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.
Analysis and Mathematical Physics – Springer Journals
Published: Jan 24, 2012
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