Complementary and Dual Variational Principles in Mechanics

Abstract

In this chapter, we consider a theory of complementary and dual variational principles associated with a large class of linear boundary- and initial-value problems of mechanics. The mathematical setting for the class of problems we wish to examine is the following abstract linear problem: find an element u of a Hilbert space U such that

$$ \Lambda {\text{u}}\,\, = {\text{F,}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{F}} \in \,{\text{U'}}\,\, $$

(4.1)

where Λ is a linear operator from U into its dual U′ which is given as a product of three linear operators,

$$ \Lambda \,\, = \,\,\,{\text{A*EA}} $$

(4.2)

Here A is a continuous linear operator from U into a Hilbert space V, E is a canonical isomorphism mapping V onto its dual V′, and A* is the adjoint of A which, by definition, maps V′ into U′. Summarizing,

$$ \left. {\begin{array}{*{20}{c}} {{\text{A:}}\,\,\,U\,\, \to \,\,V} \\ {{\text{E:}}\,\,\,V\,\, \to \,\,V'} \\ {{\text{A*:}}\,\,\,V'\,\, \to \,\,U'} \\ {\Lambda = {\text{A*EA:}}\,\,\,U\,\, \to \,\,U'} \end{array}\,\,\,\,\,\,\,\,\,} \right\} $$

(4.3)