Minimal hypersurfaces over soft obstacles

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 $\alpha$, $0<\alpha\leqslant1$.