Hypersurfaces with free boundary and large constant mean curvature: concentration along submanifolds

Abstract

Let Ω be an open bounded subset of R, m ≥ 2, with smooth boundary ∂Ω. Recall that the partitioning problem in Ω consists on finding, for a given 0 < v < meas (Ω), a critical point of the perimeter functional P( · , Ω ) in the class of sets in Ω that enclose a volume v. Here P(E , Ω ) denotes the perimeter of E relative to Ω. It is clear that whenever such a surface exits will meet ∂Ω orthogonally and will have a constant mean curvature, see Section 2.3.1. In the light of standard results in geometric measure theory, minimizers do exist for any given volume and may have various topologies (see the survey by A.Ros [17]). Actually, up to now the complete description of minimizers have been achieved only in some special cases, one can see for example [1], [16], [19] and [21]. However, the study of existence, geometric and topological properties of stationary surfaces (not necessarily minimizers) is far from being complete. Let us mention that Gruter-Jost [4], have proved the existence of minimal discs into convex bodies; while Jost in [6] proved the existence of embedded minimal surfaces of higher genus. In the particular case of the free boundary Plateau problem, some rather global existence results were obtained by M. Struwe in [22], [23] and [24]. In [2], the first author proved the existence of surfaces similar to half spheres surrounding a small volume near nondegenerate critical points of the mean curvature of ∂Ω. Here we are interested in the existence of families of stationary sets Ee for the perimeter functional relative to Ω having small volume measEe proportional to e. Our result generalizes to higher dimensional sets the one obtained by the first author in [2]. Before stating it some preliminaries are needed. We denote by V the interior normal vector field along ∂Ω. For a given smooth set E ⊂ Ω with finite perimeter, let Σ := ∂E∩Ω satisfy ∂Σ ⊂ ∂Ω and denote by N its exterior normal vector field. For a smooth vector field X in R, the flow of diffeomorphism {Ft}t∈(0,t∗) of X in Ω induces a variation {Et = Ft(E)}t of E. Set A(t) = P(Et,Ω); V (t) = meas(Et) and