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.
Categories: math.AP