Symmetries in some extremal problems between two parallel hyperplanes

Abstract: 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 $\Pi _2$ two parallel hyperplanes in $\mathbb{R}^{n+1}$ ($n \geq 2$). Suppose that $M$ 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 $\Pi _i$, $i=1,2$, under different conditions imposed on the boundary of hypersurfaces on the parallel planes: (i) when the angle of contact between $M$ and $\Pi _i$, $i=1,2$ is constant; (ii) when $\partial u / \partial \eta$ is a non-increasing function of the mean curvature of the boundary, $\partial \eta$ 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 $\partial D_i$ are symmetric to a perpendicular orthogonal to $\Pi _i$, $i=1,2$.