Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 2H / iii

Table of contents
- Solution iii

Solution

False. On the open unit disk take, for

0 < ε < 1

,

ϕ (x, y) = - (x^{2} + y^{2}) + ε (x^{4} + y^{4}) .

Its relevant fourth derivatives satisfy

ϕ_{xxxx} + ϕ_{yyyy} = 48 ε > 0.

Nevertheless, since

x^{4} + y^{4} \leq (x^{2} + y^{2})^{2}

, every point with

0 < x^{2} + y^{2} \leq 1

has

ϕ (x, y) \leq - r^{2} + ε r^{4} < 0 = ϕ (0, 0) .

Thus the maximum lies strictly inside $Ω$ .

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