Generalized elastica problems under area constraint

It was recently proved in [3, 4] that the elastic energy $E(\gamma) = \frac{1}{2} \int_{\gamma} \kappa^2 ds$ of a closed curve $\gamma$ with curvature $\kappa$ has a minimizer among all plane, simple, regular and closed curves of given enclosed area $A(\gamma)$, and that the minimum is attained only for circles. In particular, the proof used in [4] is of a geometric nature, and here we show under which hypothesis it can be extended to other functionals involving the curvature. As an example we show that the optimal shape remains a circle for the $p$-elastic energy $\int_{\gamma} {\lvert \kappa \rvert}^p ds$, whenever $p \gt 1$.

Keywords

Euler elastica, isoperimetric inequality

2010 Mathematics Subject Classification

49Q10, 51M16

Full Text (PDF format)

Received 7 June 2016

Accepted 28 July 2017

Published 5 July 2018