Geodesic
An incompressibility condition for a~certain class of integral functionals.~I
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 14, Tome 115 (1982), pp. 203-214

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One considers the integral functional $\widehat H(y),$ depending on the mapping of the domain $\Omega\subset R^m$ into $R^m$, on the set of mappings $y$, subjected to the incompressibility condition: $\det\dot y=1$. One computes its first and second variations. The obtained results are compared with the formulas arising from the formal application of the method of the undetermined Lagrange multipliers. One gives an application to problems of elasticity theory. 
@article{ZNSL_1982_115_a16,
     author = {V. G. Osmolovskii},
     title = {An incompressibility condition for a~certain class of integral {functionals.~I}},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {203--214},
     publisher = {mathdoc},
     volume = {115},
     year = {1982},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1982_115_a16/}
}
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V. G. Osmolovskii. An incompressibility condition for a~certain class of integral functionals.~I. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 14, Tome 115 (1982), pp. 203-214. http://geodesic.mathdoc.fr/item/ZNSL_1982_115_a16/