Abstract
Nonlinear functionals that appear as a product of two integrals are considered in the context of elastic curves of variable length. A technique is introduced that exploits the fact that one of the integrals has an integrand independent of the derivative of the unknown. Both the linear and the nonlinear cases are illustrated. By lengthening parameterized curves it is possible to reduce the elastic energy to zero. It is shown here that for graphs this is not the case. Specifically, there is a unique graph of minimal elastic energy among all graphs that have turned 90 degrees after traversing one unit.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 2021-2041 |
| Number of pages | 21 |
| Journal | Transactions of the American Mathematical Society |
| Volume | 359 |
| Issue number | 5 |
| DOIs | |
| State | Published - May 2007 |
Keywords
- Elastic energy
- Maximum principle
- One-dimensional Willmore equation
- Optimal graph
- Pendulum equation
- Phase constraint
- Pontrjagin
- Variable length
ASJC Scopus subject areas
- General Mathematics
- Applied Mathematics