Calculus of Variations and Geometric Measure Theory

Rectifiability of Sets of Finite Perimeter in a class of Carnot Groups of arbitrary step

Marco Marchi

created by magnani on 17 Nov 2011

30 nov 2011 -- 17:00   [open in google calendar]

Sala Seminari, Department of Mathematics, Pisa University


In 1955, De Giorgi proved his well-known rectifiability theorem, combining the theory of perimeters of Caccioppoli with the theory of rectifiable sets. More recently, the same problem has been studied in more general settings and in particular in Carnot groups. In 2003 Franchi, Serapioni and Serra Cassano extended De Giorgi’s theorem to step 2 Carnot groups by using a blow-up technique. In 2009 Ambrosio, Kleiner and Le Donne achieved a partial result of blow-up, valid for any Carnot group, but today there are no general rectifiability results. In the seminar, I will show the main result of my master’s degree thesis: the existence of an algebric condition on the Lie algebra of a Carnot group that allows to extend Franchi, Serapioni and Serra Cassano’s results to a large class of Carnot groups of arbitrary step. In my thesis, such groups are called groups of type ⋆. Their stratified Lie algebra has the following property: there exists a basis (X1,...,Xm) of the first layer such that Xj,[Xj,Xi] = 0 for i,j = 1,...,m. Besides step 2 groups, the Lie groups of unit upper triangular matrices, which appear in the Iwasawa decomposition of the linear group, are examples of groups of type ⋆.