Definition:Extremum/Functional
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Definition
Let $J \sqbrk y: S \to \R$ be a functional.
Let $y, \hat y: \R \to \R$ be real functions.
Suppose for $y = \hat y \paren x$ there exists a neighbourhood of the curve $y = \hat y \paren x$ such that the difference $J \sqbrk y - J \sqbrk {\hat y}$ does not change its sign in this neighbourhood.
Then $y = \hat y$ is called a (relative) extremum of the functional $J$.
Check if "some neighbourhood" could be more precisely defined
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Sources
- 1963: I.M. Gelfand and S.V. Fomin: Calculus of Variations ... (previous) ... (next): $\S 1.3$: The Variation of a Functional. A Necessary Condition for an Extremum