Fractional Sobolev norms and BV functions on manifolds

Abstract

The bounded variation seminorm and the Sobolev seminorm on compact manifolds are represented as a limit of fractional Sobolev seminorms. This establishes a characterization of functions of bounded variation and of Sobolev functions on compact manifolds. As an application the special case of sets of finite perimeter is considered.

Fractional perimeters on the sphere

“…If E has sufficiently regular boundary, P just coincides with the usual surface area. An analogous result was already shown for sets of finite perimeter in more general Riemannian manifolds than the sphere in [16]. Hence, this result actually does not depend on the particular structure of the sphere.…”

“…In Ludwig's proof a result of Wieacker [28,Theorem 1] for sets of finite perimeter is used which is only available in the Euclidean setting. Our convergence result holds for the larger class of sets of finite perimeter on the sphere as shown for general compact Riemannian manifolds in [16]. Thus, we decided not to include a proof of a spherical version of Wieacker's result since our objective is of conceptual nature.…”

“…It is natural to extend this notion to the setting of Riemannian manifolds. Let (M, g) be a compact, connected Riemannian manifold of dimension n, d the geodesic distance and dV g the volume element induced by the Riemannian metric g. In [16] the fractional s-perimeter of a Borel set E ⊆ M is defined as…”

Asymptotic Behaviours in Fractional Orlicz–Sobolev Spaces on Carnot Groups

“…Nevertheless, recent contributions started to attack Bourgain-Brezis-Mironescutype results even in the case where non-Euclidean geometries appear. We refer, for instance, to [25] for the case of compact Riemannian manifolds. As for many other problems, one of the non-Euclidean setting where to look for extensions is provided by Carnot groups, that are connected, simply connected and nilpotent Lie groups whose associated Lie algebra is stratified (see Section 2 for more details).…”

The anisotropic fractional isoperimetric problem with respect to unconditional unit balls

“…Here, we say that a set K ⊂ R n is a convex body if it is compact, convex, and has non-empty interior. The isotropic case, that is, K = B is the Euclidean unit ball, leads to the (Euclidean) fractional perimeter (denoted by P s (E)) which is closely connected to the theory of fractional Sobolev spaces and has been extensively studied over the last two decades (see [4,7,9,10,13,15,19,20,32] and the references therein). In particular, for bounded Borel sets E ⊂ R n the fractional isoperimetric inequality…”