We introduce a new variational method for the study of stability in the isoperimetric inequality. The method is quite general as it relies on a penalization technique combined with the regularity theory for quasiminimizers of the perimeter. Two applications are presented. First we give a new proof of the sharp quantitative isoperimetric inequality in R n . Second we positively answer to a conjecture by Hall concerning the best constant for the quantitative isoperimetric inequality in R 2 in the small asymmetry regime.2000 Mathematics Subject Classification. 52A40 (28A75, 49J45).
Abstract. In this paper we consider the Cheeger problem for non-convex domains, with a particular interest in the case of planar strips, which have been extensively studied in recent years. Our main results are an estimate on the Cheeger constant of strips, which is stronger than the previous one known from [18], and the proof that strips share with convex domains a number of crucial properties with respect to the Cheeger problem. Moreover, we present several counterexamples showing that the same properties are not valid for generic non-convex domains.
It has been recently conjectured that, in the context of the Heisenberg groupHn endowed with its Carnot–Carathéodory metric and Haar measure, the isoperimetricsets (i.e., minimizers of the H-perimeter among sets of constant Haar measure) couldcoincide with the solutions to a “restricted” isoperimetric problem within the class ofsets having finite perimeter, smooth boundary, and cylindrical symmetry. In this paper,we derive new properties of these restricted isoperimetric sets, which we call Heisenbergbubbles. In particular, we show that their boundary has constant mean H-curvature and, quitesurprisingly, that it is foliated by the family of minimal geodesics connecting two specialpoints. In view of a possible strategy for proving that Heisenberg bubbles are actuallyisoperimetric among the whole class of measurable subsets of Hn, we turn our attentionto the relationship between volume, perimeter, and -enlargements. In particular, we provea Brunn–Minkowski inequality with topological exponent as well as the fact that the Hperimeterof a bounded, open set F ⊂ Hn of class C2 can be computed via a generalizedMinkowski content, defined by means of any bounded set whose horizontal projection is the2n-dimensional unit disc. Some consequences of these properties are discussed
For a large class of equiregular sub-Riemannian manifolds, we show that length-minimizing curves have no corner-like singularities. Our first result is the reduction of the problem to the homogeneous, rank-2 case, by means of a nilpotent approximation. We also identify a suitable condition on the tangent Lie algebra implying existence of a horizontal basis of vector fields whose coefficients depend only on the first two coordinates x 1 , x 2 . Then, we cut the corner and lift the new curve to a horizontal one, obtaining a decrease of length as well as a perturbation of the end-point. In order to restore the end-point at a lower cost of length, we introduce a new iterative construction, which represents the main contribution of the paper. We also apply our results to some examples.
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