Korn inequalities for shells with zero Gaussian curvature

We consider shells with zero Gaussian curvature, namely shells with one principal curvature zero and the other one having a constant sign. Our particular interests are shells that are diffeomorphic to a circular cylindrical shell with zero principal longitudinal curvature and positive circumferential curvature, including, for example, cylindrical and conical shells with arbitrary convex cross sections. We prove that the best constant in the first Korn inequality scales like thickness to the power 3/2 for a wide range of boundary conditions at the thin edges of the shell. Our methodology is to prove, for each of the three mutually orthogonal two-dimensional cross-sections of the shell, a “first-and-a-half Korn inequality”—a hybrid between the classical first and second Korn inequalities. These three two-dimensional inequalities assemble into a three-dimensional one, which, in turn, implies the asymptotically sharp first Korn inequality for the shell. This work is a part of mathematically rigorous analysis of extreme sensitivity of the buckling load of axially compressed cylindrical shells to shape imperfections.     

Sums of norm spheres are norm shells and lower triangle inequalities are sharp

The statements in the title are explained and proved, as a little exercise in elementary normed vector space theory at the level of Chap. 5 of Dieudonné’s Foundations of Mathematical Analysis. A connection to recent moment bounds for submartingales is sketched.     

Sums of norm spheres are norm shells and lower triangle inequalities are sharp

The statements in the title are explained and proved, as a little exercise in elementary normed vector space theory at the level of Chap. 5 of Dieudonné’s Foundations of Mathematical Analysis. A connection to recent moment bounds for submartingales is sketched.     

Asymptotically sharp weight Korn’s inequality for thin-walled elastic structures

An elastic junction of several thin plates is considered. All the plates, except for one, called the basic plate, are rigidly clamped along parts of lateral surfaces. We deduce an asymptotically sharp Korn inequality which is weighted and anisotropic. The constant in this Korn inequality is independent of two parameters, the thickness h ∈ (0, 1] and relative rigidity μ ∈ (0, +∞) of the supporting and basic plates. The weight factors in the Sobolev norm on the basic plate essentially depend on the parameters h, μ and on the mutual disposition of the supporting plates. Sufficient geometric and algebraic conditions for the validity of Korn inequalities with various groups of weight factors are given. We also describe special constructions that show the impossibility to improve the obtained inequalities and the necessity of the restrictions imposed on the junction structure. Bibliography: 29 titles. Illustrations: 12 figures. __________ Translated from Problemy Matematicheskogo Analiza, No. 36, 2007, pp. 29–64.     

Model spaces for sharp isoperimetric inequalities

We obtain new sharp isoperimetric inequalities on a Riemannian manifold equipped with a probability measure, whose generalized Ricci curvature is bounded from below (possibly negatively), and generalized dimension and diameter of the convex support are bounded from above (possibly infinitely). Our inequalities are sharp for sets of any given measure and with respect to all parameters (curvature, dimension and diameter). Moreover, for each choice of parameters, we identify the model spaces which are extremal for the isoperimetric problem. In particular, we recover the Gromov–Lévy and Bakry–Ledoux isoperimetric inequalities, which state that whenever the curvature is strictly positively bounded from below, these model spaces are the n-sphere and Gauss space, corresponding to generalized dimension being n and ∞, respectively. In all other cases, which seem new even for the classical Riemannian-volume measure, it turns out that there is no single model space to compare to, and that a simultaneous comparison to a natural one parameter family of model spaces is required, nevertheless yielding a sharp result.     

New sharp inequalities for operator means

New sharp multiplicative reverses of the operator means inequalities are presented, with a simple discussion of squaring an operator inequality. As a direct consequence, we extend the operator Pólya-Szegö inequality to arbitrary operator means. Furthermore, we obtain some new lower and upper bounds for the Tsallis relative operator entropy, operator monotone functions and positive linear maps.     

Some sharp inequalities for n-monotone functions

Summary Some sharp inequalities involving n-monotone functions and their derivatives are obtained. In particular, the following generalization of the Favard-Berwald inequality is established here: \emph{If\/ is non-negative and -f is -monotone, then      

Observability inequalities for thin shells

We consider the exact controllability problem from boundary for thin shells. Under some checkable geometric assumptions on the middle surface, we establish the observability inequalities via the Bochner technique for the Dirichlet control and the Neumann control problems. We also give several examples to verify the geometric assumptions.     

The sharp Sobolev and isoperimetric inequalities split twice

This paper shows that each of the sharp (endpoint) Sobolev inequality and the isoperimetric inequality can be split into two sharp and stronger inequalities through either the 1-variational capacity or the 1-integral affine surface area. Furthermore, some related sharp analytic and geometric inequalities are also explored.     

Weak sharp solutions for generalized variational inequalities

We consider weak sharp solutions for the generalized variational inequality problem, in which the underlying mapping is set-valued, and not necessarily monotone. We extend the concept of weak sharpness to this more general framework, and establish some of its characterizations. We establish connections between weak sharpness and (1) gap functions for variational inequalities, and (2) global error bound. When the solution set is weak sharp, we prove finite convergence of the sequence generated by an arbitrary algorithm, for the monotone set-valued case, as well as for the case in which the underlying set-valued map is either Lipschitz continuous in the set-valued sense, for infinite dimensional spaces, or inner-semicontinuous when the space is finite dimensional.     

An interpolation principle for martingale inequalities

A general principle for martingale inequalities is illustrated by deriving $\text{∥M∥}{}_{\mathrm{p}}\text{? [p + (p}{}^2\text{? 2p)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{] ∥S∥}{}_{\mathrm{p}}$ for all p ? 2 from the case p = 2 where M is the maximal function of a martingale and S its square function.     

The asymptotically optimal meshsize function for bi-p degree interpolation over rectangular elements

A method is presented to recover near optimal interpolation on finite element meshes based on information in the approximation error on an initial mesh. Only a certain class of admissable meshes with rectangular elements in the computational domains are allowed. The method attempts to reach the optimal mesh in one step from the initial mesh, and is based on the notion of meshsize function components or mesh density functions. Asymptotical results showing the optimality of the recovered meshes are given, and extensive computational verification of the method in the special case of Lagrange polynomial interpolation is provided.     

Approximating solutions of variational inequalities for asymptotically nonexpansive mappings

By using viscosity approximation methods for asymptotically nonexpansive mappings in Banach spaces, some sufficient and necessary conditions for a new type of iterative sequences to converging to a fixed point which is also the unique solution of some variational inequalities are obtained. The results presented in the paper extend and improve some recent results in [C.E. Chidume, Jinlu Li, A. Udomene, Convergence of paths and approximation of fixed points of asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 138 (2) (2005) 473-480; N. Shahzad, A. Udomene, Fixed point solutions of variational inequalities for asymptotically nonexpansive mappings in Banach spaces, Nonlinear Anal. 64 (2006) 558-567; T.C. Lim, H.K. Xu, Fixed point theories for asymptotically nonexpansive mappings, Nonlinear Anal. TMA, 22 (1994) 1345-1355; H.K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl., 298 (2004) 279-291].     

Weighted rearrangement inequalities for local sharp maximal functions

Several weighted rearrangement inequalities for uncentered and centered local sharp functions are proved. These results are applied to obtain new weighted weak-type and strong-type estimates for singular integrals. A self-improving property of sharp function inequalities is established.

     

Finite convergence analysis and weak sharp solutions for variational inequalities

In this paper, we study the weak sharpness of the solution set of variational inequality problem (in short, VIP) and the finite convergence property of the sequence generated by some algorithm for finding the solutions of VIP. In particular, we give some characterizations of weak sharpness of the solution set of VIP without considering the primal or dual gap function. We establish an abstract result on the finite convergence property for a sequence generated by some iterative methods. We then apply such abstract result to discuss the finite termination property of the sequence generated by proximal point method, exact proximal point method and gradient projection method. We also give an estimate on the number of iterates by which the sequence converges to a solution of the VIP.     

Error inequalities for quintic and biquintic discrete Hermite interpolation

In this paper we shall develop a class of discrete Hermite interpolates in one and two independent variables. Further, we offer explicit error bounds in ?_∞ norm for the quintic and biquintic discrete Hermite interpolates. Some numerical examples are included to illustrate the results obtained.