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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/2007/b/7/a/iii/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 / 2007 / B / 7 / a / iii / Solution by

Codex  0 2026-10-31

A global minimizer satisfies

I (\overset{u}{ˉ}) \leq I (v)

for every admissible

v

.

A point of convexity

s

of a differentiable scalar function is one where its tangent supports the entire graph:

g (q) \geq g (s) + g^{'} (s) (q - s)

for every

q \in R

. The Weierstrass necessary condition for strong minimality, under its differentiability assumptions in the slope variable, is

f (x, \overset{u}{ˉ}, q) - f (x, \overset{u}{ˉ}, \overset{u}{ˉ}^{'}) - f_{v} (x, \overset{u}{ˉ}, \overset{u}{ˉ}^{'}) (q - \overset{u}{ˉ}^{'}) \geq 0 (q \in R)

for almost every

x

. In other words the reference slope is a point of convexity of the slope integrand. The later example is smooth, so these derivative-based conditions apply there.

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