Variational Problems with a p-Homogeneous Energy

Variational Problems with a p-Homogeneous Energy We investigate existence, uniqueness and positivity of minimizers or critical points for an energy functional which contains only p-homogeneous and linear terms, 1<p<∞. Nonlinear spectral methods are used to find necessary and sufficient conditions for the existence of a minimizer. These methods are then combined with the pointwise order relation in a Sobolev space to obtain positivity and uniqueness of minimizers or critical points. A crucial restriction on the p-homogeneous part of the energy functional is that it be given by the p-th power of an equivalent, uniformly convex norm on the underlying Sobolev space. Finally, continuous dependence of minimizers on the energy functional is established. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

Variational Problems with a p-Homogeneous Energy

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Publisher
Kluwer Academic Publishers
Copyright
Copyright © 2002 by Kluwer Academic Publishers
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1023/A:1012088127719
Publisher site
See Article on Publisher Site

Abstract

We investigate existence, uniqueness and positivity of minimizers or critical points for an energy functional which contains only p-homogeneous and linear terms, 1<p<∞. Nonlinear spectral methods are used to find necessary and sufficient conditions for the existence of a minimizer. These methods are then combined with the pointwise order relation in a Sobolev space to obtain positivity and uniqueness of minimizers or critical points. A crucial restriction on the p-homogeneous part of the energy functional is that it be given by the p-th power of an equivalent, uniformly convex norm on the underlying Sobolev space. Finally, continuous dependence of minimizers on the energy functional is established.

Journal

PositivitySpringer Journals

Published: Oct 12, 2004

References

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