Some Existence Conditions for the Compact Extrema of Variational Functionals of Several Variables in Sobolev Space W 2 1

Elimination of Hamilton-Jacobi equation in extreme variational problems

“…In the our works [5]- [7] and in the works by E.V. Bozhonok [8]- [10] a general notion of compact extremum (or K-extremum) of a functional was studied (see, also, [11]). It have been shown there that the classical, both necessary and sufficient conditions of a local extremum of variational functional in C 1 [a; b] can be extended to the case of K-extremum in The well posedness and the validity for the case of K-extremum of the variational functional (1) of the classical extreme conditions in C 1 (Euler-Lagrange equation, Legendre condition, Jacobi condition) require, as it was shown in [7], belonging coefficient R(x, y, z) in the quadratic representation (10) of the integrand f :…”