Scattering theory for the elastic wave equation in perturbed half-spaces

In this paper we consider the linear elastic wave equation with the free boundary condition (the Neumann condition), and formulate a scattering theory of the Lax and Phillips type and a representation of the scattering kernel. We are interested in surface waves (the Rayleigh wave, etc.) connected closely with situations of boundaries, and make the formulations intending to extract this connection.

The half-space is selected as the free space, and making dents on the boundary is considered as a perturbation from the flat one. Since the lacuna property for the solutions in the outgoing and incoming spaces does not hold because of the existence of the surface waves, instead of it, certain decay estimates for the free space solutions and a weak version of the Morawetz arguments are used to formulate the scattering theory.

We construct the representation of the scattering kernel with outgoing scattered plane waves. In this step, again because of the existence of the surface waves, we need to introduce new outgoing and incoming conditions for the time dependent solutions to ensure uniqueness of the solutions. This introduction is essential to show the representation by reasoning similar to the case of the reduced wave equation.

     

Nonreflecting boundary condition for the wave equation

To solve the time-dependent wave equation in an infinite two (three) dimensional domain a circular (spherical) artificial boundary is introduced to restrict the computational domain. To determine the nonreflecting boundary we solve the exterior Dirichlet problem which involves the inverse Fourier transform. The truncation of the continued fraction representation of the ratio of Hankel function, that appear in the inverse Fourier transform, provides a stable and numerically accurate approximation. Consequently, there is a sequence of boundary conditions in both two and three dimensions that are new. Furthermore, only the first derivatives in space and time appear and the coefficients are updated in a simple way from the previous time step. The accuracy of the boundary conditions is illustrated using a point source and the finite difference solution to a Dirichlet problem.     

Numerical simulation of a modified KdV equation on the whole real axis

The numerical simulation of the solution to a modified KdV equation on the whole real axis is considered in this paper. Based on the work of Fokas (Comm Pure Appl Math 58(5):639–670, 2005), a kind of exact nonreflecting boundary conditions which are suitable for numerical purposes are presented with the inverse scattering theory. With these boundary conditions imposed on the artificially introduced boundary points, a reduced problem defined on a finite computational interval is formulated. The discretization of the nonreflecting boundary conditions is studied in detail, and a dual-Petrov–Galerkin spectral method is proposed for the numerical solution to the reduced problem. Some numerical tests are given, which validate the effectiveness, and suggest the stability of the proposed scheme.Supported by the National Natural Science Foundation of China under Grant No. 10401020, the Alexander von Humboldt Foundation, and the Key Project of China High Performance Scientific Computation Research.     

Matrix-valued Schrödinger operators over local fields

We consider quantum systems that have as their configuration spaces finite dimensional vector spaces over local fields. The quantum Hilbert space is taken to be a space with complex coefficients and we include in our model particles with internal symmetry. The Hamiltonian operator is a pseudo-differential operator that is initially only formally defined. For a wide class of potentials we prove that this Hamiltonian is well-defined as an unbounded self-adjoint operator. The free part of the operator gives rise to ameasure on the Skorokhod space of paths,D[0,∞), and with respect to this measure there is a path integral representation for the semigroup associated to the Hamiltonian. We prove this Feynman-Kac formula in the local field setting as a consequence of the Hille-Yosida theory of semi-groups. The text was submitted by the authors in English.     

Bismut–Elworthy's formula and random walk representation for SDEs with reflection

We study the existence of first derivatives with respect to the initial condition of the solution of a finite system of SDEs with reflection. We prove that such derivatives evolve according to a linear differential equation when the process is away from the boundary and that they are projected to the tangent space when the process hits the boundary. This evolution, rather complicated due to the structure of the set at times when the process is at the boundary, admits a simple representation in terms of an auxiliary random walk. A probabilistic representation formula of Bismut–Elworthy's type is given for the gradient of the transition semigroup of the reflected process.     

On radiation conditions for rough surface scattering problems

^** Email: arens{at}numathics.com^*** Email: hohage{at}math.uni-goettingen.de It is well known that Sommerfeld's radiation condition is nota valid characterization of outgoing waves for scattering problemsat rough surfaces. Instead, a radiation condition called upwardpropagating radiation condition (UPRC) is commonly used. Recently,a different radiation condition called the pole condition hasbeen investigated for scattering problems at bounded obstacles.In this paper we show the equivalence between the UPRC and thepole condition. In doing so, we give a rigorous interpretationof a formula called the angular spectrum representation forDirichlet data in the space of bounded continuous functions.     

Adapted kussmaul formulation of acoustic scattering problems: an overview

Problems of exterior acoustic scattering may be conveniently formulated by means of boundary integral equations. The problem seeks to find a wave function which gives velocity potential profile, pressure density profile, etc. of the acoustic wave at points in space. At the background of the formulations are two theories viz. (Helmholtz) Potential theory and the Green's representation formula. Potential theory gives rise to the so-called indirect formulation and the Green's representation formula to the direct formulations. Classical boundary integral formulations fail at the eigenfrequencies of the interior domain. That is, if a solution is sought of the exterior problem by first solving a homogeneous boundary integral equation, one is inevitably led to the conclusion that these homogeneous boundary equations have nontrivial solutions at certain wave-numbers which are the eigenvalues of the corresponding interior problem. At lower wave-numbers, these eigenfrequencies are thinly distributed but the higher the wave-number, the denser it becomes. This is a well-known drawback for both time-harmonic acoustics and elastodynamics. This is not a physical difficulty but arises entirely as a result of a deficiency in the integral equation is representation. Why then use It? The use has many advantages notably in that the meshing region is reduced from the infinite domain exterior to the body to its finite surface. This created the need for some robust formulations. A proof of the Kussmaul [1] formulation is presented. The formulation has a hypersingular kernel in the integral operator, which creates a havoc in computation (e.g., ill conditioning). The hyper-singularity can be avoided [2], as a result a new formulation is proposed. This paper presents a broad overview of the Adapted Kussmaul Formulation (AKF).     

On the integral representation of superbiharmonic functions

We consider a nonnegative superbiharmonic function w satisfying some growth condition near the boundary of the unit disk in the complex plane. We shall find an integral representation formula for w in terms of the biharmonic Green function and a multiple of the Poisson kernel. This generalizes a Riesz-type formula already found by the author for superbihamonic functions w satisfying the condition 0 ⩽ w(z) ⩽ C(1-|z|) in the unit disk. As an application we shall see that the polynomials are dense in weighted Bergman spaces whose weights are superbiharmonic and satisfy the stated growth condition near the boundary. Research supported in part by IPM under the grant number 83310011.     

Regularity estimates for solutions to the mean curvature flow with a Neumann boundary condition

In this work we study the behaviour of compact, smooth, immersed manifolds with boundary which move under the mean curvature flow in Euclidian space. We thereby prescribe the Neumann boundary condition in a purely geometric manner by requiring a vertical contact angle between the unit normal fields of the immersions and a given, smooth hypersurface. We deduce a very sharp local gradient bound depending only on the curvature of the immersions and. Combining this with a short time existence result, we obtain the existence of a unique solution to any given smooth initial and boundary data. This solution either exists for anyt>0 or on a maximal finite time interval [0,T] such that the curvature explodes astT.This article was processed by the author using the LATEX style filepljourlm from Springer-Verlag.     

Debye sources and the numerical solution of the time harmonic Maxwell equations

In this paper, we develop a new representation for outgoing solutions to the time‐harmonic Maxwell equations in unbounded domains in ?³. This representation leads to a Fredholm integral equation of the second kind for solving the problem of scattering from a perfect conductor, which does not suffer from spurious resonances or low‐frequency breakdown, although it requires the inversion of the scalar surface Laplacian on the domain boundary. In the course of our analysis, we give a new proof of the existence of nontrivial families of time‐harmonic solutions with vanishing normal components that arise when the boundary of the domain is not simply connected. We refer to these as k‐Neumann fields, since they generalize, to nonzero wave numbers, the classical harmonic Neumann fields. The existence of k‐Neumann fields was established earlier by Kress. © 2009 Wiley Periodicals, Inc.     

Burghelea-Friedlander-Kappeler's gluing formula and the adiabatic decomposition of the zeta-determinant of a Dirac Laplacian

 In this paper we first establish the relation between the zeta-determinant of a Dirac Laplacian with the Dirichlet boundary condition and the APS boundary condition on a cylinder. Using this result and the gluing formula of the zeta-determinant given by Burghelea, Friedlander and Kappeler with some assumptions, we prove the adiabatic decomposition theorem of the zeta-determinant of a Dirac Laplacian. This result was originally proved by J. Park and K. Wojciechowski in [11] but our method is completely different from the one they presented. Received: 2 August 2002 / Revised version: 24 February 2003 Published online: 24 April 2003 Mathematics Subject Classification (2000): 58J52, 58J50     

A Condition for a Closed One-Form to Be Exact

A condition for a closed one-form to be exact, the one-form having values in Euclidean space, on a compact surface without boundary, is given in the case where the surface has suitable differentiable automorphisms. Tori and hyperelliptic curves, with holomorphic automorphisms, are in this case. A local representation formula for surfaces in Euclidean space is then globalized. A condition for a local surface of constant mean curvature to be global, can be written using a harmonic Gauss map.     

Analysis of a FEM/BEM coupling method for transonic flow computations

A sensitive issue in numerical calculations for exterior flow problems, e.g.around airfoils, is the treatment of the far field boundary conditions on a computational domain which is bounded. In this paper we investigate this problem for two-dimensional transonic potential flows with subsonic far field flow around airfoil profiles. We take the artificial far field boundary in the subsonic flow region. In the far field we approximate the subsonic potential flow by the Prandtl-Glauert linearization. The latter leads via the Green representation theorem to a boundary integral equation on the far field boundary. This defines a nonlocal boundary condition for the interior ring domain. Our approach leads naturally to a coupled finite element/boundary element method for numerical calculations. It is compared with local boundary conditions. The error analysis for the method is given and we prove convergence provided the solution to the analytic transonic flow problem around the profile exists.

     

Continuity of the scattering function and Levinson‐type formula of Klein–Gordon equation

We considered the inverse problem of scattering theory for a boundary value problem on the half line generated by Klein–Gordon differential equation with a nonlinear spectral parameter‐dependent boundary condition. We defined the scattering data, and we proved the continuity of the scattering function S(λ); in a special case, the relation for the difference of the logarithm of the scattering function, which is called the Levinson‐type formula, was obtained. Copyright © 2012 John Wiley & Sons, Ltd.     

Integral equation methods for scattering by infinite rough surfaces

In this paper, we consider the Dirichlet and impedance boundary value problems for the Helmholtz equation in a non‐locally perturbed half‐plane. These boundary value problems arise in a study of time‐harmonic acoustic scattering of an incident field by a sound‐soft, infinite rough surface where the total field vanishes (the Dirichlet problem) or by an infinite, impedance rough surface where the total field satisfies a homogeneous impedance condition (the impedance problem). We propose a new boundary integral equation formulation for the Dirichlet problem, utilizing a combined double‐ and single‐layer potential and a Dirichlet half‐plane Green's function. For the impedance problem we propose two boundary integral equation formulations, both using a half‐plane impedance Green's function, the first derived from Green's representation theorem, and the second arising from seeking the solution as a single‐layer potential. We show that all the integral equations proposed are uniquely solvable in the space of bounded and continuous functions for all wavenumbers. As an important corollary we prove that, for a variety of incident fields including an incident plane wave, the impedance boundary value problem for the scattered field has a unique solution under certain constraints on the boundary impedance. Copyright © 2003 John Wiley & Sons, Ltd.     

Stabilization through viscoelastic boundary damping: a semigroup approach

The undamped linear wave equation on a bounded domain in ℝ ⁿ with C ² boundary is considered. The interaction of the interior waves and the viscoelastic boundary material is modeled by convolution boundary conditions. It is assumed that the convolution kernel is integrable and completely monotonic. The main result is that the derivatives of all solutions tend to zero. The proof is given by an application of the Arendt-Batty-Lyubic-Vu Theorem. To this end, the model is reformulated as an abstract first order Cauchy problem in an appropriate Hilbert space, including the memory of the boundary as a state component. It is shown that the differential operator of the Cauchy problem is the generator of a contraction semigroup on the state space by establishing the range condition for the Lumer-Phillips Theorem using a generalized Lax-Milgram argument and Fredholm’s alternative. Furthermore, it is shown that neither the generator nor its adjoint have purely imaginary eigenvalues.     

A note on a finite element method dealing with corner singularities

Recently the first author and his coworker report a new finite element method for the Poisson equations with homogeneous Dirichlet boundary conditions on a polygonal domain with one re-entrant angle [7]. They use the well-known fact that the solution of such problem has a singular representation, deduce a well-posed new variational problem for regular part of solution and an extraction formula for the so-called stress intensity factor using two cut-off functions. They use Fredholm alternative and G?rding’s inequality to establish the well-posedness of the variational problem and finite element approximation, so there is a maximum bound for meshh theoretically, although the numerical experiments shows the convergence for every reasonable size ofh. In this paper we show that the method converges for everyh with reasonable size by imposing a restriction to the support of the extra cut-off function without using G?rding’s inequality. We also give error analysis with similar results.     

Interior and boundary difference equations for hyperbolic differential equations

Interior and boundary difference equations are derived for several hyperbolic partial differential equations by means of an integral method. The method is applied to a simple transport equation, to waves in a compressible, isentropic fluid, and to surface waves in shallow water. Boundary conditions treated are (a) a perfectly reflecting boundary, (b) an open boundary with outgoing waves and a specified incoming wave, and (c) a partially reflecting boundary. For open boundaries, the major assumption for the algorithms to be valid is that outgoing waves can be defined, an assumption equivalent to the most general statement of Sommerfeld's radiation condition. The difference equations obtained are conservative, second-order accurate, two time-level, explicit, and stable (for one-dimensional, time-dependent problems) for cΔt/Δx ? 1 where c is the wave speed, Δt is the temporal grid size, and Δx is the spatial grid size. Numerical calculations demonstrate the excellent accuracy of the procedure.     

Earth surface effects on active faults: An eigenvalue asymptotic analysis

We study in this paper an eigenvalue problem (of Steklov type), modeling slow slip events (such as silent earthquakes, or earthquake nucleation phases) occurring on geological faults. We focus here on a half space formulation with traction free boundary condition: this simulates the earth surface where displacements take place and can be picked up by GPS measurements. We construct an appropriate functional framework attached to a formulation suitable for the half space setting. We perform an asymptotic analysis of the solution with respect to the depth of the fault. Starting from an integral representation for the displacement field, we prove that the differences between the eigenvalues and eigenfunctions attached to the half space problem and those attached to the free space problem, is of the order of d^-2, where d is a depth parameter: intuitively, this was expected as this is also the order of decay of the derivative of the Green's function for our problem. We actually prove faster decay in case of symmetric faults. For all faults, we rigorously obtain a very useful asymptotic formula for the surface displacement, whose dominant part involves a so called seismic moment. We also provide results pertaining to the analysis of the multiplicity of the first eigenvalue in the line segment fault case. Finally we explain how we derived our numerical method for solving for dislocations on faults in the half plane. It involves integral equations combining regular and Hadamard's hypersingular integration kernels.     

Existence and Uniqueness of Green's Functions to Nonlinear Yamabe Problems

For a given finite subset S of a compact Riemannian manifold (M,g) whose Schouten curvature tensor belongs to a given cone, we establish a necessary and sufficient condition for the existence and uniqueness of a conformal metric on M\S such that each point of S corresponds to an asymptotically flat end and that the Schouten tensor of the conformal metric belongs to the boundary of the given cone. As a by-product, we define a purely local notion of Ricci lower bounds for continuous metrics that are conformal to smooth metrics and prove a corresponding volume comparison theorem. © 2022 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.