Title

Minimal Energy Problems and Applications: Ping Pong Balayage and Convexity of Equilibrium Measures

Document Type

Presentation

Presentation Date

9-16-2010

Conference Name

University of South Alabama Colloquium

Conference Location

Mobile, AL, USA

Peer Review

Invited

Abstract

This talk has two parts. Initially, I will present a survey of various minimal energy problems in potential theory having applications to different areas of research not only in mathematics, but in physics, biology, chemistry, etc. In addition to being a preview of a sequence of more detailed lectures to be given at subsequent colloquia and analysis seminars, the survey will introduce the necessary preliminaries to present the main result.

Let E be the union of finitely many intervals or arcs on the unit circle. In a joint work with David Benko we prove that the equilibrium measure of E has a convex density. This is true for both the classical logarithmic kernel, and the Riesz kernel. This seems to be a fundamental result in potential theory, maybe one for the books.

The electrostatic interpretation is the following: if we have a finite union of subintervals on the real line, or arcs on the unit circle, the electrostatic distribution of "many electrons" will have convex density on every subinterval.

Disciplines

Analysis | Mathematics

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