Geodesic
Minimal hypersurfaces over soft obstacles
Izvestiya. Mathematics , Tome 8 (1974) no. 2, pp. 379-421

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In the present work the following variational problem is discussed: minimize the area functional $$ F(u)=\int_G\sqrt{1+|\nabla u|^2}\,dx $$ in the class of all functions $W_0^{1,1}(G)$ for which $\int_{D\Subset G}u\,dx\geqslant V=\mathrm{const}$. For small enough $V$ the existence of an extremal is proved, and it is shown that it belongs to $C^{1,\alpha}(\overline G)$ with a Hölder index $\alpha$, $0\alpha\leqslant1$. 
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     author = {O. V. Titov},
     title = {Minimal hypersurfaces over soft obstacles},
     journal = {Izvestiya. Mathematics },
     pages = {379--421},
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     volume = {8},
     number = {2},
     year = {1974},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1974_8_2_a6/}
}
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O. V. Titov. Minimal hypersurfaces over soft obstacles. Izvestiya. Mathematics , Tome 8 (1974) no. 2, pp. 379-421. http://geodesic.mathdoc.fr/item/IM2_1974_8_2_a6/