40] Variational methods in the theory of conformal mapping

Abstract

Max Schiffer was twice honored by invitations to address the International Congress of Mathematicians, at Cambridge (Massachusetts) in 1950 and at Edinburgh in 1958. The articles (Schiffer, Proceedings of the International Congress of Mathematicians, Cambridge, Mass., 1950, American Mathematical Society, Providence, vol. 2, pp. 233–240, 1952; Schiffer, Proceedings of the International Congress of Mathematicians, Edinburgh, 1958, Cambridge University Press, New York, pp. 211–231, 1960) are written versions of his lectures. Both of the articles are devoted to the variational methods for which Schiffer was best known, but they are rather different in character. The paper (Schiffer, Proceedings of the International Congress of Mathematicians, Cambridge, Mass., 1950, American Mathematical Society, Providence, vol. 2, pp. 233–240, 1952) offers a broad discussion of variational methods and their relative merits, and it surveys a variety of applications. The paper (Schiffer, Proceedings of the International Congress of Mathematicians, Edinburgh, 1958, Cambridge University Press, New York, pp. 211–231, 1960), on the other hand, focuses more narrowly on applications to particular extremal problems in function theory such as coefficient problems, and it gives a fairly detailed account of technical advances in the use of variational methods. In this respect, (Schiffer, Proceedings of the International Congress of Mathematicians, Edinburgh, 1958, Cambridge University Press, New York, pp. 211–231, 1960) is somewhat dated, since the Bieberbach conjecture has now been proved [see commentaries on (Schiffer, Proc. London Math. Soc., 44(2), 432–449, 1938; Schiffer, Proc. London Math. Soc., 44(2), 450–452, 1938; Schiffer and Charzynski, Arch. Rational Mech. Anal., 5, 187–193, 1960)] and the conjecture \(\vert b_{n}\vert \leq \frac{2} {n+1}\) for functions of class Σ has been disproved for all n ≥ 3 [see commentaries on (Schiffer, Bull. Soc. Math. France, 66, 48–55, 1938; Schiffer and Garabedian, Ann. of Math., 61(2), 116–136, 1955; Schiffer et al., J. Analyse Math., 40, 203–238, 1981)]. The paper (Schiffer, Proceedings of the International Congress of Mathematicians, Edinburgh, 1958, Cambridge University Press, New York, pp. 211–231, 1960) also gives an extended description of Schiffer’s method for obtaining a lower bound for the first nontrivial Fredholm eigenvalue of a simply connected domain with analytic boundary curve. Schiffer and others returned repeatedly to this problem [see Kuhnau’s commentary on (Schiffer, Pacific J. Math., 7, 1187–1225, 1957; Schiffer, Pacific J. Math., 9, 211–269, 1959; Schiffer, Rend. Mat., 22(5), 447–468, 1963; Schiffer, Ann. Polon. Math., 39, 149–164, 1981)]. The lower bound is important numerically, because it allows an estimate on the rate at which the Neumann series converges to the solution of the classical Poincare–Fredholm integral equation associated with the solution of a Dirichlet problem. In his expository paper (Schiffer, Rend. Mat. 22(5), 447–468, 1963), Schiffer gives a clear and detailed account of this beautiful circle of ideas.