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Analysis and PDE seminar: Dmitry Khavinson

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Speaker: Dmitry Khavinson, Distinguished Professor,

Department of Mathematics, University of South Florida, Tampa, USA.

Title: Isoperimetric "sandwiches" and some free boundary boundary problems via approximation by analytic and harmonic functions

Abstract: 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. However, a rigorous proof was only achieved towards the end of the 19th century! I will start by discussing some of the history of the problem and several classical proofs of the isoperimetric inequality ( e.g., those due to Steiner, Hurwitz and Carleman).. Then we shall move on to a larger variety of isoperimetric inequalities, as , e.g., in Polya and Szego classic book of 1949, 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`` 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. Many of the classical isoperimetric problems studied this way lead to natural free boundary problems for PDE, many of which remain unsolved today. Then, as an example, I will talk about some applications to the study of shapes of electrifed droplets and small air bubbles in fluid flow. During the talks I will try not only to survey the known results and methods but focus especially on many open problems that remain. This series of talks is going to be definitely accessible to the first year graduate students, or advanced undergraduates majoring in mathematics and physics who have had a semester course in complex analysisvand a routine course in advanced calculus.