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Counterexample bank

Part C is won by knowing which tempting claims are false. 148 counterexamples; 85 free. The rest come with the Notes pack.

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#1 · ODE, PDE & Applied Mathematics › Calculus of Variations

“A weak minimum of a functional is a strong minimum” — false

Counterexample: J[y] = ∫₀^π (1 − (y′)²)y² dx at y ≡ 0

In the $C^{1} -$ ball the integrand is non-negative so J ≥ 0; in the $C^{0} -$ ball a steeply oscillating small y makes J negative.

calculus of variations