Why are the most stable phase boundaries round?

Dr. Evans M. Harrell

School of Mathematics
Georgia Institute of Technology

January 24, 1997

Abstract

In a popular article, Hilbert once discussed eleven geometric ways to characterize the sphere, and many additional special properties of the sphere are known today. Recently, E. Harrell and M. Loss have discovered new characterizations of the sphere and circle as the most stable shapes for phase boundaries in certain materials. Instabilities come from negative eigenvalues of a differential equation involving curvature. Counting the eigenvalues requires an interplay between operator theory and elementary geometry of curves and surfaces.