Coercive singular perturbations II: Reduction and convergence

An algebra of pseudodifferential singular perturbations is introduced. It provides a constructive machinery in order to reduce an elliptic singularly perturbed operator (in $\text{R}$ⁿ or on a smooth manifold without border) to a regular perturbation. The technique developed is applied to some singularly perturbed boundary value problems as well. Special attention is given to a singular perturbation appearing in the linear theory of thin elastic plates. A Wiener-Hopf-type operator containing the small parameter reduces this singular perturbation to a regular one. It also gives rise to a natural recurrence process for the construction of high-order asymptotic formulae for the solution of the perturbed problem. The method presented can be extended to the general coercive singular perturbations.     

Coercive singular perturbations: Eigenvalue problems and bifurcation phenomena

Summary The method based upon a constructive reduction of coercive singular perturbations to regular ones, introduced in 1977 (see [4]) and developed later on (see [9–11]) is applied for computing the asymptotic expansions for eigenvalues of coercive singular perturbations, when the small parameter goes to zero. The same method turns out to be useful for investigating the asymptotic behaviour of solutions to quasi-linear coercive singular perturbations in the neighbourhood of the bifurcation points. It can be applied to classes of quasi-linear singular perturbations whose principal linear part in local representation is coercive and the nonlinear part is analytic in some ball in the solution space with values in the data space. The results are summarized in [7, 8].     

Supersymmetry and singular perturbations

First it is shown in an abstract framework that supersymmetric quantum theory may give rise to very singular perturbations, which make perturbation theories invalid. Then, the abstract results are applied to a model of 1-dimensional supersymmetric quantum mechanics (Witten's model) to obtain some families of singular Hamiltonians. Some concrete examples, which reveal their singularness, are discussed in detail.     

Elliptic difference singular perturbations

Summary Difference approximations for differential singular perturbations with small parameter ɛ are considered. We point out ellipticity and coerciveness conditions which arenecessary andsufficient for a two-sided a priori estimate to hold for the solution of difference singular perturbation uniformly with respect to the ratio of both small parameters: the original one ɛ and the meshsize h. Entrata in Redazione il 27 maggio 1978.     

Smooth approximation of singular perturbations

We consider a certain subclass of self-adjoint extensions of the symmetric operator

     

Corner singularities and singular perturbations

A corner singularity expansion is developed for a singularly perturbed elliptic boundary value problem. The problem is set in a sector of the plane. In the expansion, particular attention is paid to the singular perturbation parameter. The result is used to give pointwise bounds on derivatives of the solution. These bounds show the influence of both the boundary layers and the corner singularity.

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Sommario In questo lavoro viene proposto uno sviluppo della singolarità angolare per un problema al contorno ellittico singolarmente perturbato. Il problema è posto in un settore del piano. Nello svilluppo viene posta particolare attenzione al parametro della perturbazione singolare. Il risultato è usato per fornire stime puntuali alle derivate della soluzione. Queste stime evidenziano sia l'influenza dello stato limite sia quella della singolarità angolare.

     

Nash-Moser iteration and singular perturbations

We present a simple and easy-to-use Nash-Moser iteration theorem tailored for singular perturbation problems admitting a formal asymptotic expansion or other family of approximate solutions depending on a parameter ε→0. The novel feature is to allow loss of powers of ε as well as the usual loss of derivatives in the solution operator for the associated linearized problem. We indicate the utility of this theorem by describing sample applications to (i) systems of quasilinear Schrödinger equations, and (ii) existence of small-amplitude profiles of quasilinear relaxation systems in the degenerate case that the velocity of the profile is a characteristic mode of the hyperbolic operator.     

Pseudo-differential energy estimates of singular perturbations

In this paper we study singular limits of hyperbolic systems, which exhibit large time oscillations, and in particular pseudo-differential energy estimates that enable us to obtain uniform existence time and a priori bounds on the solutions. © 1997 John Wiley & Sons, Inc.     

One-dimensional perturbations of singular unitary operators

Under some natural restrictions, we prove that any one-dimensional perturbation of a singular unitary operator on a Hilbert space is unitarily equivalent to a model operator on a space determined (in a certain way) by two functions from the Hardy space H². Bibliography: 3 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 232, 1996, pp. 118–122. Translated by V. V. Kapustin.     

Some singular perturbations of periodic operators

We consider two examples of singular perturbations of a periodic differential operator on the axis. The first example is a rapidly oscillating potential with compact support, and the second example is the delta potential with a small complex coupling constant. We investigate the structure and asymptotic behavior of the spectrum of the perturbed operators in detail. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 2, pp. 207–218, May, 2007.     

Analytic singular perturbations of elliptic systems

We study a singular perturbation problem for a system defined under a variational form. We show the analytic dependence of the solution of the equation with respect to a small, nonnull parameter ε, and make explicit the terms of the power series. This result improves a theorem of Chap. I of J. L. Lions (“Perturbations singulières dans les problèmes aux limites et en contrôle optimal,” Springer-Verlag, Berlin 1973) in which the variational forms are supposed to be symmetric and no analycity result is given. We give an application to the study of a stationary thermical system with a small convection coefficient.     

On singular perturbations of superlinear elliptic systems

We study the existence, multiplicity and shape of positive solutions of the system −ε²Δu+V(x)u=K(x)g(v), −ε²Δv+V(x)v=H(x)f(u) in RN, as ε→0. The functions f and g are power-like nonlinearities with superlinear and subcritical growth at infinity, and V, H, K are positive and locally Hölder continuous.     

Rank one perturbations and singular integral operators

We consider rank one perturbations Aα=A+α(⋅,φ)φ of a self-adjoint operator A with cyclic vector φ∈H−1(A) on a Hilbert space H. The spectral representation of the perturbed operator Aα is given by a singular integral operator of special form. Such operators exhibit what we call ‘rigidity’ and are connected with two weight estimates for the Hilbert transform. Also, some results about two weight estimates of Cauchy (Hilbert) transforms are proved. In particular, it is proved that the regularized Cauchy transforms Tε are uniformly (in ε) bounded operators from L²(μ) to L²(μα), where μ and μα are the spectral measures of A and Aα, respectively. As an application, a sufficient condition for Aα to have a pure absolutely continuous spectrum on a closed interval is given in terms of the density of the spectral measure of A with respect to φ. Some examples, like Jacobi matrices and Schrödinger operators with L² potentials are considered.     

Distributional convergence in planar dynamics and singular perturbations

Motivated by applications to singular perturbations, the paper examines convergence rates of distributions induced by solutions of ordinary differential equations in the plane. The solutions may converge either to a limit cycle or to a heteroclinic cycle. The limit distributions form invariant measures on the limit set. The customary gauges of topological distances may not apply to such cases and do not suit the applications. The paper employs the Prohorov distance between probability measures. It is found that the rate of convergence to a limit cycle and to an equilibrium are different than the rate in the case of heteroclinic cycle; the latter may exhibit two paces, depending on a relation among the eigenvalues of the hyperbolic equilibria. The limit invariant measures are also exhibited. The motivation is stemmed from singularly perturbed systems with non-stationary fast dynamics and averaging. The resulting rates of convergence are displayed for a planar singularly perturbed system, and for a general system of a slow flow coupled with a planar fast dynamics.