Definition:Sequence/Minimizing/Functional

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Definition

Let $y$ be a real mapping defined on a space $\MM$.

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Let $J \sqbrk y$ be a functional such that:

$\exists y \in \MM: J \sqbrk y < \infty$

$\ds \exists \mu > -\infty: \inf_y J \sqbrk y = \mu$

Let $\sequence {y_n}$ be a sequence such that:

$\ds \lim_{n \mathop \to \infty} J \sqbrk {y_n} = \mu$

Then the sequence $\sequence {y_n}$ is called a minimizing sequence (of the functional $J \sqbrk y$).

Limit Minimizing Function of Functional

Let $\sequence {y_n}$ be a minimizing sequence of a functional $J$.

Suppose:

$\ds \lim_{n \mathop \to \infty} y_n = \hat y$

and

$\ds \lim_{n \mathop \to \infty} J \sqbrk {y_n} = J \sqbrk {\hat y}$

Then $\hat y$ is the limit minimizing function of $J \sqbrk {y_n}$ and $J \sqbrk {\hat y} = \mu$.

Sources

• 1963: I.M. Gelfand and S.V. Fomin: Calculus of Variations ... (previous) ... (next): $\S 8.39$: Minimizing Sequences