On quasiclassical solutions of an inverse problem in the calculus of variations. (English. Russian original) Zbl 0605.47062

Filippov, V. M.

Sov. Math., Dokl. 32, 646-649 (1985); translation from Dokl. Akad. Nauk SSSR 285, 53-56 (1985).

The usual basis for a variational method for solving a nonlinear equation \(N(u)=f\) is a functional \(\phi\) such that \((\nabla \phi)(u)=N(u)-f\) for all \(u\in D(N)\). The existence of such a \(\phi\) is guaranteed only under restrictive conditions on N. However, all that is really needed is that a functional F exists such that \((\delta F)(\bar u)=0\) if and only if \(\bar u\in D(N)\) and \(N(\bar u)=f\), where \(\delta\) is the Gateaux variation. Several examples of such functionals are given. Based on one of them, the variational theory for finding generalized solutions of \(N(u)=f\) is developed.

Reviewer: H.W.Engl

MSC:

47J25	Iterative procedures involving nonlinear operators
49K27	Optimality conditions for problems in abstract spaces

Keywords:

variational method for solving a nonlinear equation; Gateaux variation; variational theory for finding generalized solutions