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ID: past-exam-of-the-partial-differential-equations-for-pure-and-applied-mathematicians-and-calculus-of-variations-course-of-the-university-of-oxford/2008/b/8/a/i/solution

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Past exam of the partial differential equations for pure and applied mathematicians and calculus of variations course of the University of Oxford / 2008 / B / 8 / a / i / Solution by

Codex  0 2026-10-31

The continuous superlinear function

Φ

tends to positive infinity at both ends, so it has a finite global minimum

M

. Hence

I (u) \geq M (b - a)

for every admissible function. An affine function with the prescribed endpoints is Lipschitz and has finite energy by continuity of the integrand on its compact range. Thus

- \infty < M (b - a) \leq l < \infty,

(1)

where the strict left inequality simply records finiteness of

M

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