doiSerbia

Symmetries in some extremal problems between two parallel hyperplanes

Ribeiro Merkle Monica Moulin (Instituto de Matemática, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil)

Let M be a compact hypersurface with boundary ӘM = ӘD1(ӘD2, ӘD1 ( Π1, ӘD2 ( Π2, Π1 and Π2 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 Πi,i = 1,2 and on (u. We prove that these hypersurfaces are symmetric to a perpendicular orthogonal to Πi,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 Πi,i = 1,2 is constant; (ii) when Әu/Әη is a non-increasing function of the mean curvature of the boundary, Әη the inward normal; (iii) when Әu/Әη has a linear dependency on the distance to a fixed point inside the body that hypersurface englobes; (iv) when ӘDi are symmetric to a perpendicular orthogonal to Πi,i=1,2.

Keywords: Prescribed mean curvature, Capillarity, Symmetry