[54] Variation of domain functionals

Abstract

This survey article is a written version of an invited address given at a meeting of the American Mathematical Society. It can be viewed as a sequel to (Schiffer, Amer. J. Math., 68, 417–448, 1946), which has a similar title. Comparing [54] with (Schiffer, Amer. J. Math., 68, 417–448, 1946), one is amazed to see how far the variational method had progressed and how much Schiffer had accomplished, in the short span of 8 years. In Schiffer (Amer. J. Math., 68, 417–448, 1946), he was just starting to look beyond univalent functions, broadening his outlook by adapting the variation of Green’s function to problems involving transfinite diameter, harmonic measures, the Bergman kernel function, and Riemann surfaces. Eight years later, he had completed books on two of those topics, one with Bergman and Schiffer (Kernel Functions and Elliptic Differential Equations in Mathematical Physics, 1953) on kernel functions and their application to boundary value problems for elliptic partial differential equations, the other with Spencer and Schiffer (Functionals of Finite Riemann Surfaces, 1954) applying variational methods to extremal problems for Riemann surfaces. He had written a series of important papers with Garabedian and Schiffer (Trans. Amer. Math. Soc., 65, 187–238, 1949; Ann. of Math., 52(2), 164–187, 1950; Ann. of Math., 56(2), 560–602, 1952; J. Rational Mech. Anal., 2, 137–171, 1953; J. Analyse Math., 2, 281–368, 1953; Proc. Amer. Math. Soc., 5, 206–211, 1954) on application of variational techniques to existence theorems and three-dimensional problems of mathematical physics. He had also completed major papers with Szegő and Schiffer (Trans. Amer. Math. Soc., 67, 130–205, 1949) on virtual mass and polarization and with Pólya and Schiffer (J. Analyse Math., 3, 245–346, 1954) on variation by transplantation, an ingenious method for verifying that a suspected extremal function is indeed extremal.