 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/iii/paper-107/1/b/solution

Codex  0 Created 2026-10-03 Updated 2026-10-05

For

x, z \in B_{ρ} (y)

, the open balls satisfy

B_{ρ} (x) \subset B_{3 ρ} (z) \subset B_{4 ρ} (y)

u (x) = \frac{1}{ω _{n} ρ ^{n}} \int_{B_{ρ} (x)} u \leq \frac{1}{ω _{n} ρ ^{n}} \int_{B_{3 ρ} (z)} u = 3^{n} u (z) .

Taking the supremum over

x

and the infimum over

z

B_{ρ} (y) sup u \leq 3^{n} B_{ρ} (y) in f u .

This includes a zero infimum, which forces

u = 0

on the smaller ball.

 New to topics? Read the docs here!