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Bruno Franchi Raul Serapioni Francesco Serra Cassano 《Journal of Geometric Analysis》2011,21(4):1044-1084
We study the notion of intrinsic Lipschitz graphs within Heisenberg groups, focusing our attention on their Hausdorff dimension
and on the almost everywhere existence of (geometrically defined) tangent subgroups. In particular, a Rademacher type theorem
enables us to prove that previous definitions of rectifiable sets in Heisenberg groups are natural ones. 相似文献
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We describe intrinsically regular submanifolds in Heisenberg groups Hn. Low dimensional and low codimensional submanifolds turn out to be of a very different nature. The first ones are Legendrian surfaces, while low codimensional ones are more general objects, possibly non-Euclidean rectifiable. Nevertheless we prove that they are graphs in a natural group way, as well as that an area formula holds for the intrinsic Hausdorff measure. Finally, they can be seen as Federer-Fleming currents given a natural complex of differential forms on Hn. 相似文献
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Gabriella Arena Raul Serapioni 《Calculus of Variations and Partial Differential Equations》2009,35(4):517-536
We characterize intrinsic regular submanifolds in the Heisenberg group as intrinsic differentiable graphs.
G. Arena is supported by MIUR (Italy), by INDAM and by University of Trento.
R. Serapioni is supported by MIUR (Italy), by GALA project of the Sixth Framework Programme of European Community and by University
of Trento. 相似文献
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A Carnot group is a connected, simply connected, nilpotent Lie group with stratified Lie algebra. We study the notions of intrinsic graphs and of intrinsic Lipschitz graphs within Carnot groups. Intrinsic Lipschitz graphs are the natural local analogue inside Carnot groups of Lipschitz submanifolds in Euclidean spaces, where “natural” emphasizes that the notion depends only on the structure of the algebra. Intrinsic Lipschitz graphs unify different alternative approaches through Lipschitz parameterizations or level sets. We provide both geometric and analytic characterizations and a clarifying relation between these graphs and Rumin’s complex of differential forms. 相似文献
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We give examples of discontinuous solutions and of unbounded solutions of linear isotropic degenerate elliptic equations. Discontinuous solutions exist even when both the maximum eigenvalue and the inverse of the minimum eigenvalue of the matrix of the coefficients are in the intersection of all the Lp spaces. 相似文献
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Bruno?FranchiEmail author Raul?Serapioni Francesco?Serra?Cassano 《Journal of Geometric Analysis》2003,13(3):421-466
In this article we study codimension 1 rectifiable sets in Carnot groups and we extend classical De Giorgi ’s rectifiability
and divergence theorems to the setting of step 2 groups. Related problems in higher step Carnot groups are discussed, pointing
on new phenomena related to the blow up procedure.
First author was supported by University of Bologna, Italy, funds for selected research topics; second and third authors were
supported by MURST, Italy, and University of Trento, Italy. 相似文献
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Sunto Viene risolto il problema di Cauchy Dirichlet relativo all'operatore parabolico degenere u/t–/xi(aij(x, t) u/xj), in opportune ipotesi di integrabilità per gli autovalori di aij(x, t). Vengono inoltre forniti controesempi circa l'impossibilità di risultati di regolarità per le soluzioni deboli mostrando in tal modo che operatori parabolici degeneri hanno un comportamento radicalmente differente da quello dei corrispondenti operatori ellittici degeneri.
Both the authors were supported in part by a grant of the italian C.N.R. 相似文献
Both the authors were supported in part by a grant of the italian C.N.R. 相似文献
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In this paper, we fully extend to the Heisenberg group endowed with its intrinsic Carnot-Carathéodory metric and perimeter
the classical De Giorgi's rectifiability divergence theorems.
Received: 27 March 2000 / Revised version: 13 December 2000 / Published online: 24 September 2001 相似文献
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