weylness of 2 × 2 operator matrices

Let and be complex separable infinite‐dimensional Hilbert spaces. Given the operators and , we define where is an unknown element. In this paper, a necessary and sufficient condition is given for to be a right Weyl (left Weyl, or Weyl) operator for some . Moreover, some relevant properties and illustrating examples are also given.     

Hardy spaces for Bessel–Schrödinger operators

Consider the Bessel operator with a potential on , namely We assume that and is a nonnegative function. By definition, a function belongs to the Hardy space if Under certain assumptions on V we characterize the space in terms of atomic decompositions of local type. In the second part we prove that this characterization can be applied to for with no additional assumptions on the potential V.     

Well‐posedness of non‐autonomous linear evolution equations in uniformly convex spaces

This paper addresses the problem of well‐posedness of non‐autonomous linear evolution equations in uniformly convex Banach spaces. We assume that for each t is the generator of a quasi‐contractive, strongly continuous group, where the domain D and the growth exponent are independent of t . Well‐posedness holds provided that is Lipschitz for all . Hölder continuity of degree is not sufficient and the assumption of uniform convexity cannot be dropped.     

Littlewood–Paley characterizations of higher‐order Sobolev spaces via averages on balls

In this article, the authors characterize higher‐order Sobolev spaces , with , and , or with , and , via the Lusin area function and the Littlewood–Paley ‐function in terms of ball averages, where denotes the maximal integer not greater than . Moreover, the authors also complement the above results in the endpoint cases of p via establishing some weak type estimates. These improve and develop the corresponding known results for Sobolev spaces with smoothness order .     

Solvable extensions of negative Ricci curvature of filiform Lie groups

We give necessary and sufficient conditions of the existence of a left‐invariant metric of strictly negative Ricci curvature on a solvable Lie group the nilradical of whose Lie algebra is a filiform Lie algebra . It turns out that such a metric always exists, except for in the two cases, when is one of the algebras of rank two, or , and is a one‐dimensional extension of , in which cases the conditions are given in terms of certain linear inequalities for the eigenvalues of the extension derivation.     

Rigidity of ‐quasi Einstein manifolds

This paper deals with the study on ‐quasi Einstein manifolds. First, we give some characterizations of an ‐quasi Einstein manifold admitting closed conformal or parallel vector field. Then, we obtain some rigidity conditions for this class of manifolds. We prove that an ‐quasi Einstein manifold with a closed conformal vector field has a warped product structure of the form , where I is a real interval, is an ‐dimensional Riemannian manifold and q is a smooth function on I . Finally, a non‐trivial example of an ‐quasi Einstein manifold verifying our results in terms of the potential function is presented.