On some singular perturbation problems in the theory of lubrication

Some singular perturbation problems occurring in the theory of hydrodynamic lubrication are studied. We examine in detail the case of the very long or very short journal bearing; the asymptotic problem of small eccentricity ratio is also considered.     

On some singular perturbation problems arising in stochastic control

We discuss the problem of the existence of periodic and almost periodic solutions in distribution of semilinear stochastic equations on a separable Hilbert space.

Under a dissipativity condition we prove that the translation of the mean square bounded solution is periodic or almost periodic. Similar results hold in the affine case under mean square stability of the linear part of the equation.     

On the numerical integration of a class of singular perturbation problems

A three-point difference scheme recently proposed in Ref. 1 for the numerical solution of a class of linear, singularly perturbed, two-point boundary-value problems is investigated. The scheme is derived from a first-order approximation to the original problem with a small deviating argument. It is shown here that, in the limit, as the deviating argument tends to zero, the difference scheme converges to a one-sided approximation to the original singularly perturbed equation in conservation form. The limiting scheme is shown to be stable on any uniform grid. Therefore, no advantage arises from using the deviating argument, and the most accurate and efficient results are obtained with the deviation at its zero limit.     

On the solvability of periodic boundary value problems with impulse

In this work we obtain some new results concerning the existence of solutions to an impulsive first-order, nonlinear ordinary differential equation with periodic boundary conditions. The ideas involve differential inequalities and Schaefer's fixed-point theorem.     

On the energy of inviscid singular flows

It is known that the energy of a weak solution to the Euler equation is conserved if it is slightly more regular than the Besov space . When the singular set of the solution is (or belongs to) a smooth manifold, we derive various Lp-space regularity criteria dimensionally equivalent to the critical one. In particular, if the singular set is a hypersurface the energy of u is conserved provided the one-sided non-tangential limits to the surface exist and the non-tangential maximal function is L³ integrable, while the maximal function of the pressure is L3/2 integrable. The results directly apply to prove energy conservation of the classical vortex sheets in both 2D and 3D at least in those cases where the energy is finite.     

On the linearization of some singular, nonlinear elliptic problems and applications

This paper deals with the spectrum of a linear, weighted eigenvalue problem associated with a singular, second order, elliptic operator in a bounded domain, with Dirichlet boundary data. In particular, we analyze the existence and uniqueness of principal eigenvalues. As an application, we extend the usual concepts of linearization and Frechet derivability, and the method of sub and supersolutions to some semilinear, singular elliptic problems.     

On sensitive elliptic singular perturbation problems and logarithmic oscillations: the case of shells with edges

This work deals with singular perturbation problems depending on small positive parameter ?. The limit problem as ? → 0 has no solution within the classical theory of PDEs, which uses distribution theory. A very particular and less‐known phenomenon appears: large oscillations. These problems exhibit some kind of instability; very small and smooth variations of the data imply large singular perturbations of the solution. That kind of problems appears in elasticity for highly compressible two‐dimensional bodies and thin shells with elliptic middle surface with a part of the boundary free. Here, we consider certain properties of that oscillations and extend the theory to shells with edges. Copyright © 2012 John Wiley & Sons, Ltd.     

On some problems in perturbation theory of smooth invariant tori of dynamical systems

Problems related to perturbation theory of smooth invariant tori of dynamical systems in an-dimensional Euclidean spaceR ⁿ are considered. The clarification of these problems plays an important role for perturbation theory suggested by the author in [1] and extends the scope of its application.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 12, pp. 1665–1699, December, 1994.     

On sampling theory associated with the resolvents of singular Sturm-Liouville problems

This paper is concerned with the sampling theory associated with resolvents of eigenvalue problems. We introduce sampling representations for integral transforms whose kernels are Green's functions of singular Sturm-Liouville problems provided that the singular points are in the limit-circle situation, extending the results obtained in the regular problems.

     

Smoothness of transition maps in singular perturbation problems with one fast variable

This paper deals with the smoothness of the transition map between two sections transverse to the fast flow of a singularly perturbed vector field (one fast, multiple slow directions). Orbits connecting both sections are canard orbits, i.e. they first move rapidly towards the attracting part of a critical surface, then travel a distance near this critical surface, even beyond the point where the orbit enters the repelling part of the critical surface, and finally repel away from the surface. We prove that the transition map is smooth. In a transcritical situation however, where orbits from an attracting part of one critical manifold follow the repelling part of another critical manifold, the smoothness of the transition map may be limited, due to resonance phenomena that are revealed by blowing up the turning point! We present a polynomial example in R³.     

Unbounded components of the singular set of the distance function in

Given a closed set , the set of all points at which the metric projection onto is multi-valued is nonempty if and only if is nonconvex. The authors analyze such a set, characterizing the unbounded connected components of . For compact, the existence of an asymptote for any unbounded component of is obtained.

     

Minimal distance upper bounds for the perturbation of least squares problems in Hilbert spaces

Let X and Y be Hilbert spaces, and let T : X → Y be a bounded linear operator with closed range. In this paper, we present an optimal perturbation result on the least squares solutions to the operator equation Tx = y under the most general condition.     

On the monotonicity of some singular integral operators

The paper deals with the monotonicity of singular integral operators of the form where is the Cauchy singular integral operator on the interval (0,1) of the real axis and q is a power or logarithmic function. Under suitable assumptions, such singular integral operators are proved to be monotone and maximal monotone in spaces with power weights. Moreover, two related integral equations with weakly singular kernels of logarithmic type are studied. Copyright © 2012 John Wiley & Sons, Ltd.     

On the order of singular optimal control problems

In singular optimal control problems, the functional form of the optimal control function is usually determined by solving the algebraic equation which results by successively differentiating the switching function until the control appears explicitly. This process defines the order of the singular problem. Order-related results are developed for singular linear-quadratic problems and for a bilinear example which gives new insights into the relationship between singular problem order and singular are order.Dedicated to R. BellmanThis work was supported by the National Science Foundation under Grant No. ENG-77-16660.     

Existence of solutions to the nonlinear,singular second order Bohr boundary value problems

This paper investigates the existence of solutions for nonlinear systems of second order, singular boundary value problems (BVPs) with Bohr boundary conditions. A key application that arises from this theory is the famous Thomas–Fermi equations for the model of the atom when it is in a neutral state. The methodology in this paper uses an alternative and equivalent BVP, which is in the class of resonant singular BVPs, and thus this paper obtains novel results by implementing an innovative differential inequality, Lyapunov functions and topological techniques. This approach furnishes new results in the area of singular BVPs for a priori bounds and existence of solutions, where the BVP has unrestricted growth conditions and subject to the Bohr boundary conditions. In addition, the results can be relaxed and hold for the non-singular case too.     

On the choice of the function spaces for some state-constrained control problems

We study a quadratic parabolic control problem with pointwise final state constraints. As the set of admissible states has an empty interior, the existence of Lagrange multipliers cannot be proved directly. We obtain, however some optimality conditions by expressing the fact that among a space of regular perturbations of the optimal control, the null perturbation is optimal. We show that the qualification hypothesis can be effectively checked in some examples and that the information given by the optimality conditions is useful because it allows to get some regularity results for the optimal control.     

On the asymptotic distribution of the eigenvalues of singular Sturm-Liouville problems with an indefinite weight function

We investigate spectrum and asymptotic distribution of the eigenvalues of certain singular Sturm-Liouville eigenvalue problems on the half-line having a weight function that changes sign.