#### Publication Date

2021

#### Document Type

Dissertation/Thesis

#### First Advisor

Linner, Anders

#### Degree Name

Ph.D. (Doctor of Philosophy)

#### Legacy Department

Department of Mathematical Sciences

#### Abstract

Consider constant-speed planar curves $\gamma =(x,y):[0,1]\to\mathbb{R}^2$ subject to $\gamma (0)=(0,0)$ and a prescribed value of $x(1)$, but with $y(1)$ unconstrained. We analyze the existence and the stability of critical curves of the elastic energy. Elastic curves are here thought of as points in an infinite-dimensional Sobolev manifold where the intrinsic gradient of the elastic energy vanishes. The admissible curves subject to constraints are members of a Riemannian submanifold using the ambient metric. A curve is stable with respect to the negative gradient flow if it is global or local minimum, and unstable otherwise. We analyze the stability of the classical Euler-Bernoulli elastica with vanishing curvature at the endpoints, and show that all are unstable. Our geometric treatment takes full advantage of the Riemannian structure. There is a smooth scalar field defined on the whole manifold that corresponds to the classical Lagrange multiplier values when evaluating the scalar field at the critical points. We illustrate the delicate nature of the stability issues here with natural examples in the presence of non-convex isoperimetric constraints. In the case of the constraint $x(1)=0$, a half of the ‘Euler figure eight’ is a candidate to be a critical point. It is of course possible to rotate the curve and still satisfy the constraints given in this case. Only the two ‘vertical’ positions turn out to be critical, and this has implications concerning mountain-pass issues when examining traversals between the ‘upwards’ and the ‘downwards’ globally minimizing straight-line segments, while insisting that the maximal elastic energy of the curves during the transition is as small as possible. The numerically delicate nature of stability involving isoperimetric constraints in the infinite dimensional Sobolev space setting is illustrated.

#### Recommended Citation

Rexford, Scott W., "Steepest Descent, Elastic Energy, and Stability in infinite Dimensional Hilbert Manifolds" (2021). *Graduate Research Theses & Dissertations*. 7592.

https://huskiecommons.lib.niu.edu/allgraduate-thesesdissertations/7592

#### Extent

167 pages

#### Language

eng

#### Publisher

Northern Illinois University

#### Rights Statement

In Copyright

#### Rights Statement 2

NIU theses are protected by copyright. They may be viewed from Huskie Commons for any purpose, but reproduction or distribution in any format is prohibited without the written permission of the authors.

#### Media Type

Text