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 $\realnum$ ($n \geq 2$). 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 $\Pi _i$, $i=1,2$ and on $\nabla u$. 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 $\Pi _i$, $i=1,2$.