A variational problem involving the distributional determinant

We deal with bounded W 1,n−1-maps defined in n-dimensional domains, whose graph has finite area and finite boundary mass. We show that the singular part of the distributional determinant is concentrated on a countable set of points. A related variational problem is then considered. Finally, we study the analogous problem involving the distributional minors of fixed order. Let u : Ω → R be a vector valued map defined on an n-dimensional bounded domain Ω ⊂ R. If u is essentially bounded and belongs to the Sobolev space W 1,n−1(Ω,Rn), the distributional determinant Det∇u, introduced by J.M. Ball in [4], is well-defined in the distributional sense by