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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Extremal points of a functional
on the set of convex functions


Authors: T. Lachand-Robert and M. A. Peletier
Journal: Proc. Amer. Math. Soc. 127 (1999), 1723-1727
MSC (1991): Primary 49K99
Published electronically: February 11, 1999
MathSciNet review: 1646197
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Abstract: We investigate the extremal points of a functional $\int f(\nabla u)$, for a convex or concave function $f$. The admissible functions $u:\Omega\subset \mathbf{R}^N\to \mathbf{R}$ are convex themselves and satisfy a condition $u_2\leq u \leq u_1$. We show that the extremal points are exactly $u_1$ and $u_2$ if these functions are convex and coincide on the boundary $\partial\Omega$. No explicit regularity condition is imposed on $f$, $u_1$, or $u_2$. Subsequently we discuss a number of extensions, such as the case when $u_1$ or $u_2$ are non-convex or do not coincide on the boundary, when the function $f$ also depends on $u$, etc.


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Additional Information

T. Lachand-Robert
Affiliation: Université Pierre et Marie Curie, Laboratoire d’Analyse Numérique, 75252 Paris Cedex 05, France
Email: lachand@ann.jussieu.fr

M. A. Peletier
Affiliation: University of Bath, Claverton Down, Bath BA2 7AY United Kingdom
Email: M.A.Peletier@bath.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9939-99-05209-0
PII: S 0002-9939(99)05209-0
Keywords: Extremal points, convexity constraint, non-convex minimization
Received by editor(s): September 10, 1997
Published electronically: February 11, 1999
Additional Notes: Part of this work was carried out during a visit of the second author to Université Pierre et Marie Curie under the contract of the European Union 921 CHRX CT 94.
Communicated by: Jeffrey B. Rauch
Article copyright: © Copyright 1999 American Mathematical Society