Uniqueness Theorems for Variational Problems by the Method of Transformation Groups

• Main Author: Reichel, Wolfgang (Author, http://id.loc.gov/vocabulary/relators/aut)
• Corporate Author: SpringerLink (Online service)
• Format: Electronic eBook
• Language: English
• Published: Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 2004.
• Edition: 1st ed. 2004.
• Series: Lecture Notes in Mathematics, 1841
• Subjects: Calculus of variations.; Partial differential equations.; Calculus of Variations and Optimal Control; Optimization.; Partial Differential Equations.
• Physical Description: XIV, 158 p. online resource.
• ISBN: 9783540409151
• ISSN: 0075-8434 ;
• DOI: 10.1007/b96984

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.