Some remarks about the density of smooth functions in weighted Sobolev spaces.
Gehring's lemma for metrics and higher integrability of the gradient for minimizers of noncoercive variational functionals
We prove a higher integrability result - similar to Gehring's lemma - for the metric space associated with a family of Lipschitz continuous vector fields by means of sub-unit curves. Applications are given to show the higher integrability of the gradient of minimizers of some noncoercive variational functionals.
Rectifiability and parameterization of intrinsic regular surfaces in the Heisenberg group
We construct an intrinsic regular surface in the first Heisenberg group equipped wiht its Carnot-Carathéodory metric which has euclidean Hausdorff dimension . Moreover we prove that each intrinsic regular surface in this setting is a -dimensional topological manifold admitting a -Hölder continuous parameterization.
Rectifiability and perimeter in step 2 groups
Comparison of Hausdorff measures with respect to the Euclidean and the Heisenberg metric.
We compare the Hausdorff measures and dimensions with respect to the Euclidean and Heisenberg metrics on the first Heisenberg group. The result is a dimension jump described by two inequalities. The sharpness of our estimates is shown by examples. Moreover a comparison between Euclidean and H-rectifiability is given.
Rectifiability and perimeter in step 2 Groups
We study finite perimeter sets in step 2 Carnot groups. In this way we extend the classical De Giorgi’s theory, developed in Euclidean spaces by De Giorgi, as well as its generalization, considered by the authors, in Heisenberg groups. A structure theorem for sets of finite perimeter and consequently a divergence theorem are obtained. Full proofs of these results, comments and an exhaustive bibliography can be found in our preprint (2001).
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