Multi-peak solutions for some singular perturbation problems

We consider the problem where is a smooth domain in , not necessarily bounded, is a small parameter and f is a superlinear, subcritical nonlinearity. It is known that this equation possesses a solution that concentrates, as approaches zero, at a maximum of the function , the distance to the boundary. We obtain multi-peak solutions of the equation given above when the domain presents a distance function to its boundary d with multiple local maxima. We find solutions exhibiting concentration at any prescribed finite set of local maxima, possibly degenerate, of d. The proof relies on variational arguments, where a penalization-type method is used together with sharp estimates of the critical values of the appropriate functional. Our main theorem extends earlier results, including the single peak case. We allow a degenerate distance function and a more general nonlinearity. Received September 3, 1998 / Accepted February 29, 1999     

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.     

A note on a parameterized singular perturbation problem

We consider a uniform finite difference method on Shishkin mesh for a quasilinear first-order singularly perturbed boundary value problem (BVP) depending on a parameter. We prove that the method is first-order convergent except for a logarithmic factor, in the discrete maximum norm, independently of the perturbation parameter. Numerical experiments support these theoretical results.     

A review on some contributions to perturbation theory, singular limits and well-posedness

In the beginning of the 1990s we devoted a sequence of papers to perturbation theory, singular limits and well-posedness problems. In particular, the strong well-posedness of the initial-boundary value problem for the compressible Euler equations was demonstrate for the first time. Our method also allowed singular limit results in the strong norm, even under assumptions weaker than the current ones in the literature (where the strong norm is not reached). It is worth noting that, until now, the above method and results have not been substantially improved. Hence an introduction to it still looks timely. Actually, in a forthcoming paper, by returning to this method, we improve (in a very substantial way) some important results recently appeared in the literature.     

On a class of singular boundary value problems with singular perturbation

In this paper we investigate a class of singular second order differential equations with singular perturbation subject to three-point boundary value conditions, whose solution exhibits a couple of boundary layers at two endpoints. We first establish a lower–upper solutions theorem by using the Schauder fixed point theorem. By the asymptotic expansions and the lower–upper solutions theorem we obtain the existence, asymptotic estimates and uniqueness for the proposed problem. Several examples are given for illustrating our results.     

A note on the perturbation of singular values

In this note a variant of the classical perturbation theorem for singular values is given. The bounds explain why perturbations will tend to increase rather than decrease singular values of the same order of magnitude as the perturbation.     

A family of singular semi-classical functionals

The distributional equation of a semi-classical linear functional allows us an efficient study of other characterizations and properties of the semi-classical OPS/functionals. In [5] an extensive survey of this approach has been presented. A particular case of semi-classical OPS/functionals are the classical ones. For the distributional equation of a classical functional a regularity condition holds. For compatibility reasons in [5] it is assumed that this condition holds also for all semi-classical functionals. Here we give a counterexample, and we show that the regularity condition does not hold in general for semi-classical functionals.     

Results on the relative perturbation of the singular values of a matrix

A highly accurate computation of the singular values of a matrix is a topic of current interest in the literature. In this paper we develop general bounds on relative perturbation of singular values. These bounds permit slight improvements in a unified derivation of some previous inequalities. The main result is a better criterion to neglect off-diagonal elements in the bidiagonal singular values decomposition.The present paper has been developed under the M. U. R. S. T. 40% National Program and the Interuniversity Center of Numerical Analysis and Computational Mathematics.     

Remarks on elliptic singular perturbation problems

We show the effectiveness of viscosity-solution methods in asymptotic problems for second-order elliptic partial differential equations (PDEs) with a small parameter. Our stress here is on the point that the methods, based on stability results [3], [16], apply without hard PDE calculations. We treat two examples from [11] and [23]. Moreover, we generalize the results to those for Hamilton—Jacobi—Bellman equations with a small parameter.H. Ishii was supported in part by the AFOSR under Grant No. AFOSR 85-0315 and the Division of Applied Mathematics, Brown University.     

On a singular perturbation problem in magnetohydrodynamics

Zusammenfassung Die Strömung einer zähen, inkompressiblen und elektrisch leitenden Flüssigkeit über einen rotationssymmetrischen Körper wird studiert mit Hilfe einer singulären Methode der Störungsrechnung. Eine asymptotische, im ganzen Strömungsfeld gültige Lösung wird gegeben für grosse Hartmann-ZahlenM.Die Resultate ergeben folgendes Strömungsbild: Zwei Totwasser-Bereiche von der LängeO (M) und der BreiteO (1) werden vor und nach dem Körper geformt. Sie sind begrenzt durch eine zylindrische Schubschicht, die vom grössten Durchmesser des Körpers aus parabolisch stromaufwärts und stromabwärts anwächst. In einer Entfernung der GrössenordnungO (M) geht diese Schubschicht in eine Wirbelstrasse über, die sich parabolisch ins Unendliche erstreckt. Die Einzelheiten des Strömungsbildes werden analytisch aufgezeigt. Die Wirbelstrasse wird mit derjenigen der klassischen Navier-Stokes-Theorie verglichen.     

Multigrid convergence for a singular perturbation problem

So far there has been no analysis of multigrid methods applied to singularly perturbed Dirichlet boundary-value problems. Only for periodic boundary conditions does the Fourier transformation (mode analysis) apply, and it is not obvious that the convergence results carry over to the Dirichlet case, since the eigenfunctions are quite different in the two cases. In this paper we prove a close relationship between multigrid convergence for the easily analysable case of periodic conditions and the convergence for the Dirichlet case.     

Existence of singular extremals and singular functionals in reachable spaces

Given a linear, infinite dimensional control system with point target and "full" control we show that singular extremals for the minimum norm problem exist except in certain exceptional cases ("singular" means "not satisfying Pontryagin's maximum principle"). Existence of singular extremals implies existence of certain functionals (also called singular) in the space of reachable states. Received March 5, 2001; accepted April 10, 2001.     

Iterative solution of a singular convection-diffusion perturbation problem

An iterative domain decomposition method is developed to solve a singular perturbation problem. The problem consists of a convection-diffusion equation with a discontinuous (piecewise-constant) diffusion coefficient, and the problem domain is decomposed into two subdomains, on each of which the coefficient is constant. After showing that the boundary value problem is well posed, we indicate a specific numerical implementation of the iterative technique that combines the finite element method on one subdomain with the method of matched asymptotic expansions on the other subdomain. This procedure extends work by Carlenzoli and Quarteroni, which was originally intended for a boundary layer problem with an outer region and an inner region. Our extension carries over to a problem where the domain consists of the outer and inner boundary layer regions plus a region in which the diffusion coefficient is constant and significant in magnitude. An unexpected benefit of our new implementation is its efficiency, which is due to the fact that at each iteration the problem needs to be solved explicitly only on one subdomain. It is only when the final approximation on the entire domain is desired that the matched asymptotic expansions approximation need be computed on the second subdomain. Two-dimensional convergence results and numerical results illustrating the method for a two-dimensional test problem are given.Received: February 12, 2004     

On singular perturbation of a semilinear hyperbolic equation

We consider a hyperbolic version of Eells-Sampson's equation: . This equation is semilinear with respect to space derivative and time derivative. Letu (x) be the solution with initial data u(0) and (0), and putv (t,x)=u (t,x). We show that when the resistance ,V (t,x) converges to a solution of the original parabolic Eells-Sampson's equation: . Note thatv _t(0)= (0) diverges when . We show that this phenomena occurs in more general situations.This article was processed by the author using the Springer-Verlag Pjourlg macro package.