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Extremal Points of a Functional on the Set of Convex Functions

T. Lachand-Robert and M. A. Peletier
Proceedings of the American Mathematical Society
Vol. 127, No. 6 (Jun., 1999), pp. 1723-1727
Stable URL: http://www.jstor.org/stable/119483
Page Count: 5
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Extremal Points of a Functional on the Set of Convex Functions
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Abstract

We investigate the extremal points of a functional ∫ f(∇ u),, for a convex or concave function f. The admissible functions u: $\Omega \subset {\bf R}^{N}\rightarrow {\bf R}$ are convex themselves and satisfy a condition u 2≤ u ≤ u1. We show that the extremal points are exactly u1 and u2 if these functions are convex and coincide on the boundary ∂ Ω . No explicit regularity condition is imposed on f, u1, or u2. Subsequently we discuss a number of extensions, such as the case when u1 or u2 are non-convex or do not coincide on the boundary, when the function f also depends on u, etc.

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