G. F. Roach

Characteristic inequalities of uniformly convex and uniformly smooth Banach spaces

Abstract Let X be a real Banach space with dual X ∗ and moduli of convexity and smoothness δ X ( e ) and ϱ X ( τ ), respectively. For 1 p denotes the duality mapping from X into 2 X ∗ with gauge function t p − 1 and j p denotes an arbitrary selection for J p . Let A = { φ : R + → + : φ (0) = 0, φ ( t ) is strictly increasing and there exists c > 0 such that φ(t) ⩾ cδ X ( t 2 )} and F = { ϑ : R + → + : ϑ (0) = 0, ϑ ( t ) is convex, nondecreasing and there exists K > 0 such that ϑ ( τ ) ⩽ Kϱ X ( τ )}. It is proved that X is uniformly convex if and only if there is a φ ϵ A such that ∥ x + y ∥ p ⩾ ∥ x ∥ p + p 〈 j p x , y 〉 + σ φ ( x , y ) ∀ x , y ϵ X and X is uniformly smooth if and only if there is a ϑ ∈ F such that ∥ x + y ∥ p ⩽ ∥ x ∥ p + p 〈 j p x , y 〉 + σ ϑ ( x , y ) ∀ x , y ϵ X , where, for given function f , σ f ( x , y ) is defined by σ f (x,y) = p∫ 0 1 ∥ x+ ty ∥ ∨ ∥ x ∥) p t f t ∥ y ∥ ∥ x+ty ∥ ∨ ∥ x ∥ dt These inequalities which have various applications can be regarded as general Banach space versions of the well-known polarization identity occurring in Hilbert spaces.

On Modified Green Functions in Exterior Problems for the Helmholtz Equation

The question of non-uniqueness in boundary integral equation formulations of exterior problems for the Helmholtz equation has recently been resolved with the use of additional radiating multipoles in the definition of the Green function. The present note shows how this modification may be included in a rigorous formalism and presents an explicit choice of coefficients of the added terms that is optimal in the sense of minimizing the least-squares difference between the modified and exact Green functions.

A necessary and sufficient condition for convergence of steepest descent approximation to accretive operator equations

Abstract A necessary and sufficient condition is established which ensures the strong convergence of the steepest descent approximation to a solution of equations involving quasi-accretive operators defined on a uniformly smooth Banach space.

Weak Solutions of Fluid–Solid Interaction Problems

This paper is concerned with various variational formulations for the fluid–solid interaction problems. The basic approach here is a coupling of field and boundary integral equation methods. In particular, Gardings inequalities are established in appropriate Sobolev spaces for all the formulations. Existence and uniqueness results of the corresponding weak solutions are given under suitable assumptions.

A constructive method for identification of an impenetrable scatterer

Abstract This paper presents a constructive algorithm for solving the problem of reconstructing the shape of a scattering object from measurements of the scattered far field when the unknown object is illuminated by a known incident wave. The problem is recast as an optimization problem with a penalty term. The cost functional consists of a term which assesses the difference between the measured far field and the far field of the solution of the field equation to a particular surface and the penalty term which measures the error in satisfying the boundary conditions on that surface. A complete family of radiating solutions of the Helmholtz equation is employed to construct approximate solutions by solving finite dimensional minimization problems. Existence of solutions of the original problem as well as the finite dimensional approximations is established. Moreover convergence of the approximate solutions to a solution of the original problem is proven. Some preliminary numerical results are presented to indicate the viability of the method.

The null field method and modified Green functions

A relation is established between the null field or T-matrix method and the method of modified Green functions for solving the Dirichlet and Neumann problems for the Helmholtz equation in an exterior domain. It is shown that elements of the T-matrix, formed by using appropriately chosen expansion functions, are precisely the coefficients of a representation of the Green function in terms of these same expansion functions. The linear independence of these functions is established together with the invertibility of a number of infinite dimensional matrices and their finite dimensional approximations, which is crucial to the T -matrix approach.

Iterative solutions of boundary integral equations in acoustics

It is shown that exterior problems for the Helmholtz equation may be solved iteratively for all frequencies. A general class of boundary-value problems for the Helmholtz equation is reduced to boundary integral equations. Using modified Green’s functions these boundary integral equations are known to be uniquely solvable. Even though the boundary integral operators are not self-adjoint they may be transformed into a form appropriate for iterative solution by a method developed for linear operator equations. Rates of convergence of the iteration are given and an example is presented to show the method.

An over-relaxation method for the iterative solution of integral equations in scattering problems

Abstract A simple iterative method for solving many of the integral equations arising in scattering problems is presented. By introducing a relaxation parameter the equation is changed to one which may be solved as a Neumann series. An explicit choice of the relaxation parameter is proposed which does not require detailed knowledge of the spectrum nor does the method require the symmetrization of the, in general, non-selfadjoint integral operators that occur. Convergence of the method is demonstrated in examples where the Neumann series for the original equation either diverges or converges at a much lower rate.

On the relationship between boundary value problems

Representations of the solutions to the principal boundary value problems of mathematical physics, can always be obtained provided certain boundary integral equations can be solved. It is fair to say that much effort has been devoted towards ensuring that the associated boundary integral equations are equations of the second rather than the first kind. (In this connection we would cite the work of Kleinman and Roach [S] as a source reference.) However, it is frequently the case that methods leading to boundary integral equations of the first kind are more instructive. Nevertheless, these methods have been very largely neglected due to the difficulty of handling the equations involved. Recently, new developments have been made in solving certain integral equations of the first kind [l-5, g-101 and in consequence a more unified and instructive analysis of boundary value problems would appear to be available. We shall demonstrate this here by considering the Dirichlet, and Neumann problems for the Laplace equation. The approach we shall adopt will be centered on the use of Green’s identities rather than on layer theoretic methods. The reason for this is that in general the density function required in the layer theoretic approach is not a natural physical quantity but rather an ingenious mathematical artifact. On the other hand the Green’s Theorem approach is concerned only with quantities of immediate physical significance.

A transmission problem for the reduced wave equation in inhomogeneous media with an infinite interface

In this paper, we consider the transmission problem in Rn (n ≽ 2) for the reduced wave equation in inhomogeneous media with an infinite interface. A new radiation condition involving a surface integral over the interface, which is different from the well-known Rellich-Sommerfeld radiation condition, is introduced to ensure the uniqueness of the solutions. It is then used together with the limiting absorption principle to establish the unique existence of solutions to the problem in the various media.