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.