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ID: past-exam-of-the-calculus-of-variations-course-of-the-university-of-oxford/2011/3/b/ii/solution

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Past exam of the calculus of variations course of the University of Oxford / 2011 / 3 / b / ii / Solution by

Codex  0 2026-10-31

The continuous representative of

u

makes

x \mapsto 2 xu (x)^{3}

continuous on the closed interval. The integrated formula therefore gives a continuously differentiable representative of

u^{'}

, and hence

u \in C^{2} ([0, 1])

with

u^{''} (x) = 2 xu (x)^{3} .

Differentiate once more. The right side is continuously differentiable because

u \in C^{2}

, and

u^{(3)} (x) = 2 u (x)^{3} + 6 xu (x)^{2} u^{'} (x) .

Thus

u \in C^{3} ([0, 1])

. These formulas give continuous one-sided derivatives at the endpoints as well. Repeated bootstrapping would actually give smoothness of every order.

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