The structure of null Lagrangians

It is said that L(x,u, Del u) is a null Lagrangian if and only if the corresponding integral functional E(u)= integral _OmegaL(x,u, Del u) dx has the property that E(u+ phi )=E(u) For all phi in C₀ ^infinity( Omega ), for any choice of u in C¹( Omega ). In the homogeneous case, corresponding to L(x,u, Del u)= Phi ( Del u), it is known that a necessary and sufficient condition for L to be a null Lagrangian is that Phi ( Del u) is an affine combination of subdeterminants of Del u of all orders. The authors show that all inhomogeneous null Lagrangians may be constructed from these homogeneous ones by introducing appropriate potentials.