Partial Regularity of Cartesian Currents which Minimize Certain Variational Integrals

Abstract

This paper deals with the local regularity of minimizers of functionals of the type

$${\cal{F}}(u;\Omega ) = \int_\Omega {F(x,u(x),M(Du(x)))dx}$$

((1.1))

where Ω is an open set in ℝⁿ, u: Ω → ℝ^N, n ≥ 2, N ≥ 2, M(Du(x)) stands for all minors of the Jacobian matrix Du of u and $F(x,u,M):\Omega \times {\Re^\ell } \to {\Re^ + }\ell : = \left( {\matrix{{n + N} \cr n \cr }} \right) - 1$ is a smooth function which is convex in M.