Uniqueness Theorems for Variational Problems by the Method of Transformation Groups

Wolfgang Reichel (auth.)3540218394, 9783540218395

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.

Table of contents :
1. Introduction….Pages 1-7
2. Uniqueness of critical points (I)….Pages 9-26
3. Uniqueness of critical points (II)….Pages 27-57
4. Variational problems on Riemannian manifolds….Pages 59-87
5. Scalar problems in Euclidean space….Pages 89-125
6. Vector problems in Euclidean space….Pages 127-138
Appendix….Pages 139-143
References….Pages 145-149