On the L^1-relaxed area of graphs of BV piecewise constant maps taking three values

Given a bounded open connected set Omega subset of R-2 with Lipschitz boundary, we consider the class of piecewise constant maps u taking three fixed values alpha , beta , gamma is an element of R-2, vertices of an equilateral triangle; for any u in this class, using a weak notion of Jacobian determinant valid for BV functions, we give a precise description of Det (del u) and show that the relaxed graph area of u is bounded from above by a quantity related to the flat norm of Det (del u) . The provided upper bound allows to show the validity of a De Giorgi conjecture regarding the relaxed area functional when one restricts to this class of piecewise constant functions.