Analysis of local minima for constrained minimization problems

TL;DR

This paper investigates the conditions under which certain constrained vector fields are local minimizers in calculus of variations, applying the findings to analyze the stability of liquid crystal configurations.

Contribution

It develops necessary and sufficient conditions for local minimizers in constrained variational problems using projection methods, extending existing unconstrained results.

Findings

Derived conditions for weak and strong local minimizers

Applied results to liquid crystal stability analysis

Quantified stability of cholesteric states under boundary conditions

Abstract

We consider vectorial problems in the calculus of variations with an additional pointwise constraint. Our admissible mappings ${\bf n}:\mathbb{R}^k\rightarrow \mathbb{R}^d$ satisfy ${\bf n}(x)\in M$, where $M$ is a manifold embedded in Euclidean space. The main results of the paper all formulate necessary or sufficient conditions for a given mapping ${\bf n}$ to be a weak or strong local minimizer. Our methods involve using projection mappings in order to build on existing, unconstrained, local minimizer results. We apply our results to a liquid crystal variational problem to quantify the stability of the unwound cholesteric state under frustrated boundary conditions.