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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 subsymmetry", i.e., a oneparameter group G of transformations of V, which strictly reduces the values of {cal L}. The "method of transformation groups" is applied to secondorder elliptic boundary value problems on Riemannian manifolds. Further applications include problems of geometric analysis and elasticity.
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