Use the <reflection extension from a half-space>
$$
(Eu)(x,y)=u(|x|,y).
$$
It is plainly linear and restricts to $u$ on $U$. For a smooth $u$, the <chain rule> gives
$$
D_x(Eu)(x,y)=\operatorname{sgn}(x)(D_xu)(|x|,y),
\qquad
D_y(Eu)(x,y)=(D_yu)(|x|,y).
$$
A change of variables therefore gives
$$
\|Eu\|_{W^{1,p}(\mathbb R^2)}^p
=2\|u\|_{W^{1,p}(U)}^p
$$
for $p<\infty$, with the evident equality of essential suprema for $p=\infty$. Approximate a general $u$ by the smooth functions from part c. The estimate makes their reflections Cauchy in $W^{1,p}(\mathbb R^2)$, and their limit defines a bounded <Sobolev extension operator> with $\|E\|\leq2^{1/p}$.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 105 / 2 / d / Solution

For $U=(0,\infty)\times\mathbb R^{n-1}$, the formula $Eu(x_1,x')=u(|x_1|,x')$ defines a bounded <Sobolev extension operator>. Its weak normal derivative changes sign across the boundary, while its tangential weak derivatives are reflected unchanged.

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