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

Abstract

Let f be a smooth convex homogeneous function of degreep, 1 < p < ∞, on ${\mathbb{C} \setminus \{0\}.}$ We show that if u is a minimizer for the functional whose integrand is ${f(\nabla v ), v}$ in a certain subclass of the Sobolev space W ^1,p(Ω), and ${\nabla u \not = 0 }$ at ${z \in \Omega,}$ then in a neighborhood of z, ${ \log f (\nabla u ) }$ is a sub, super, or solution (depending on whether p > 2, p < 2, or p = 2) to L where

$$L \zeta=\sum_{k,j=1}^{2}\frac{\partial}{\partial x_k}\left( f_{\eta_k \eta_j}(\nabla u(z)) \frac{\partial \zeta }{ \partial x_j }\right),$$

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.