Maximum Principle for convex function

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Maximum Principle: Suppose that $f:(a, b) \rightarrow \mathbf{R}$ is convex and has a maximum at some $c$ in $(a, b)$ . Then, $f$ is constant (use subdifferentials).

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admin · Answer 1 · 2021-05-04T04:51:12+00:00

If $c$ is a maximum of $f$ , then $f(c) \geq f(x)$ for $a<x<b$ . Let $p$ be a subdifferential of $f$ at $c: f(x) \geq f(c)+p(x-c)$ for $a<x<b$ . Combining these inequalities yields $f(x) \geq f(x)+p(x-c)$ or $0 \geq p(x-c)$ for $a<x<b .$ Hence $p=0$ hence $f(x) \geq f(c)$ from the subdifferential inequality. Thus $f(x)=f(c)$ for $a<x<b$ .

Maximum Principle for convex function

Published by admin on 2021年5月4日2021年5月4日

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