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

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 $${\mathbb{C} \setminus \{0\}.}$$ 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Mathematical Physics Springer Journals

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

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Publisher
Springer Journals
Copyright
Copyright © 2012 by Springer Basel AG
Subject
Mathematics; Analysis; Mathematical Methods in Physics
ISSN
1664-2368
eISSN
1664-235X
DOI
10.1007/s13324-012-0023-8
Publisher site
See Article on Publisher Site

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.

Journal

Analysis and Mathematical PhysicsSpringer Journals

Published: Jan 24, 2012

References