Existence and Regularity in the Oval Problem

Series: 
Math Physics Seminar
Tuesday, November 12, 2013 - 16:00
1 hour (actually 50 minutes)
Location: 
Skiles 006
,  
University of Tennessee, Knoxville
,  
The oval problem asks to determine, among all closed loops in${\bf R}^n$ of fixed length, carrying a Schrödinger operator${\bf H}= -\frac{d^2}{ds^2}+\kappa^2$ (with curvature $\kappa$ andarclength $s$), those loops for which the principal eigenvalue of${\bf H}$ is smallest. A 1-parameter family of ovals connecting the circlewith a doubly traversed segment (digon) is conjectured to be the minimizer.Whereas this conjectured solution is an example that proves a lack ofcompactness and coercivity in the problem, it is proved in this talk(via a relaxed variation problem) that a minimizer exists; it is eitherthe digon, or a strictly convex planar analytic curve with positivecurvature. While the Euler-Lagrange equation of the problem appearsdaunting, its asymptotic analysis near a presumptive singularity givesuseful information based on which a strong variation can excludesingular solutions as minimizers.