 Solution

ID: past-exam-of-the-calculus-of-variations-course-of-the-university-of-oxford/2010/2/b/iii/solution

Use

u_{A} (x) = A sin (π x / b)

, which is admissible for every real

A

. Direct integration gives

I_{λ} (u_{A}) = \frac{A ^{2} b}{2} (\frac{π ^{2}}{b ^{2}} - λ^{2}) .

The coefficient is strictly negative, so the energy tends to minus infinity as

∣ A ∣ \to \infty

. Therefore

u \in A in f I_{λ} (u) = - \infty.

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