Uniqueness Theorems for Variational Problems by the Method of Transformation Groups

A classical problem in the calculus of variations is the investigation of critical points of functionals {cal L} on normed spaces V. The present work addresses the question: Under what conditions on the functional {cal L} and the underlying space V does {cal L} have at most one critical point?

A sufficient condition for uniqueness is given: the presence of a "variational sub-symmetry", i.e., a one-parameter group G of transformations of V, which strictly reduces the values of {cal L}. The "method of transformation groups" is applied to second-order elliptic boundary value problems on Riemannian manifolds. Further applications include problems of geometric analysis and elasticity.