Symmetries in some extremal problems between two parallel hyperplanes

Abstract

Let M be a compact hypersurface with boundary ?M = ?D1(?D2, ?D1 ( ?1, ?D2 ( ?2, ?1 and ?2 two parallel hyperplanes in Rn+1 (n ? 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 ?i,i = 1,2 and on (u. We prove that these hypersurfaces are symmetric to a perpendicular orthogonal to ?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 ?i,i = 1,2 is constant; (ii) when ?u/?? is a non-increasing function of the mean curvature of the boundary, ?? the inward normal; (iii) when ?u/?? has a linear dependency on the distance to a fixed point inside the body that hypersurface englobes; (iv) when ?Di are symmetric to a perpendicular orthogonal to ?i,i=1,2.