The stability of the equilibrium of holonomic conservative systems

The problem of the stability of the equilibrium positions and steady motions of holonomic conservative systems has been fairly completely treated in a number of reviews [44,58,9]. However, investigations are continuing in this field and a number of new important results have recently been obtained (in 1982–1992). This review analyses these results and compares them with previous ones.     

Interface problem in holonomic elastoplasticity

The three-dimensional interface problem with the homogeneous Lamé system in an unbounded exterior domain and holonomic material behaviour in a bounded interior Lipschitz domain is considered. Existence and uniqueness of solutions of the interface problem are obtained rewriting the exterior problem in terms of boundary integral operators following the symmetric coupling procedure. The numerical approximation of the solutions consists in coupling of the boundary element method (BEM) and the finite element method (FEM). A Céa-like error estimate is presented for the discrete solutions of the numerical procedure proving its convergence.     

Finite element analysis of a holonomic elastic-plastic problem

Summary A thorough analysis of the finite element method is given for a holonomic elastic-plastic problem. An inequality of the Cea's lemma type is proved, which is the basis of error estimates for various finite element solutions. Difficulty caused by a non-differentiable term in the problem can be overcome by using two convergent procedures, an iterative procedure and a regularization procedure. An a-posteriori quantitative error estimate is derived for the regularized solution.The work was done while the author was at the Department of Mathematics, University of Maryland, College Park. The research was partially supported by the National Science Foundation grant CCR-88-20279     

A stability estimate for an elliptic problem

We investigate the Caucy problem for linear elliptic operators withC ^∞-coefficients at a regular domain ℝ ⊂ ℝ, which is a classical example of an ill-posed problem. The Cauchy data are given at the manifold Γ⊂∂Ω and our goal is to obtain a stability estimate inH ₄(Ω).     

A two-point problem for first-order systems

Let f be a continuous function from [a, b] ×\mathbbRⁿ [a, b] \times \mathbb{R}^n into \mathbbRⁿ \mathbb{R}^n . In this paper we prove that the problem¶¶ { llu^￠ = f(t,u)+ lu(a)=u(b)=0  \left \{ \begin{array}{ll}u^{\prime}= f(t,u)+ \lambda \\[3pt]u(a)=u(b)=0\end{array}\right.\ ¶¶ has a (classical) solution for a wide class of functions f. Next we point out a particular case.     

A unified approach to the modelling of holonomic and nonholonomic mechanical systems

An alternative technique, called projection method, for solving constrained system problems is presented. This approach can be used to derive equations of motion of both holonomic and nonholonomic systems, and the dynamic equations can be expressed in generalized velocities and/or quasi-velocities. Compared against the other methods of classical mechanics (Lagrange's, Gibbs-Appell, Kane's,...), the present method turns out to be extraordinarily short, elementary and general. As such, it deserves to be promoted as a generally accepted method in academic and engineering applications. Three examples are reported to illustrate advantages of the technique     

A special stability problem for linear multistep methods

The trapezoidal formula has the smallest truncation error among all linear multistep methods with a certain stability property. For this method error bounds are derived which are valid under rather general conditions. In order to make sure that the error remains bounded ast , even though the product of the Lipschitz constant and the step-size is quite large, one needs not to assume much more than that the integral curve is uniformly asymptotically stable in the sense of Liapunov.The preparation of this paper was partly sponsored by the Office of Naval Research and the US Army Research Office (Durham). Reproduction in whole or in part is permitted for any purpose of the US Government.     

A stability result for the magnetic shaping problem

We consider the bidimensional magnetic shaping problem without surface tension and study its stability when the boundary of the domain has cusp points. We show in particular that one has stability when the curvature of the smooth parts of is negative.     

A Cousin problem for systems of PDE

This article formulates an additive Cousin type problem for systems of partial differential equations. The obstruction to the solution of this problem, like the obstruction in the classical Cousin problem, lies in a first cohomology module, viz. in an Ext¹ module associated to the problem. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A contribution to stability theory for nonlinear Neumann problem

In this paper we study the stability of the solutions of some nonlinear Neumann problems, under perturbations of the domains in the Hausdorff complementary topology. We consider the problem $${{\left\{\begin{array}{c}-\text{ div}\;\left(a\left( x,\nabla u_{\Omega}\right)\right)=0 \;\text{in}\; \Omega \\ {a\left( x, \nabla u_{\Omega}\right) \cdot \nu=0\; \text{on}\; \partial\Omega}\end{array}\right.}}$$ where ${{\mathbf{R}^n \times \mathbf{R}^n \rightarrow \mathbf{R}^n}}$ is a Caratheodory function satisfying the standard monotonicity and growth conditions of order p, 1?<?p?<???. If ?? _h is a uniformly bounded sequence of connected open sets in R ⁿ , n ??? 2, we prove that if ${{\Omega_{h}^{c} \rightarrow \Omega^{c}}}$ in the Hausdorff metric, ${|\Omega_{h}| \rightarrow |\Omega|}$ and the geodetic distances satisfy the inequality ${d_{\Omega}\left( x,y\right) \leq \liminf_{h} d_{\Omega_{h}} \left( x,y\right)}$ for every ${x, y \in \Omega,}$ then ${\nabla u_{\Omega_h} \rightarrow\nabla u_{\Omega}}$ strongly in L ^p , provided that W ^{1, ??}(??) is dense in the space L ^{1, p} (??) of all functions whose gradient belongs to L ^p (??, R ⁿ ).     

Nonautonomous differential systems: several functions in the stability problem

Results of asymptotic stability and instability for nonautonomous differential systems with unbounded right hand side are obtained by using several auxiliary functions and topological properties of the Ω-limit set of solutions. The theorem established is applied to prove the asymptotic stability of the stationary solution of the Lienard equation with unbounded dampings.