Geodesic
A uniqueness theorem for the dual problem associated to a~variational problem with linear growth
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 31, Tome 271 (2000), pp. 83-91

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Uniqueness is proved for solutions of the dual problem which is associated to the minimum problem $\int_\Omega f(\nabla u)dx\to\min$ among mappings $u\colon\mathbb R^n\supset\Omega\to\mathbb R^N$ with prescribed Dirichlet boundary data and for smooth strictly convex integrands $f$ of linear growth. No further assumptions on $f$ or its conjugate function $f^*$ are imposed, in particular $f^*$ is not assumed to be strictly convex. One special solution of the dual problem is seen to be a mapping into the image of $\nabla f$ which immediately implies uniqueness. 
@article{ZNSL_2000_271_a5,
     author = {M. Bildhauer},
     title = {A uniqueness theorem for the dual problem associated to a~variational problem with linear growth},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {83--91},
     publisher = {mathdoc},
     volume = {271},
     year = {2000},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2000_271_a5/}
}
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M. Bildhauer. A uniqueness theorem for the dual problem associated to a~variational problem with linear growth. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 31, Tome 271 (2000), pp. 83-91. http://geodesic.mathdoc.fr/item/ZNSL_2000_271_a5/