Menahem Schiffer

CONVEXITY OF DOMAIN FUNCTIONALS

Abstract : A rigorous theory is developed for variation of domain functions in a space of 3 dimensions as well as in the plane. The classical Hadamard variational formulas in space are discussed, and the so-called interior variational method is generalized to 3 dimensions. Interior variations of a 3- dimensional domain D are defined by means of differential mappings of D which depend on a small parameter Epsilon. The first-order shifts in terms of Epsilon of the Greens function, Neumanns function, and eigen-values, which result from this variation of D, are calculated rigorously by referring all varied quantities back to the original D through the infinitesimal mappings. Since proof is possible that the varied domain functions can be expanded in powers of Epsilon, the perturbation method is employed to calculate the second variations of these functions. Second variation expressions are obtained for the capacity, virtual mass, and eigenvalues corresponding to various particular ways in which D can be shifted. The variational theory is specialized to the case of 2 independent variables to show the existence of vortex sheets in axially symmetric, irrotational flow of an incompressible fluid. An external characterization of vortex sheets in 3-dimensional space without symmetry of any kind is sketched heuristically. In a study of the eigen functions and eigenvalues of the vibrating membrane, the second variation is used to show that under certain conformal mappings of a domain, which depend on a suitable parameter, the inverse square of the principal frequency of the domain becomes a convex function of the parameter.

A new proof of the Bieberbach conjecture for the fourth coefficient

or one of its rotations. The conjecture was proved to be true for n = 2, 3 and 4 by BIEBERBACH [1], LOEWNER [11] and GARABEDIAN & SCHIFFER [5], respectively. Alternate proofs for the case n = 4 have been provided in the papers [19], [4] and [14]. Recent evidence in support of the conjecture has been obtained by GARABEDIAN, ROSS & SCHIFFER [4], GARABEDIAN & SCHIFFER [6] and BOMBIERI [2]. In these papers it was shown that Rean<n if f (z) is sufficiently close to K(z) in various topologies. The author [14] proved that, at the Koebe point, these topologies are all equivalent. It is the purpose of this paper to prove the Bieberbach Conjecture for the sixth coefficient.

CONVESITY OF FUNCTIONALS BY TRANSPLANTATION

Abstract : The dependence of various functionals on their domain of definition is discussed. The functionals are defined by certain extremum problems. The methods of transplanting extremum functions and of variation are applied to the problem of utilizing the knowledge of the functional for a few special domains to obtain knowledge about the same functional in the general case. Functionals such as torsional rigidity, virtual mass, outer conformal radius, and electrostatic capacity are treated. A discussion is given of a theorem of Poincare which permits an easy simultaneous estimation of the N first eigenvalues of a general type of eigenvalue problem. The convexity of various combinations of eigenvalues is studied for the case in which the domain of definition is deformed by stretching or by conformal transformation. The usefulness of the fact that the initial domain D(1) has symmetry properties is indicated. The invariance of the class of harmonic functions in D under conformal mapping can be used to derive convexity statements for some functionals connected with the Greens function for Laplaces equation. A numerical application is given for the torsional rigidity of isosceles triangles and rectangles.

Kerr geometry as complexified Schwarzschild geometry

We present a simple derivation of the Schwarzschild and Kerr geometries by simplifying the Einstein free space field equations for the algebraically special form of metric studied by Kerr. This results in a system of two partial differential equations, the Laplace and eikonal equations, for a complex generating function. The metric tensor is a simple explicit functional of this generating function. The simplest solution generates the Schwarzschild geometry, while a displacement of the origin by ia in this solution generates the Kerr geometry.

A variational method for functions schlicht in an annulus

Abstract : Consideration is given to the family of function f(z) regular analytic and schlicht in the annulus R and satisfying the following conditions: (1) f(z) maps R onto the unit disk minus some continuum G, and (2) G contains the origin. Extremal problems (maximum modulus on the inner boundary and maximum displacement on the outer boundary of R) are solved using a specific method of variation within the family. The variation leads from every given f(z) to a large set of comparison functions within the family. The use of the variational method is further illustrated in finding the maximum diameter of the continuum G for all functions in the family F.

On the mean-value property of harmonic functions

THEOREM. Let D be a simply connected plane domain of finite area and t a point of D such that, for every function u harmonic in D and integrable over D, the mean value of u over the area of D equals u(t). Then D is a disc and t is its center. PROOF. Let f(z) be any function analytic in D with finite quadratic integral ffD If(z) I 2dxdy; by the Schwarz inequality, the real and imaginary parts of f(z) are each (absolutely) integrable over D, and so

Robin functions and distortion of capacity under conformal mapping

Under normalized conformal mappings of a multiply connected domain ω it is found that the sharp lower bound for the capacity of the image of a given set is the Robin capacity of A with respect to ω. The result is generalized to a sharp inequality for quadratic forms involving Greens function and the Robin function associated with A and ω. AMS No. 30C85, 30C35, 30C70, 31A15

THE THEORY OF THE SECOND VARIATION IN EXTREMUM PROBLEMS FOR UNIVALENT FUNCTIONS

Abstract : An attempt is made to improve the method of variations by considering further necessary conditions for the extremum function which arises from a study of the second variation. The formulas for the second variation of coefficients of univalent functions are, in general, so involved as to be impractical for a finer study of the extremum function. However, using the fact that this function satisfies a differential equation due to the first variational condition, success was reached in simplifying the expressions considerably. It was shown that a whole new set of necessary extremum conditions can be obtained to test every competing solution of the first variational condition. The new extremum conditions have the form of quadratic inequalities which are similar in type to those occurring through the method of contour integration. The characteristic difference lies in the fact that the quadratic inequalities have only to hold in the case of the extremum function, while in the other case the inequalities are asserted for all univalent functions. Nevertheless, it seems that the theory of the second variation is more closely connected with the method of contour integration than is that of the first variation. It might be possible to combine both methods for a unified approach to the general coefficient problem. (Author)

An extremal problem for the Fredholm eigenvalues

Here nt denotes the inner normal to C at t, and the integration is with respect to arc length. The integral equation (1) with a sign change is useful in constructive methods of conformal mapping [11], and with transposed kernel it leads to a solution of the Neumann boundary value problem. It can be shown [3] that the eigenvalues of the kernel k(z, t) are all real, are symmetric with respect to the origin, and lie outside the unit circle. The only exception is the trivial eigenvalue 1 corresponding to constant eigenfunctions. The integral equation (1) can be solved by successive approximations as follows. Let ~bo = u and

Conformal Mappings onto Nonoverlapping Regions

Let f(ζ) = a + dζ… be analytic and univalent in the unit disk |ζ| < 1, mapping it conformally onto some domain D. We shall call a = f(0) the center and |d| = |f′(0)| the inner radius of D with respect to a. Roughly speaking, our problem is to find n functions