# Isoperimetric "sandwiches", free boundary boundary problems and approximation by analytic and harmonic functions

The isoperimetric problem, posed by the Greeks, proposes to find among all simple closed curves the one that surrounds the largest area. The isoperimetric theorem then states that the curve is a circle. It is frst mentioned in the writings of Pappus in the third century A.D. and is attributed there to Zenodorus. Steiner in 1838 was the first to attempt a "rigorous" proof. However, first truly rigorous proofs were only achieved in the beginning of the 20th century (e.g., Caratheodory, Hurwitz, Carleman, ...). We shall discuss a variety of isoperimetric inequalities, as, e.g., in Polya and Szego 1949 classics, but deal with them via a relatively novel approach based on approximation theory. Roughly speaking, this approach can be characterized by a recently coined term "isoperimetric sandwiches". A certain quantity is introduced, usually as a degree of approximation to a given simple function, e.g., z* , |x|2, by either analytic or harmonic functions in some norm. Then, the estimates from below and above of the approximate distance are obtained in terms of simple geometric characteristics of the set, e.g., area, perimeter, capacity, torsional rigidity, etc. The resulting "sandwich" yields the relevant isoperimetric inequality. Several of the classical isoperimetric problems approached in

this way lead to natural free boundary problems for PDE, many of which remain unsolved today. (An example of such free boundary problem is the problem of a shape of an electrifed droplet , or a small air bubble in fluid flow. Another example is identifying a cross-section of laminary flow of viscous fluid that exhibits constant pressure on the pipe walls, J. Serrin's problem.)

On the maximal ideal spaces of some algebras of analytic