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

@article{Akman2012OnTL, title={On the logarithm of the minimizing integrand for certain variational problems in two dimensions}, author={Murat Akman and John L. Lewis and Andrew Vogel}, journal={Analysis and Mathematical Physics}, year={2012}, volume={2}, pages={79-88} }

AbstractLet 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 W1,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…

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