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.

1 aBellettini, Giovanni1 aElshorbagy, Alaa uhttp://cvgmt.sns.it/media/doc/paper/4260/manif_with_bound_dist.pdf01475nas a2200121 4500008004100000245011300041210006900154520101400223100001801237700002501255700002201280856005101302 2015 en d00aSemicartesian surfaces and the relaxed area of maps from the plane to the plane with a line discontinuity0 aSemicartesian surfaces and the relaxed area of maps from the pla3 aWe 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.

1 aTealdi, Lucia1 aBellettini, Giovanni1 aPaolini, Maurizio uhttp://urania.sissa.it/xmlui/handle/1963/3448300375nas a2200121 4500008004100000245004500041210004500086260001800131100002500149700002300174700002100197856003500218 1995 en d00aSpecial functions of bounded deformation0 aSpecial functions of bounded deformation bSISSA Library1 aBellettini, Giovanni1 aCoscia, Alessandra1 aDal Maso, Gianni uhttp://hdl.handle.net/1963/978