Classical Dirichlet Research Articles

Can One See the Shape of a Surface?

Some ten years ago Professor Mark Kac published in these pages an article [3] entitled Can one hear the shape of a Drum?. Under this provocative title Kac raised a fascinating question: To what extent is the shape of a smooth closed surface (a drum) determined by the eigenvalues of the classical Dirichlet problem for the interior? Since these eigenvalues are just the resonance frequencies for the acoustic wave motion in the interior, the question asks: to what extent is the shape of a surface determined by the sound of its interior? This problem is a representative example of a class of problems, called inverse problems, which are quite different in character from the more familiar direct problems of partial differential equations. Generally speaking, direct problems ask us to develop a solution from the data, Whereas inverse problems ask us to recover the data from the solution. Inverse problems have attracted a consideKable and increasing interest in recent years because of their many immediate practical applications in our physical world. A splendid introduction may be found in Prof. Keller's recent article [20]. Typically they are very hard problems; Kac's problem, for example, remains unsolved today, though it has given rise to a considerable literature. Here we should like to raise a companion question: To what extent is the shape of a surface determined by the eigenfunctions of the classical Dirichlet problem for the exterior? We must note first of all that information about the shape of the drum is not to be found in the eigenvalues of the exterior problem, since, as we shall see, these consist of all negative real numbers, no matter what the shape. So whatever information about the shape can be recovered from the solution must be somehow contained in the eigenfunctions of the exterior problem. In particular, we expect on the basis of pract,ical experience that this information should be available in the asymptotic behavior of these eigenfunctions at great distances from the surface, since it is this behavior that determines what we when the surface is illuminated by a given source of light, radar, or sonar waves. That is, we expect to find that the shape of a surface is determined by the sight of its exterior. Thus we can see the shape of a drum, even if we can't hear it.

Perturbation theory of scattering from irregular bodies

A perturbation solution is derived for the following problem: A time harmonic wave of amplitude ψ, propagating in a medium with wave number k, is incident on an irregular volume V, inside of which the propagation constant k′(r) can be an arbitrary function of | r |, where r is a position vector with origin inside V. The boundary conditions are that both ψ and its normal derivative ∂ψ ∂n may be discontinuous across the surface of V. Special cases of these conditions correspond to acoustic scattering, to B-wave scattering from a dielectric cylinder, or to the classical Dirichlet (ψ = 0) or Neumann ( ∂ψ ∂n = 0 ) surface conditions. An integral equation is derived that satisfies the appropriate differential equations both outside and inside the body, and satisfies the boundary conditions as well. This equation is reduced to a set of linear algebraic equations by expansion in a certain basis set and these linear equations are then solved in a perturbation approximation for the case that the surface of the body differs from a sphere or cylinder by a small parameter λ. Comparison is made with formulae in the literature, and except for some minor discrepancies, which are here corrected, there is general agreement.

On the average order of a class of arithmetical functions, II

We consider a large class of arithmetical functions generated by Dirichlet series satisfying a very general functional equation with gamma factors. In our previous paper we obtained a “one-sided” Ω result, but here a “two-sided” Ω result is obtained in most cases. Unfortunately, the method fails in the classical circle and Dirichlet divisor problems.

A class of related Dirichlet and initial value problems

A class of related Dirichlet and initial value problems

Potential theory on a rhombic lattice

Of concern are complex valued functions defined on the lattice points of the complex plane. The lattice can be the usual lattice of square blocks but of main interest is an irregular lattice with squares replaced by rhombs. A function is defined to be discrete analytic if the difference quotient across one diagonal of a rhomb equals the difference quotient across the other diagonal. Based on this definition discrete analogs of the following concepts in classical function theory are developed: Laplace equation, Cauchy-Riemann equations, differentiation, contour integration, Morera's theorem, and harmonic polynomials. The theory is more than an analogy because, for a common class of boundary value problems, it proves possible to obtain upper and lower bounds for the classical Dirichlet integral in terms of discrete harmonic functions.

Arithmetical Functions of a Greatest Common Divisor, III. Cesàro's Divisor Problem

Let σ1(n) denote the sum of the tth powers of the divisors of n, σ(n) = σ1(n). Also placewhere γ is Euler's constant, ζ(s) is the Riemann ζ-function and x ≧ 2. The function Δ(x) is the remainder term arising in the divisor problem for σ((m, n)). Cesàro proved originally [1], [6, p. 328] that Δ(x) = o(x2 log x). More recently in I [2, (3.14)] it was shown by elementary methods that . This estimate was later improved to in II [3, (3.7)]. In the present paper (§ 3) we obtain a much more substantial reduction in the order of Δ(x), by showing that Δ(x) can be expressed in terms of the remainder term in the classical Dirichlet divisor problem. On the basis of well known results for this problem, it follows easily that . The precise statement of the result for σ((m, n)) is contained in (3.2).

Differential Forms in Manifolds with Boundary

The theory of harmonic forms in Riemannian manifolds may be regarded as a generalization of potential theory. It is therefore natural that the boundary value problems of this theory which generalize the classical Dirichlet and Neumann problems should play an important role in the theory. Such theorems were conjectured by Tucker (4a). The present paper is concerned with some theorems regarding closed forms, which are necessary for the proof of the boundary value existence theorems for harmonic fields in a Riemannian manifold with boundary. In an accompanying paper, by D. C. Spencer and the author, the results of this paper will be used in the proofs of these theorems. We consider closed forms in a finite regular manifold with boundary, and prove results which generalize two theorems of G. de Rham (3) for a closed manifold. To extend the technique of de Rham, we make use of the relative, or modular, homology theory introduced by Lefschetz (2). The main results of the paper are contained in Theorems 2 and 6. Theorems 1, 3, and 5 are analogues of de Rham's first theorem, and Theorems 2, 4 and 6 of his second theorem. Theorems 3 to 6 inclusive were conjectured in substantially their present form by Tucker (4b).

A Generalization of Epstein Zeta Functions

§ 1. An Epstein zeta function (in its simplest form) is a function represented by the Dirichlet's series where a1… ak are real and n1, n2, … nk run through integral values. The properties of this function are well known and the simplest of them were proved by Epstein [2, 3]. The aim of this note is to define a general class of Dirichlet's series, of which the above can be viewed as an instance, and to discuss the problem of analytic continuation of such series.

A Correction: On the Classical Dirichlet Problem for General Domains

A Correction: On the Classical Dirichlet Problem for General Domains

On the Analytic Continuation of Functions Represented by a Class of Dirichlet's Series

Let† ∑ ∞ p φ (n) e -rf(n) (1.1) , (p, a positive integer), p where s = δ + it, be a Dirichlet's series. Then the main object of this paper is to study the analytic continuation of functions represented by a class of Dirichlet's Series for which a relation of the type.