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.
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