A simple approach to asymptotic expansions for Fourier integrals of singular functions

In this work, we are concerned with the derivation of full asymptotic expansions for Fourier integrals as s → ∞, where s is real positive, [a, b] is a finite interval, and the functions f(x) may have different types of algebraic and logarithmic singularities at x = a and x = b. This problem has been treated in the literature by techniques involving neutralizers and Mellin transforms. Here, we derive the relevant asymptotic expansions by a method that employs simpler and less sophisticated tools.     

Green's functions for elliptic and parabolic systems with Robin-type boundary conditions

The aim of this paper is to investigate Green's function for parabolic and elliptic systems satisfying a possibly nonlocal Robin-type boundary condition. We construct Green's function for parabolic systems with time-dependent coefficients satisfying a possibly nonlocal Robin-type boundary condition assuming that weak solutions of the system are locally Hölder continuous in the interior of the domain, and as a corollary we construct Green's function for elliptic system with a Robin-type condition. Also, we obtain Gaussian bound for Robin Green's function under an additional assumption that weak solutions of Robin problem are locally bounded up to the boundary. We provide some examples satisfying such a local boundedness property, and thus have Gaussian bounds for their Green's functions.     

Lp-convergence for expansions in terms of the eigenfunctions of a Sturm — Liouville problem

For the operator L_v=–(x^2ay). x [0, 1], y(0)=y(1)=0 with 0 < 1/2, or ¦y¦ < , y(1)=0 with 1/2 <1, we investigate the effect which the singularity of the Sturm-Liouville operator derived from this self-adjoint expression has on L_p-convergence of expansions in terms of the eigenfunctions of this operator. We will prove that the orthonormalized system of eigenfunctions forms a basis in L_p [0, 1] for 2/(2–) < p < 2/.Translated from Matematicheskii Zametki, Vol. 3, No. 6, pp. 683–692, June, 1968.The author is grateful to V. M. Tikhomirov for his many valuable remarks and his constant attention to this work.     

Green's functions for polynomials in the Laplacian

A Green's function is constructed for an arbitrary polynomial in theN-dimensional Laplacian operator, subject only to the condition that no root of the polynomial may be real and negative.     

A very simple proof of the LSI for high temperature spin systems

We present a very simple proof that the $O(n)$ model satisfies a uniform logarithmic Sobolev inequality (LSI) if the difference between the largest and the smallest eigenvalue of the coupling matrix is less than n. This condition applies in particular to the SK spin glass model at inverse temperature $\beta <1/4$. It is the first result of rapid relaxation for the SK model and requires significant cancellations between the ferromagnetic and anti-ferromagnetic spin couplings that cannot be obtained by existing methods to prove Log-Sobolev inequalities. The proof also applies to more general bounded and unbounded spin systems. It uses a single step of zero range renormalisation and Bakry–Emery theory for the renormalised measure.     

Green's functions for continuously supported plates

The Green's functions for an infinite elastic plate, attached respectively to a Pasternak or a Kerr base model, and subjected to a concentrated force, are obtained in terms of Bessel functions. It is shown, that for each base model, depending on the plate and base parameters, the solutions may be of different form. The method of images is then utilized to generate closed form solutions for the semi-infinite and quarter plates with simply supported boundaries. Paper also presents a generalization of Bessel functions of the Kelvin type and a discussion of their properties. They were needed for the solution of some of the equations under consideration.Research supported by National Science Foundation Grant MSM-8308919.     

Linear extensions of dynamical systems on a torus and Green's functions

We found new structures of linear extensions of dynamical systems on a torus which have a unique Green's function.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 9, pp. 1219–1224, 1990.     

An initial-value method for the determination of Green's functions

The convergence of an initial-value method for computing the Green's function of a class of second-order differential operators is established. The proof relies on an interpolation procedure which is shown to generalize the Nyström method for Fredholm integral equations. The approximate Green's function is related to the solution of a discrete summation equation. The results of Anselone and Moore on collectively compact operators are then applied.This research was partially supported by UNLV Grant No. 001-060-4573.     

Asymptotic expansions for parabolic systems

We consider a parabolic system in a half space. A theorem, similar to one proved by Meyers and Pazy for elliptic equations outside the unit ball is proved, namely, if the coefficients, the right side, and the initial conditions of the parabolic system have asymptotic expansions at infinity with respect to the space variable, then so does the solution of the corresponding Cauchy problem. Some generalizations and examples are given.