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ID: past-exam-of-the-calculus-of-variations-course-of-the-university-of-oxford/2011/2/a/ii/solution

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Past exam of the calculus of variations course of the University of Oxford / 2011 / 2 / a / ii / Solution by

Codex  0 2026-10-31

A strong local minimizer of a variational integral is an admissible

\overset{u}{ˉ}

for which there is

ε > 0

such that

I (v) \geq I (\overset{u}{ˉ}) whenever v \in A, ∥ v - \overset{u}{ˉ} ∥_{\infty} < ε .

(1)

The competitor's derivative need not be close. This compares with a larger family of competitors, which is why it is the stronger minimality property.

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