Extremal points of a functional on the set of convex functions

Extremal points of a functional on the set of convex functions

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\RN\to\bR 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...

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society Vol. 127; no. 6; pp. 1723 - 1727
Main Authors: Lachand-Robert, T., Peletier, M. A.
Format: Journal Article
Language:English
Published: American Mathematical Society 01-06-1999
Subjects:
Calculus of variations
Convexity
Mathematical functions
Mathematical theorems
convexity constraint
non-convex minimization
Extremal points
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Summary: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\RN\to\bR 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.
ISSN:0002-9939
1088-6826
DOI:10.1090/S0002-9939-99-05209-0