Computational Methods and Function Theory 6 (2006), No. 2, 253--274
Copyright Heldermann Verlag 2006
Catherine Bénéteau
beneteca@shu.edu , Department of Mathematics and Computer Science, Seton Hall University, South Orange, New Jersey 07079, U.S.A.
Dmitry Khavinson
dmitry@uark.edu, dkhavins@nsf.gov , Department of Mathematics, University of Arkansas, Fayetteville, Arkansas 72701, U.S.A.
In this survey paper, we examine the isoperimetric inequality from an analytic point of view. We use as a point of departure the concept of analytic content in approximation theory: this approach reveals ties to overdetermined boundary problems and hydrodynamics. In particular, we look at problems connected to determining the shape of an electrified droplet or equivalently, that of an air bubble in fluid flow. We also discuss the connection with the Schwarz function and quadrature domains. Finally, we survey some known generalizations to higher dimensions and list many open problems that remain. This paper is an expanded version of the plenary talk given by the second author at the fifth CMFT Conference in Joensuu, Finland, in June 2005.
Keywords: Analytic content, free boundary problems, hydrodynamics, soperimetric inequality, quadrature domains, rational approximation.
MSC 2000: Primary 30H05; Secondary 35R35.
[FullText-pdf (345 K)] [FullText-ps (563 K)]