Symmetries in some extremal problems between two parallel hyperplanes

Let $M$ be a compact hypersurface with boundary $\partial M=\partial D_1 \cup \partial D_2$,$\partial D_1 \subset \Pi _1$,$\partial D_2 \subset \Pi _2$,$\Pi_1$and$\Pi _2$twoparallelhyperplanesin$$\mathbb{R}$^{n+1}$($n \geq 2$)\mathrm{.}Supposethat$M$ is contained in the slab determined by these hyperplanes and that the mean curvature $H$ of $M$ depends only on the distance $u$ to $\Pi _i$,$i=1,2$. We prove that these hypersurfaces are symmetric to a perpendicular orthogonal to ${\mathrm{\Pi }}_i$, $i=1,2$, under different conditions imposed on the boundary of hypersurfaces on the parallel planes: (i) when the angle of contact between $M$ and ${\mathrm{\Pi }}_i$, $i=1,2$ is constant; (ii) when $\mathrm{\partial }u\mathrm{/}\mathrm{\partial }\eta$ is a non-increasing function of the mean curvature of the boundary, $\mathrm{\partial }\eta$ the inward normal; (iii) when $\partial u / \partial \eta$ has a linear dependency on the distance to a fixed point inside the body that hypersurface englobes; (iv) when $\mathrm{\partial }{D}_i$ are symmetric to a perpendicular orthogonal to ${\mathrm{\Pi }}_i$, $i=1,2$.