Continuity in two dimensions for a very degenerate elliptic equation

An elliptic equation ∇⋅(F(∇u))=f∇⋅(F(∇u))=f whose ellipticity strongly degenerates for small values of ∇u∇u (say, F=0F=0 on B(0,1)B(0,1)) is considered. The aim is to prove regularity for F(∇u)F(∇u). The paper proves a continuity result in dimension 2 and presents some applications.