Inverse extremal problem for variational functionals

We investigate an inverse extremal problem for the variational functionals: to describe, under certain conditions, all types of variational functionals having a local extremum (in case of the space $C^1[a;b]$) or a compact extremum (in case of the Sobolev space $W^{1,2}[a;b]=H^1[a;b]$) at a given point of the corresponding function space. The non-locality conditions for a compact extrema of variational functionals are described as well.