Geodesic
Regularity for dual solutions and for weak cluster points of minimizing sequences of variational problems with linear growth
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 30, Tome 259 (1999), pp. 46-66

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The minimum problem $\int_{\Omega}f(\nabla u)dx\longrightarrow\min$ among mappings $u:\mathbb R^n\supset\Omega\to\mathbb R^N$ with prescribed Dirichlet boundary data and for integrands $f$ of linear growth in general fails to have solutions in the Sobolev space $W^1_1$. We therefore concentrate on the dual variational problem which admits a unique maximizer $\sigma$ and prove partial Hölder continuity of $\sigma$. Moreover, we study smoothness properties of $L^1$-limits of minimizing sequences of the original problem. 
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     author = {M. Fuchs and M. Bildhauer},
     title = {Regularity for dual solutions and for weak cluster points of minimizing sequences of variational problems with linear growth},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {46--66},
     publisher = {mathdoc},
     volume = {259},
     year = {1999},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1999_259_a2/}
}
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M. Fuchs; M. Bildhauer. Regularity for dual solutions and for weak cluster points of minimizing sequences of variational problems with linear growth. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 30, Tome 259 (1999), pp. 46-66. http://geodesic.mathdoc.fr/item/ZNSL_1999_259_a2/