A Poincare inequality with applications to volume-constrained area-minimizing surfaces

We derive a Poincare-type inequality which is satisfied along the boundary of any set of finite perimeter which is stable with respect to perimeter among volume-preserving perturbations, provided the singular set is of sufficiently low Hausdorff dimension. This inequality in particular yields the connectivity of stable solutions in convex domains, instability of cones within a ball and regularity in all dimensions for local minimizers in certain settings within a ball. We also derive an analogous inequality satisfied by stable critical points for semilinear elliptic problems with Neumann boundary conditions either with or without a mass constraint.