01524nas a2200157   4500008004100000245011100041210006900152300001000221490000700231520095900238653002001197100002501217700001801242700002201260856008401282       2016                        en  d00aOn the area of the graph of a piecewise smooth map from the plane to the  plane with a curve discontinuity0 aarea of the graph of a piecewise smooth map from the plane to th  a29-630 v223 a<p>In this paper we provide an estimate from above for the value of the relaxed area functional for a map defined on a bounded domain of the plane with values in the plane and discontinuous on a regular simple curve with two endpoints. We show that, under suitable assumptions, the relaxed area does not exceed the area of the regular part of the map, with the addition of a singular term measuring the area of a disk type solution of the Plateau's problem spanning the two traces of the map on the jump. The result is valid also when the area minimizing surface has self intersections. A key element in our argument is to show the existence of what we call a semicartesian parametrization of this surface, namely a conformal parametrization defined on a suitable parameter space, which is the identity in the first component. To prove our result, various tools of parametric minimal surface theory are used, as well as some result from Morse theory.</p>10aArea functional1 aBellettini, Giovanni1 aTealdi, Lucia1 aPaolini, Maurizio  uhttps://www.esaim-cocv.org/articles/cocv/abs/2016/01/cocv140065/cocv140065.html01380nas a2200133   4500008004300000245007200043210006900115260001500184520093400199100001901133700001801152700002501170856005101195       2015                        en_Ud  00aAnisotropic mean curvature on facets and relations with capillarity0 aAnisotropic mean curvature on facets and relations with capillar  bde Gruyter3 a<p>We discuss the relations between the anisotropic calibrability of a facet F of a solid crystal E, and the capillary problem on a capillary tube with base F. When F is parallel to a facet of the Wulff shape, calibrability is equivalent to show the existence of an anisotropic subunitary vector field in $F, with suitable normal trace on the boundary of the facet, and with constant divergence equal to the anisotropic mean curvature of F. When the Wulff shape is a cylynder, assuming E convex at F, and F (strictly) calibrable, such a vector field is obtained by solving the capillary problem on F in absence of gravity and with zero contact angle. We show some examples of facets for which it is possible, even without the strict calibrability assumption, to build one of these vector fields. The construction provides, at least for convex facets of class C^{1,1}, the solution of the total variation flow starting at 1_F.</p>1 aAmato, Stefano1 aTealdi, Lucia1 aBellettini, Giovanni  uhttp://urania.sissa.it/xmlui/handle/1963/34481