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.