We characterize arbitrary codimensional smooth manifolds $\mathcal{M}$ with boundary embedded in $\mathbb{R}^n$ using the square distance function and the signed distance function from $\mathcal{M}$ and from its boundary. The results are localized in an open set.

UR - http://cvgmt.sns.it/media/doc/paper/4260/manif_with_bound_dist.pdf ER - TY - RPRT T1 - Semicartesian surfaces and the relaxed area of maps from the plane to the plane with a line discontinuity Y1 - 2015 A1 - Lucia Tealdi A1 - Giovanni Bellettini A1 - Maurizio Paolini AB -We address the problem of estimating the area of the graph of a map u, defined on a bounded planar domain O and taking values in the plane, jumping on a segment J, either compactly contained in O or having both the end points on the boundary of O. We define the relaxation of the area functional w.r.t. a sort of uniform convergence, and we characterize it in terms of the infimum of the area among those surfaces in the space spanning the graphs of the traces of u on the two side of J and having what we have called a semicartesian structure. We exhibit examples showing that the relaxed area functional w.r.t the L^1 convergence may depend also on the values of u far from J, and on the relative position of J w.r.t. the boundary of O; these examples confirm the non-local behaviour of the L^1 relaxed area functional, and justify the interest in studying the relaxation w.r.t. a stronger convergence. We prove also that the L^1 relaxed area functional in non-subadditive for a rather class of maps.

UR - http://urania.sissa.it/xmlui/handle/1963/34483 N1 - The preprint is compsed of 37 pages and is recorded in PDF format U1 - 34670 U2 - Mathematics U4 - 1 U5 - MAT/05 ER - TY - JOUR T1 - Special functions of bounded deformation Y1 - 1995 A1 - Giovanni Bellettini A1 - Alessandra Coscia A1 - Gianni Dal Maso PB - SISSA Library UR - http://hdl.handle.net/1963/978 U1 - 3476 U2 - Mathematics U3 - Functional Analysis and Applications ER -