On the variational formulation for linear initial value problems

Summary The paper shows the key place of the choice of the bilinear form in order to give a variational formulation to a given problem. In particular it is shown how the use of a convolution bilinear form makes possible a variational formulation for linear initial value problems. A critical survey of the three main methods that was devised in the past to solve the same problem is done.

---

Sunto. La nota mette in evidenza il ruolo fondamentale della scelta di una forma bilineare al fine di dare formulazione variazionale ad un dato problema. In particolare è mostrato come l'uso di una forma bilineare di convoluzione renda possibile la formulazione variazionale dei problemi ai valori iniziali. Si fa un esame critico dei tre principali metodi che sono stati escogitati nel passato per risolvere lo stesso problema.

---

---

This work has been sponsored by the Consiglio Nazionale delle Ricerche.

---

Entrata in Redazione il 19 novembre 1971.     

On a new class of boundary value problems for the Sturm-Liouville operator

We consider the spectral problem generated by the Sturm-Liouville operator with complex-valued potential q(x) ∈ L ₂(0, π) and degenerate boundary conditions. We show that the set of potentials q(x) for which there exist associated functions of arbitrarily high order in the system of root functions is everywhere dense in L ₁(0, π).     

On the simplex method and a class of linear complementary problems

We consider the linear complementarity problem of finding vectors w?Rⁿ, z?Rⁿ satisfying w ? Mz = q, w ? 0, z ? 0, w^Tz = 0. We show that if the off diagonal elements of M are nonpositive, then the above problem is solved by applying the simplex method to the problem Minimize z₀ subject to w ? Mz ? e_nz₀ = q, (z₀, w, z) ? 0, where e_n is a column vector of 1's. In fact the sequence of basic feasible solutions obtained by the simplex method and by Lemke's algorithm are the same. We also obtain necessary and sufficient conditions for the problem to have solutions for all q.     

On a class of boundary value problems involving the p-biharmonic operator

A nonlinear boundary value problem involving the p-biharmonic operator is investigated, where p>1. It describes various problems in the theory of elasticity, e.g., the shape of an elastic beam where the bending moment depends on the curvature as a power function with exponent p−1. We prove existence of solutions satisfying a quite general boundary condition that incorporates many particular boundary conditions which are frequently considered in the literature.     

On a class of operator equations arising in infinite dimensional replicator dynamics

We introduce a class of infinite dimensional replicator dynamics for the study of equilibrium selection in non-cooperative games and study their properties. Examples from economic theory are provided.     

On the number of solutions to a class of linear complementarity problems

In this note, we consider the linear complementarity problemw = Mz + q, w 0, z 0, w ^T z = 0, when all principal minors ofM are negative. We show that for such a problem for anyq, there are either 0, 1, 2, or 3 solutions. Also, a set of sufficiency conditions for uniqueness is stated.The work of both authors is partially supported by a grant from the National Science Foundation, MCS 77-03472.     

A coupled far-field formulation for time-periodic numerical problems in fluid dynamics

Consider uniform flow past an oscillating body generating a time-periodic motion in an exterior domain, modelled by a numerical fluid dynamics solver in the near field around the body. A far-field formulation, based on the Oseen equations, is presented for coupling onto this domain thereby enabling the whole space to be modelled. In particular, examples for formulations by boundary elements and infinite elements are described.     

On spectra of a certain class of quadratic operator pencils with one-dimensional linear part

We consider a class of quadratic operator pencils that occur in many problems of physics. The part of such a pencil linear with respect to the spectral parameter describes viscous friction in problems of small vibrations of strings and beams. Patterns in the location of eigenvalues of such pencils are established. If viscous friction (damping) is pointwise, then the operator in the linear part of the pencil is one-dimensional. For this case, rules in the location of purely imaginary eigenvalues are found. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 5, pp. 702–716, May, 2007.     

On linear extremal problems in the class of schlicht functions

The form of the Schiffer differential equation puts severe restrictions on the class of functions that can occur as extremal functions for arbitrary coefficient-functionals of finite degree. Theorem 1 characterizes the algebraic extremal functions for coefficient-functionals of finite degree. Furthermore it is shown that the extremal function either is an algebraic function or it must possess a non-isolated singularity, or must have a transcendental branch-point. The results are closely related to the Malmquist-Yosida theorems. However Nevanlinna's Theory of Value Distribution is not the mainly used tool but the special form of the Schiffer differential equation and the multiplicity of certain values together with the Great Picard Theorem are exploited.     

A class of operator equilibrium problems

In this paper, we consider a class of operator equilibrium problems (OEP for short) with operator solutions and derive a Minty type lemma for this class of problems. Further, using this lemma and KKM theorem, we establish some existence theorems for OEP. The theorems presented in this paper generalize, improve and unify many known results.     

On estimates of the spectrum of a linear operator

The paper is concerned with new approaches to the analysis of spectra of linear operators. New algorithms are proposed to calculate the coefficients of the minimal polynomial of a matrix; they are based on the well-known Krylov’s method, SSA decomposition, and the “Caterpillar” method of recurrent translation. The extension obtained is capable of dealing with matrices of infinite order; this has great value in solving queuing problems. Results from numerical experiments for matrices of various orders are given.     

On a class of nonlinear problems involving a p(x)-Laplace type operator

We study the boundary value problem in Ω, u = 0 on ∂Ω, where Ω is a smooth bounded domain in ℝ ^N . Our attention is focused on two cases when , where m(x) = max{p ₁(x), p ₂(x)} for any x ∈ or m(x) < q(x) < N · m(x)/(N − m(x)) for any x ∈ . In the former case we show the existence of infinitely many weak solutions for any λ > 0. In the latter we prove that if λ is large enough then there exists a nontrivial weak solution. Our approach relies on the variable exponent theory of generalized Lebesgue-Sobolev spaces, combined with a ℤ₂-symmetric version for even functionals of the Mountain Pass Theorem and some adequate variational methods.     

On continuous dependence for a class of dynamical problems in linear thermoelasticity

Given that the temperature dependent elasticities of a linear anisotropic elastic material satisfy certain Lipschitz type conditions, it is shown that the displacement vector depends continuously, in an appropriate norm, on temperature deviations from a fixed but arbitrary steady state temperature distribution.     

On a certain class of operator equations with small parameter and regularization of ill-posed problems

The research was financially supported by the International Science Foundation (Grant NMW000).     

Boundary value problems for a class of elliptic operator pencils

In this paper operator pencilsA(x, D, ) are studied which act on a manifold with boundary and satisfy the condition of N-ellipticity with parameter, a generalization of the notion of ellipticity with parameter as introduced by Agmon and Agranovich-Vishik. Sobolev spaces corresponding to the Newton polygon are defined and investigated; in particular it is possible to describe their trace spaces. With respect to these spaces, an a priori estimate is proved for the Dirichlet boundary value problem connected with an N-elliptic pencil.Supported in part by the Deutsche Forschungsgemeinschaft and by Russian Foundation of Fundamental Research, Grant 00-01-00387.     

Approximate variational-iterative solutions for a class of nonlinear operator problems

A general theory for the approximate solution of nonlinear operator equations where the solution lies in some real or complex Hilbert space is developed. It is shown that the techniques can be applied to a variety of problems taken from the theories of plasma physics, colloid theory, and the theory of current flow in nerve membranes.     

Preconditioning a class of fourth order problems by operator splitting

We develop preconditioners for systems arising from finite element discretizations of parabolic problems which are fourth order in space. We consider boundary conditions which yield a natural splitting of the discretized fourth order operator into two (discrete) linear second order elliptic operators, and exploit this property in designing the preconditioners. The underlying idea is that efficient methods and software to solve second order problems with optimal computational effort are widely available. We propose symmetric and non-symmetric preconditioners, along with theory and numerical experiments. They both document crucial properties of the preconditioners as well as their practical performance. It is important to note that we neither need H ^s -regularity, s > 1, of the continuous problem nor quasi-uniform grids.