Existence and multiplicity results for hemivariational inequalities with two parameters

A multiplicity result of Ricceri, based on a minimax inequality, is applied to locally Lipschitz functionals and the existence of one or two non-trivial solutions for a class of two-parameter hemivariational inequalities is proved. An application is given to elliptic problems with discontinuous nonlinearities on unbounded domains.     

Quantitative stability analysis of optimal solutions in PDE-constrained optimization

PDE-constrained optimization problems under the influence of perturbation parameters are considered. A quantitative stability analysis for local optimal solutions is performed. The perturbation directions of greatest impact on an observed quantity are characterized using the singular value decomposition of a certain linear operator. An efficient numerical method is proposed to compute a partial singular value decomposition for discretized problems, with an emphasis on infinite-dimensional parameter and observation spaces. Numerical examples are provided.     

Polynomial hulls and an optimization problem

We say that a subset of Cⁿ is hypoconvex if its complement is the union of complex hyperplanes. We say it is strictly hypoconvex if it is smoothly bounded hypoconvex and at every point of the boundary the real Hessian of its defining function is positive definite on the complex tangent space at that point. Let B_n be the open unit ball in Cⁿ.Suppose K is a C^∞ compact manifold in ∂B₁ × Cⁿ, n > 1, diffeomorphic to ∂B₁ × ∂B_n, each of whose fibers K_z over ∂B₁ bounds a strictly hypoconvex connected open set. Let K be the polynomialhull of K. Then we show that K∖K is the union of graphs of analytic vector valued functions on B₁. This result shows that an unnatural assumption regarding the deformability of K in an earlier version of this result is unnecessary. Next, we study an H^∞ optimization problem. If pis a C^∞ real-valued function on ∂B₁× Cⁿ, we show that the infimum γ_ρ = inf_ƒ∈H ^∞ (B₁)ⁿ ‖ρ(z, ƒ (z))‖_∞ is attained by a unique bounded ƒ provided that the set (z, w) ∈ ∂B₁ × C ⁿ|ρ(z, w) ≤ γ_ρ has bounded connected strictly hypoconvex fibers over the circle.     

Variational Principles for Eigenvalues of Self-adjoint Operator Functions

Variational principles for eigenvalues of certain functions whose values are possibly unbounded self-adjoint operators T() are proved. A generalised Rayleigh functional is used that assigns to a vector x a zero of the function T()x, x), where it is assumed that there exists at most one zero. Since there need not exist a zero for all x, an index shift may occur. Using this variational principle, eigenvalues of linear and quadratic polynomials and eigenvalues of block operator matrices in a gap of the essential spectrum are characterised. Moreover, applications are given to an elliptic eigenvalue problem with degenerate weight, Dirac operators, strings in a medium with a viscous friction, and a Sturm-Liouville problem that is rational in the eigenvalue parameter.     

Convex Minimization over the Fixed Point Set of Demicontractive Mappings

This paper deals with a viscosity iteration method, in a real Hilbert space , for minimizing a convex function over the fixed point set of , a mapping in the class of demicontractive operators, including the classes of quasi-nonexpansive and strictly pseudocontractive operators. The considered algorithm is written as: x _n+1 := (1 − w) v _n + w T v _n , v _n := x _n − α _n Θ′(x _n ), where w ∈ (0,1) and , Θ′ is the Gateaux derivative of Θ. Under classical conditions on T, Θ, Θ′ and the parameters, we prove that the sequence (x _n ) generated, with an arbitrary , by this scheme converges strongly to some element in Argmin _Fix(T) Θ.      

Differential form methods in the theory of variational systems and Lagrangian field theories

This paper will attempt to unify diverse material from physics and engineering in terms of differential forms on manifolds. A variational system will be defined by means of a scalar-valued differential form on a manifold and an ideal in the Grassmann algebra of differential forms on that manifold to serve as constraints. Two types of extremal submanifolds will be defined. The first-called the Euler-Lagrange extremals-will be defined by a method that is the generalization of the classical methods in the calculus of variations. The second—a generalization of a method used by Cartan in his treatise Leçons sur les invariants intégraux-will define extremals as integral submanifolds of an exterior differential system invariently attached to the variational system. As examples, the variational systems attached to string theories in Riemannian manifolds and Yang-Mills fields will be discussed from this differential form point of view. Finally, as application, the differential geometric properties and definition of energy will be presented from the differential form point of view.This work was supported by a grant from the Applied Mathematics program of the National Science Foundation.     

An Extended Beam Theory for Smart Materials Applications Part II: Static Formation Problems

In this paper we further develop the theory of the extended Timoshenko beam model, as first introduced in Part I [5] of this work, with particular emphasis on applications of the model in formation theory [11], [12]. We begin with formal development of the equilibrium equations of static formation theory in the context of the extended Timoshenko model, giving a rigorous discussion of existence, uniqueness, and regularity of weak solutions with appropriate assumptions on the coefficients. We continue to obtain the fundamental duality relationship in the context of weak solutions and indicate its usefulness in investigations of approximate formability. Optimal formation problems and corresponding necessary conditions for optimality are discussed. We conclude with a discussion of a particular problem of joint optimization of controls and actuator densities in the context of a prismatic extended Timoshenko beam and we present the results of some computational studies. Accepted 14 November 1996     

Linearly perturbed polyhedral normal cone mappings and applications

Under a mild regularity assumption, we derive an exact formula for the Fréchet coderivative and some estimates for the Mordukhovich coderivative of the normal cone mappings of perturbed polyhedra in reflexive Banach spaces. Our focus point is a positive linear independence condition, which is a relaxed form of the linear independence condition employed recently by Henrion et al. (2010) [1], and Nam (2010) [3]. The formulae obtained allow us to get new results on solution stability of affine variational inequalities under linear perturbations. Thus, our paper develops some aspects of the work of Henrion et al. (2010) [1] Nam (2010) [3] Qui (in press) [12] and Yao and Yen (2009) [6] and [7].     

Commutability of homogenization and linearization at identity in finite elasticity and applications

We prove under some general assumptions on elastic energy densities (namely, frame indifference, minimality at identity, non-degeneracy and existence of a quadratic expansion at identity) that homogenization and linearization commute at identity. This generalizes a recent result by S. Müller and the second author by dropping their assumption of periodicity. As a first application, we extend their Γ-convergence commutation diagram for linearization and homogenization to the stochastic setting under standard growth conditions. As a second application, we prove that the Γ-closure is local at identity for this class of energy densities.