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| The oval problem asks to determine, among all closed loops in
${\bf R}^n$ of fixed length, carrying a Schr\"odinger operator
${\bf H}= -\frac{d^2}{ds^2}+\kappa^2$ (with curvature $\kappa$ and
arclength $s$), those loops for which the principal eigenvalue of
${\bf H}$ is smallest. A 1-parameter family of ovals connecting the circle
with a doubly traversed segment (digon) is conjectured to be the minimizer.
Whereas this conjectured solution is an example that proves a lack of
compactness and coercivity in the problem, it is proved in this talk
(via a relaxed variation problem) that a minimizer exists; it is either
the digon, or a strictly convex planar analytic curve with positive
curvature. While the Euler-Lagrange equation of the problem appears
daunting, its asymptotic analysis near a presumptive singularity gives
useful information based on which a strong variation can exclude
singular solutions as minimizers. |
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