On the logarithm of the minimizing integrand for certain variational problems in two dimensions

Let f be a smooth convex homogeneous function of degreep, 1 < p < ∞, on (Formula presented.) We show that if u is a minimizer for the functional whose integrand is (Formula presented.) in a certain subclass of the Sobolev space W^1,p(Ω), and (Formula presented.) then in a neighborhood of z, (Formula presented.) is a sub, super, or solution (depending on whether p > 2, p < 2, or p = 2) to L where (Formula presented.) we then indicate the importance of this fact in previous work of the authors when f(η) = |η|^p and indicate possible future generalizations of this work in which this fact will play a fundamental role.