Analysis of local minima for constrained minimization problems

Abstract

We consider vectorial problems in the calculus of variations with an additional pointwise constraint. Our admissible mappings $\mathbf{n}:{\mathbb{R}}^k\to {\mathbb{R}}^d$ satisfy $\mathbf{n}(x)\in M$, 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 $\mathbf{n}$ 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.