Interpolation error estimates in W 1,p for degenerate Q 1 isoparametric elements

Optimal order error estimates in H ¹, for the Q ₁ isoparametric interpolation were obtained in Acosta and Durán (SIAM J Numer Anal37, 18–36, 1999) for a very general class of degenerate convex quadrilateral elements. In this work we show that the same conlusions are valid in W ^1,p for 1≤ p < 3 and we give a counterexample for the case p ≥ 3, showing that the result cannot be generalized for more regular functions. Despite this fact, we show that optimal order error estimates are valid for any p ≥ 1, keeping the interior angles of the element bounded away from 0 and π, independently of the aspect ratio. We also show that the restriction on the maximum angle is sharp for p ≥ 3.     

Constraint preconditioning for linear systems in elasticity

The mixed variational formulation of elastic deformation of a material body leads to a constraint minimisation (or saddle‐point) problem. Using the PEERS finite element space to discretise the problem one gets an indefinite system of linear equations. To solve this system, preconditioners with similar saddle‐point patterns will be applied. The results of numerical tests with this preconditioning strategy show its efficiency for linear systems with up to more than half a million unknowns.     

Resolvent estimates in homogenisation of periodic problems of fractional elasticity

We provide operator-norm convergence estimates for solutions to a time-dependent equation of fractional elasticity in one spatial dimension, with rapidly oscillating coefficients that represent the material properties of a viscoelastic composite medium. Assuming periodicity in the coefficients, we prove operator-norm convergence estimates for an operator fibre decomposition obtained by applying to the original fractional elasticity problem the Fourier–Laplace transform in time and Gelfand transform in space. We obtain estimates on each fibre that are uniform in the quasimomentum of the decomposition and in the period of oscillations of the coefficients as well as quadratic with respect to the spectral variable. On the basis of these uniform estimates we derive operator-norm-type convergence estimates for the original fractional elasticity problem, for a class of sufficiently smooth densities of applied forces.     

Quadratic control for linear periodic systems

We propose a new quadratic control problem for linear periodic systems which can be finite or infinite dimensional. We consider both deterministic and stochastic cases. It is a generalization of average cost criterion, which is usually considered for time-invariant systems. We give sufficient conditions for the existence of periodic solutions.Under stabilizability and detectability conditions we show that the optimal control is given by a periodic feedback which involves the periodic solution of a Riccati equation. The optimal closed-loop system has a unique periodic solution which is globally exponentially asymptotically stable. In the stochastic case we also consider the quadratic problem under partial observation. Under two sets of stabilizability and detectability conditions we obtain the separation principle. The filter equation is not periodic, but we show that it can be effectively replaced by a periodic filter. The theory is illustrated by simple examples.This work was done while this author was a visiting professor at the Scuola Normale Superiore, Pisa.     

Uniform saturation in linear inequality systems

Redundant constraints in linear inequality systems can be characterized as those inequalities that can be removed from an arbitrary linear optimization problem posed on its solution set without modifying its value and its optimal set. A constraint is saturated in a given linear optimization problem when it is binding at the optimal set. Saturation is a property related with the preservation of the value and the optimal set under the elimination of the given constraint, phenomena which can be seen as weaker forms of excess information in linear optimization problems. We say that an inequality of a given linear inequality system is uniformly saturated when it is saturated for any solvable linear optimization problem posed on its solution set. This paper characterizes the uniform saturated inequalities and other related classes of inequalities. This work was supported by the MCYT of Spain and FEDER of UE, Grant BFM2002-04114-C02-01.     

Residual-based a posteriori error estimates for symmetric conforming mixed finite elements for linear elasticity problems

A posteriori error estimators for the symmetric mixed finite element methods for linear elasticity problems with Dirichlet and mixed boundary conditions are proposed. Reliability and efficiency of the estimators are proved. Finally, numerical examples are presented to verify the theoretical results.     

Stability criteria for linear periodic Hamiltonian systems

In this paper, we obtain new stability criteria for linear periodic Hamiltonian systems. A Lyapunov type inequality is established. Our results improve the existing works in the literature.     

Uniform persistence of the periodic predator-prey lotka-volterra systems

In this paper we give the sufficient and necessary conditions of the uniform persistence for periodic predator-prey Lotka- Volterra systems. That is, the system is uniformly persistent if and only if any one of the semi-trivial periodic solutions is linearly unstable.     

Error estimates for certain approximate solutions of problems in the linear theory of elasticity

Zusammenfassung Eine Näherungslösung eines linearen Elastizitätsproblems kann manchmal erreicht werden mittels einer (kleinen) Änderung des Ausdrucks für die elastische Energie. Obere Schranken für die Fehler einer solchen Näherungslösung werden abgeleitet. Das Ergebnis hat auch Bedeutung vom grundsätzlichen Gesichtspunkt, weil die «exakte» Form des Energieausdrucks für wirkliche elastische Systeme niemals genau bekannt ist.

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Dedicated to Professor Dr. Hans Ziegler on the occasion of his 60th birthday     

Stability estimates of linear stochastic systems

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 9, pp. 1267–1273, September, 1989.     

Stability of linear almost-Hamiltonian periodic systems

A linear Hamiltonian system with periodic coefficients is subject to a small “dissipative” perturbation that makes it asymptotically stable. The conditions under which the perturbation remains dissipative for all Hamiltonian systems sufficiently close to the original one are discussed.     

Stability criteria for linear periodic impulsive Hamiltonian systems

In this paper we obtain stability criteria for linear periodic impulsive Hamiltonian systems. A Lyapunov type inequality is established. Our results improve also the ones previously obtained for systems without impulse effect.     

Resolvent estimates in W−1,p related to strongly coupled‐linear parabolic systems with coupled nonsmooth capacities

We investigate linear parabolic systems with coupled nonsmooth capacities and mixed boundary conditions. We prove generalized resolvent estimates in W^{?1, p} spaces. The method is an appropriate modification of a technique introduced by Agmon to obtain L^p estimates for resolvents of elliptic differential operators in the case of smooth boundary conditions. Moreover, we establish an existence and uniqueness result. Copyright © 2007 John Wiley & Sons, Ltd.     

Gradient estimates below duality exponent for a class of linear elliptic systems

We provide sharp estimates in Lorentz spaces for the solution of the Dirichlet problem associated to the system $\left\{ \begin{array}{ll} A(u)\equiv-D_i (A_{ij}(x) D_j u)=f\\ u \in W^{1,1}_{0}(\Omega, \mathbb {R}^N) \end{array} \right.$ where Ω is an open bounded subset of ${\mathbb R^n}We provide sharp estimates in Lorentz spaces for the solution of the Dirichlet problem associated to the system

{ ll A(u) o -Di (Aij(x) Dj u)=fu ? W1,10(W, \mathbb RN) \left\{ \begin{array}{ll} A(u)\equiv-D_i (A_{ij}(x) D_j u)=f\\ u \in W^{1,1}_{0}(\Omega, \mathbb {R}^N) \end{array} \right.      20. W 2,p -a priori estimates for the emergent Poincaré Problem      Dian K. Palagachev 《Journal of Global Optimization》2008,40(1-3):305-318 We derive W 2,p (Ω)-a priori estimates with arbitrary p ∈(1, ∞), for the solutions of a degenerate oblique derivative problem for linear uniformly elliptic operators with low regular coefficients. The boundary operator is given in terms of directional derivative with respect to a vector field ℓ that is tangent to ∂Ω at the points of a non-empty set ε ⊂ ∂Ω and is of emergent type on ∂Ω.

W 2,p -a priori estimates for the emergent Poincaré Problem

We derive W ^2,p (Ω)-a priori estimates with arbitrary p ∈(1, ∞), for the solutions of a degenerate oblique derivative problem for linear uniformly elliptic operators with low regular coefficients. The boundary operator is given in terms of directional derivative with respect to a vector field ℓ that is tangent to ∂Ω at the points of a non-empty set ε ⊂ ∂Ω and is of emergent type on ∂Ω.