Forward and inverse scattering on manifolds with asymptotically cylindrical ends

We study an inverse problem for a non-compact Riemannian manifold whose ends have the following properties: On each end, the Riemannian metric is assumed to be a short-range perturbation of the metric of the form ²(dy)+h(x,dx), h(x,dx) being the metric of some compact manifold of codimension 1. Moreover one end is exactly cylindrical, i.e. the metric is equal to ²(dy)+h(x,dx). Given two such manifolds having the same scattering matrix on that exactly cylindrical end for all energies, we show that these two manifolds are isometric.     

On the gluing problem for Dirac operators on manifolds with cylindrical ends

Combining elements of the b-calculus and the theory of elliptic boundary value problems, we solve the gluing problem for b-determinants of Dirac type operators on manifolds with cylindrical ends. As a corollary of our proof, we derive a gluing formula for the b-eta invariant and also a relative invariant formula relating the b-spectral invariants on a manifold with cylindrical end to the spectral invariants with the augmented APS boundary condition on the corresponding compact manifold with boundary.     

Wodzicki residue for operators on manifolds with cylindrical ends

We define the Wodzicki Residue TR(A) for A belonging to a space of operators with double order, denoted \({L^{m_1,m_2}_{\rm cl}}\). Such operators are globally defined initially on \({\mathbb{R}^n}\) and then, more generally, on a class of non-compact manifolds, namely, the manifolds with cylindrical ends. The definition is based on the analysis of the associate zeta function ζ(A, z). Using this approach, under suitable ellipticity assumptions, we also compute a two terms leading part of the Weyl formula for a positive selfadjoint operator \({A\in L^{m_1,m_2}_{\rm cl}}\) in the case m ₁ = m ₂.     

On the operator of exterior derivation on the Riemannian manifolds with cylindrical ends

We formulate some conditions for the normal and compact solvability of the operator of exterior derivation on the cylindrical manifolds equipped with some Riemannian metrics. Some analogous results were obtained in the particular case of warped cylinders [1].     

Embedded minimal surfaces with an infinite number of ends

The research described in this paper was supported by research grant DE-FG02-86ER250125 of the Applied Mathematical Science subprogram of the Office of Energy Research, U.S. Department of Energy, and National Science Foundation grants DMS-8503350 and DMS-8611574     

Aubry-mather theory and the inverse spectral problem for planar convex domains

The inverse spectral problem is concerned with the question to what extent the spectrum of a domain determines its geometry. We find that, associated to a convex domain Ω in ℝ², there is a convexfunction which is a length spectrum invariant under continuous deformations. It includes several geometric quantities, such as the lengths and Lazutkin parameters of caustics, as well as the asymptotic invariants discovered by Marvizi and Melrose. Via a Poisson relation, we also find invariants determined by the Laplace spectrum of Ω.     

A mixed problem for the equation of internal gravity waves in an infinite cylindrical domain

The existence of a unique classical solution to the mixed problem for the equation describing internal gravity waves in a cylindrical domain is proved. The behavior of the solution is studied at t → +∞.     

Interactions in an infinite medium of planar cracks with completely rigid planar inclusions

The problem of determining the interactions in an infinite medium of planar cracks with absolutely rigid inclusions leads to a system of integral equations, the regular kernals of which represent the interaction. The system of integral equations is completely determined under boundary conditions for the equilibrium of the inclusions as a rigid body. An approximate solution for the system of integral equations is used. The dependence of the magnitude of the external load on parameters characterizing the distribution in the medium of disc-shaped cracks and inclusions is graphically presented.Translated from Matematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 29, pp. 63–68, 1989.     

A multidimensional inverse problem for an equation with memory

Translated from Sibirskii Zhurnal, Vol. 35, No. 3, pp. 574–582, May–June, 1994.     

Sequence spaces and inverse of an infinite matrix

We give here some properties of the sets _α(uΔ) generalizing the space of generalized difference sequencesl ^∞(uΔ). Then we study spaces related to the sets of sequences that are strongly convergent or strongly bounded. Next we define from the sets of spaces that are (N,q) summable or bounded the sets of spaces that are (N,q)α-bounded orr-bounded. Then we give some properties of these spaces using Banach space of the forms _α.     

The capillary problem for an infinite trough

Consider an infinite trough (or wedge) with dihedral angle 2, 0 < < and a quantity of fluid inside contacting the edge. In equilibrium the free interface of the fluid will be a surface of constant mean curvature meeting the planar walls at a constant angle determined from physical considerations. One obvious configuration is for the free surface to be a section of a round circular cylinder parallel to the axis of the wedge whose position is determined by the angles and . For + > /2 the cylinder configuration is unstable and bifurcation occurs. We exhibit the full family of bifurcating solutions starting with the round cylinder solution and proceeding through a beading up process into a series of spherical sections suitably positioned. Furthermore, if the edge of the wedge is a re-entrant corner ( > /2) then there are further bifurcating families. One is a secondary bifurcation from the family initially constructed while the other is a primary bifurcation from the cylinder which are less symmetric than the initial families.This paper was completed while the author was on a subbatical from the University of Toledo and was a visitor at Stanford University and also with SFB 256, University of Bonn.     

An inverse problem for an elliptic equation with an affine term

We consider the following elliptic boundary value problem: on , u = 0 on where is a smooth bounded planar domain. We show that for a large class of domains and for any such that is not identically constant there exist at most finitely many different pairs of coefficients such that the problem has a solution with the normal flux on . Received: 4 February 1999     

Radial vibrations of an infinite medium with a cylindrical cavity

Zusammenfassung Die vorliegende Arbeit bringt die vollständige theoretische Berechnung der Radialschwingungen eines unendlichen elastischen Mediums mit zylindrischem Ausschnitt. Die zylindrische Körperwand steht unter einem mit der Zeit allgemein veränderlichen Druck, und am Anfang sind alle Körperteile frei von Verschiebungen und Geschwindigkeiten.     

A probabilistic approach to Martin boundaries for manifolds with ends

Summary We use martingale methods and coupling arguments to prove results of Li and Tam (1987) and Donnelly (1986) characterizing positive and bounded harmonic functions, respectively, on certain manifolds with ends.Research supported by a grant from NSA/NSF     

An inverse Sturm-Liouville problem for an impedance

A numerical method for reconstructing an impedance in a Sturm-Liouville operator from finitely many eigenvalues is investigated. The method constructs an impedance that has the given eigenvalues by finding a zero of a nonlinear finite dimensional map. A Newton scheme is investigated and numerical examples are considered.