Geodesic
Inverse extremal problem for variational functionals
Eurasian mathematical journal, Tome 1 (2010) no. 4, pp. 95-115.

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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. 
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I. V. Orlov. Inverse extremal problem for variational functionals. Eurasian mathematical journal, Tome 1 (2010) no. 4, pp. 95-115. http://geodesic.mathdoc.fr/item/EMJ_2010_1_4_a4/

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