Isoperimetric, Sobolev and Poincaré inequalities on hypersurfaces in sub-Riemannian Carnot groups

Details Isoperimetric, Sobolev and Poincaré inequalities on hypersurfaces in sub-Riemannian Carnot groups Isoperimetric, Sobolev and Poincaré inequalities on hypersurfaces in sub-Riemannian Carnot groups

2009-10-29

arxiv.org/abs/0910.5656v1

Mathematics

Analysis of PDEs

61 pages

Scientific paper

In this paper we shall study smooth submanifolds immersed in a k-step Carnot group G of homogeneous dimension Q. Among other results, we shall prove an isoperimetric inequality for the case of a $C^2$-smooth compact hypersurface S with - or without - boundary $\partial S$; S and $\partial S$ are endowed with their homogeneous measures, actually equivalent to the intrinsic (Q-1)-dimensional and (Q-2)-dimensional Hausdorff measures with respect to some homogeneous metric $\varrho$ on G; see Section 5. This generalizes a classical inequality, involving the mean curvature of the hypersurface, proven by Michael and Simon [63] and, independently by Allard [1]. In particular, from this result one may deduce some related Sobolev-type inequalities; see Section 7. The strategy of the proof is inspired by the classical one. In particular, we shall begin by proving some linear isoperimetric inequalities. Once this is proven, one can deduce a local monotonicity formula and then conclude the proof by a covering argument. We stress however that there are many differences, due to our different geometric setting. Some of the tools which have been developed ad hoc in this paper are, in order, a ``blow-up'' theorem, which also holds for characteristic points, and a smooth Coarea Formula for the HS-gradient; see Section 3 and Section 4. Other tools are the horizontal integration by parts formula and the 1st variation of the H-perimeter already developed in [68], [69], and here generalized to hypersurfaces having non-empty characteristic set. Some natural applications of these results are in the study of minimal and constant horizontal mean curvature hypersurfaces. Moreover we shall prove some purely horizontal, local and global Poincar\'e-type inequalities as well as some related facts and consequences; see Section 4 and Section 5.

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