Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 107 / 2 / b

Table of contents
- Solution b

Solution

For compactly supported smooth variations

(ϕ, ψ)

, differentiating at

t = 0

gives

0 = \frac{1}{2} \frac{d}{d t} E [X + tϕ, Y + t ψ]_{t = 0} = \int_{Ω} [\frac{\nabla X \cdot \nabla ϕ + \nabla Y \cdot \nabla ψ}{X ^{2}} - \frac{( ∣\nabla X ∣ ^{2} + ∣\nabla Y ∣ ^{2} ) ϕ}{X ^{3}}] .

This is the weak formulation. Taking

ψ = 0

or

ϕ = 0

separates its two equations. When

X, Y

are smooth, integration by parts transfers each derivative from the test function; the fundamental lemma of the calculus of variations then recovers exactly the two pointwise equations in part (a).

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