On the estimation of nonlinear time series models

In the present paper, a framework for parametric estimation in nonlinear time series is developed. Strong consistency and asymptotic normality of minimum Hellinger distance estimates for a determined class of nonlinear models are investigated. The main Interest for these estimates is motivated by their robustness under perturbations as it has been emphazized in Beran [2]. The first part of the paper is devoted to the study of some probabilistic properties which ensure the existence and the optimal properties of the estimates     

Some Remarks on Korn Inequalities

A recent joint paper with Doina Cioranescu and Julia Orlik was concerned with the homogenization of a linearized elasticity problem with inclusions and cracks(see[Cioranescu, D., Damlamian, A. and Orlik, J., Homogenization via unfolding in periodic elasticity with contact on closed and open cracks, Asymptotic Analysis, 82, 2013, 201–232]). It required uniform estimates with respect to the homogenization parameter. A Korn inequality was used which involves unilateral terms on the boundaries where a nopenetration condition is imposed. In this paper, the author presents a general method to obtain many diverse Korn inequalities including the unilateral inequalities used in [Cioranescu, D., Damlamian, A. and Orlik, J., Homogenization via unfolding in periodic elasticity with contact on closed and open cracks, Asymptotic Analysis, 82, 2013, 201–232]. A preliminary version was presented in [Damlamian, A., Some unilateral Korn inequalities with application to a contact problem with inclusions, C. R. Acad. Sci. Paris, Ser. I,350, 2012, 861–865].     

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.     

Book Reviews

Book reviewed in this article: Contemporary Perspectives in Biology, by Robert W. Korn and Ellen J. Korn, Bellarmine College. John Wiley and Sons, Inc., New York, N. Y. and Investigations into Biology, by Robert W. Korn and Ellen J. Korn, Bellarmine College. Geometry for Elementary Teachers, by John E. Young, Southeast Missouri State College and Grace A. Bush, Kent State University.     

Korn inequality on irregular domains

In this paper, we study the weighted Korn inequality on some irregular domains, e.g., s-John domains and domains satisfying quasihyperbolic boundary conditions. Examples regarding sharpness of the Korn inequality on these domains are presented. Moreover, we show that Korn inequalities imply certain Poincaré inequality.     

On stationary solutions of a nonlinear system of reactor dynamic equations with distributed parameters

We analyze a nonlinear stationary model of reactor dynamics with distributed parameters. We find sufficient conditions for the existence of bifurcation points in this system and study the behavior of solutions in a neighborhood of the bifurcation points. We prove the existence of countably many bifurcation points in the case of a homogeneous medium and obtain constructive estimates for the distance between the bifurcation points.     

Improved interpolation inequalities,relative entropy and fast diffusion equations

We consider a family of Gagliardo–Nirenberg–Sobolev interpolation inequalities which interpolate between Sobolev?s inequality and the logarithmic Sobolev inequality, with optimal constants. The difference of the two terms in the interpolation inequalities (written with optimal constant) measures a distance to the manifold of the optimal functions. We give an explicit estimate of the remainder term and establish an improved inequality, with explicit norms and fully detailed constants. Our approach is based on nonlinear evolution equations and improved entropy–entropy production estimates along the associated flow. Optimizing a relative entropy functional with respect to a scaling parameter, or handling properly second moment estimates, turns out to be the central technical issue. This is a new method in the theory of nonlinear evolution equations, which can be interpreted as the best fit of the solution in the asymptotic regime among all asymptotic profiles.     

Optimal bounds from below of the critical load for elastic solids subject to uniaxial compression

In the recent papers [1,2] we studied a new procedure based on the Korn inequality for determining sufficient conditions for the Hadamard stability, aimed at determining optimal lower bound estimates for the critical load in bifurcation problems. Here, we discuss the effectiveness of our approach for the classical representative problem of uniaxial compression of a Mooney-Rivlin circular cylinder. We find that our lower bound estimate is effective and advantageous for applications, since it is easily implementable in numerical codes. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A general theorem on error estimates with application to a quasilinear elliptic optimal control problem

A theorem on error estimates for smooth nonlinear programming problems in Banach spaces is proved that can be used to derive optimal error estimates for optimal control problems. This theorem is applied to a class of optimal control problems for quasilinear elliptic equations. The state equation is approximated by a finite element scheme, while different discretization methods are used for the control functions. The distance of locally optimal controls to their discrete approximations is estimated.     

Estimates of the deviations from the exact solutions for variational inequalities describing the stationary flow of certain viscous incompressible fluids

This paper is concerned with computable and guaranteed upper bounds of the difference between exact solutions of variational inequalities arising in the theory of viscous fluids and arbitrary approximations in the corresponding energy space. Such estimates (also called error majorants of functional type) have been derived for the considered class of nonlinear boundary‐value problems in (Math. Meth. Appl. Sci. 2006; 29:2225–2244) with the help of variational methods based on duality theory from convex analysis. In the present paper, it is shown that error majorants can be derived in a different way by certain transformations of the variational inequalities that define generalized solutions. The error bounds derived by this techniques for the velocity function differ from those obtained by the variational method. These estimates involve only global constants coming from Korn‐ and Friedrichs‐type inequalities, which are not difficult to evaluate in case of Dirichlet boundary conditions. For the case of mixed boundary conditions, we also derive another form of the estimate that contains only one constant coming from the following assertion: the L² norm of a vector‐valued function from H¹(Ω) in the factor space generated by the equivalence with respect to rigid motions is bounded by the L² norm of the symmetric part of the gradient tensor. As for some ‘simple’ domains such as squares or cubes, the constants in this inequality can be found analytically (or numerically), we obtain a unified form of an error majorant for any domain that admits a decomposition into such subdomains. Copyright © 2009 John Wiley & Sons, Ltd.     

Some nonlinear elliptic systems with right-hand side integrable data with respect to the distance to the boundary

In this paper, we study the very weak solutions to some nonlinear elliptic systems with right-hand side integrable data with respect to the distance to the boundary. Firstly, we study the existence of the approximate solutions. Secondly, a priori estimates are given in the framework of weighted spaces. Finally, we prove the existence, uniqueness and regularity of the very weak solutions.     

Convergence Analysis of the Nonconforming Finite Element Discretization of Stokes and Navier–Stokes Equations with Nonlinear Slip Boundary Conditions

This work is concerned with the nonconforming finite approximations for the Stokes and Navier–Stokes equations driven by slip boundary condition of “friction” type. It is well documented that if the velocity is approximated by the Crouzeix–Raviart element of order one, whereas the discrete pressure is constant elementwise that the inequality of Korn does not hold. Hence, we propose a new formulation taking into account the curvature and the contribution of tangential velocity at the boundary. Using the maximal regularity of the weak solution, we derive a priori error estimates for the velocity and pressure by taking advantage of the enrichment mapping and the application of Babuska–Brezzi’s theory for mixed problems.     

Analogs of Korn’s Inequality on Heisenberg Groups

We give two analogs of Korn’s inequality on Heisenberg groups. First, the norm of the horizontal differential is estimated in terms of the symmetric part of the differential. Second, Korn’s inequality is treated as a coercive estimate for a differential operator whose kernel coincides with the Lie algebra of the isometry group. For this purpose, we construct a differential operator whose kernel coincides with the Lie algebra of the isometry group on Heisenberg groups and prove a coercive estimate for the operator.

     

On Korn’s Inequality

For β ∈ R, the authors consider the evolution system in the unknown variables u and α { ttu＋ xxxxu＋ xxtα＋（β＋｜｜ xu｜｜L2^2） xxu=f, ttα- xxα- xxtα- xxtu=0} describing the dynamics of type III thermoelastic extensible beams, where the dissipation is entirely contributed by the second equation ruling the evolution of the thermal displacement α. Under natural boundary conditions, the existence of the global attractor of optimal regularity for the related dynamical system acting on the phase space of weak energy solutions is established.     

On Korn's Inequality

The author first reviews the classical Korn inequality and its proof. Following recent works of S. Kesavan, P. Ciarlet, Jr., and the author, it is shown how the Korn inequality can be recovered by an entirely different proof. This new proof hinges on appropriate weak versions of the classical Poincare and Saint-Venant lemma. In fine, both proofs essentially depend on a crucial lemma of J. L. Lions, recalled at the beginning of this paper.     

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.     

The asymptotically sharp Korn interpolation and second inequalities for shells

We consider shells in three-dimensional Euclidean space that have bounded principal curvatures. We prove Korn's interpolation (or the so-called first and a half¹) and the second inequalities on that kind of shells for $\mathit{u}\in W^{1,2}$ vector fields, imposing no boundary or normalization conditions on u. The constants in the estimates are optimal in terms of the asymptotics in the shell thickness h, having the scalings h or $O(1)$. The Korn interpolation inequality reduces the problem of deriving any linear Korn type estimate for shells to simply proving a Poincaré-type estimate with the symmetrized gradient on the right-hand side. In particular, this applies to linear geometric rigidity estimates for shells, i.e. Korn's fist inequality without boundary conditions.     

高阶波动方程的时空估计与低能量散射

本文研究了高阶波动方程的低能量散射理论，基本工具是高阶线性波动方程解的时空估计．与经典的二阶波动方程解的时空估计证明不同，我们采用泛函分析的方法与待定指标技巧，首次给出了高阶线性波动方程的时空估计，藉此与非线性函数在齐次Ｓｏｂｏｌｅｖ空间中的估计，获得了高阶波动方程的低能量散射结论．与此同时，也得到了具临界增长的高阶波动方程的柯西问题在低能量条件下的整体存在唯一性．     

Justification of the asymptotic theory of thin rods. Integral and pointwise estimates

We present a general (without any condition on symmetry) and simplified procedure of obtaining onedimensional equations describing strains of thin rods that can be anisotropic, nonhomogeneous and have periodic structure as well. The presented asymptotics is justified with the help of the weighted Korn inequality, i.e., the difference of the exact solution and an asymptotic solution to the problem of elasticity theory is estimated in the energetic integral metric. Uniform (by the maximum of modulus) estimates for the error of approximation of 3-dimensional displacement fields and stresses are also obtained. As is shown, it is impossible to obtain the pointwise closeness with respect to stresses if the influence of the boundary layer near the end-walls of the rod is not taken into account. Bibliography: 44 titles. Translated fromProblemy Matematicheskogo Analiza, No. 17, 1997, pp. 101–152.     

Estimates in the invariance principle in terms of truncated power moments

We obtain estimates for the distributions of errors which arise in approximation of a random polygonal line by a Wiener process on the same probability space. The polygonal line is constructed on the whole axis for sums of independent nonidentically distributed random variables and the distance between it and the Wiener process is taken to be the uniform distance with an increasing weight. All estimates depend explicitly on truncated power moments of the random variables which is an advantage over the earlier estimates of Komlos, Major, and Tusnady where this dependence was implicit.