Singular perturbations in the interaction representation,II

Perturbations of $\text{(?Δ)}{}^{\mathrm{<mtext>m</mtext><mtext>2</mtext>}}\text{in}\text{L}{}^2\text{(R}{}^{\mathrm{k}}\text{),}\text{for}\text{k ? 1}$ and suitable m, by distributions V for which $\text{V}\text{?}\text{(k) = 0(¦k¦}{}^{\mathrm{\alpha }}\text{),}\text{where}\text{α =}\text{(m + 1 ? ε)}\text{2 ? k}\text{, 0 < ε ? m + 1 ? 2k}$, are shown to correspond to self-adjoint operators H_v, in such a way that H_v depends continuously on V, and agrees with H + V when V is sufficiently regular. These results extend joint work with Irving E. Segal [J. Functional Analysis38 (1980), 71–98], in which perturbations of $\text{(}\text{?id}\text{dx}\text{)}{}^{\mathrm{m}}$ by distributions V with bounded Fourier transforms in L²(R¹) were considered.     

Singular perturbations in foliations

Summary We define a constraint system , [0,₀), which is a kind of family of vector fields on a manifold. This is a generalized version of the family of the equations , [0,₀),x ^m ,y ⁿ . Finally, we prove a singular perturbation theorem for the system , [0,₀).Dedicated to Professor Kenichi Shiraiwa on his 60th birthday     

Singular perturbations of elliptic problems

Summary The paper treates applications of singular perturbations of variational inequalities, to differential problems. Some informations on the boundary layer phenomenon are obtained. Entrata in Redazione il 20 ottobre 1971.     

Singular perturbations of abstract wave equations

Given, on the Hilbert space H₀, the self-adjoint operator B and the skew-adjoint operators C₁ and C₂, we consider, on the Hilbert space H?D(B)⊕H₀, the skew-adjoint operator

     

Singular perturbations of some nonlinear problems

We deal with singular perturbations of nonlinear problems depending on a small parameter ε > 0. First we consider the abstract theory of singular perturbations of variational inequalities involving some nonlinear operators, defined in Banach spaces, and describe the asymptotic behavior of these solutions as ε → 0. Then these abstract results are applied to some boundary value problems. Bibliography: 15 titles.     

Singular perturbations to semilinear stochastic heat equations

We consider a class of singular perturbations to the stochastic heat equation or semilinear variations thereof. The interesting feature of these perturbations is that, as the small parameter ε tends to zero, their solutions converge to the ‘wrong’ limit, i.e. they do not converge to the solution obtained by simply setting ε?=?0. A similar effect is also observed for some (formally) small stochastic perturbations of a deterministic semilinear parabolic PDE. Our proofs are based on a detailed analysis of the spatially rough component of the equations, combined with a judicious use of Gaussian concentration inequalities.     

Beyond all orders: Singular perturbations in a mapping

Summary We consider a family ofq-dimensional (q>1), volume-preserving maps depending on a small parameterε. Asε → 0⁺ these maps asymptote to flows which attain a heteroclinic connection. We show that for smallε the heteroclinic connection breaks up and that the splitting between its components scales withε likeε ^γexp[-β/ε]. We estimateβ using the singularities of theε → 0⁺ heteroclinic orbit in the complex plane. We then estimateγ using linearization about orbits in the complex plane. These estimates, as well as the assertions regarding the behavior of the functions in the complex plane, are supported by our numerical calculations. Deceased.     

Singular perturbations of integral equations with degenerate kernels

Singularly perturbed Fredholm equations of the second kind are investigated. The kernels are allowed to have a jump discontinuity which vanishes at a point along the diagonal. Sufficient conditions for existence and uniqueness of sohtions are found and the behavior of the solutions is studied.