Definition:Weak Extremum
Jump to navigation
Jump to search
This article needs to be linked to other articles. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding these links. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{MissingLinks}} from the code. |
Definition
Let $y, \hat y \in S: \R \to \R$ be real functions.
Let $J \sqbrk y: S \to \R$ be a functional.
Suppose there exists $\epsilon \in \R_{> 0}$ such that for $\norm {y - \hat y}_1 < \epsilon$ the expression $J \sqbrk y - J \sqbrk {\hat y}$ has the same sign for all $y$.
Here $\norm{\, \cdot \,}_1$ denotes the norm of in the space $C^1$.
Then $y = \hat y$ is a weak extremum of the functional $J \sqbrk y$.
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