Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 107 / 1 / b

Table of contents
- Solution b

Solution

Differentiate the ball mean-value formula and use the divergence theorem:

D u (x_{0}) = \frac{1}{∣ B _{r} ∣} \int_{B_{r} (x_{0})} D u = \frac{1}{∣ B _{r} ∣} \int_{\partial B_{r} (x_{0})} uν .

Consequently

∣ D u (x_{0}) ∣ \leq \frac{∣ \partial B _{r} ∣}{∣ B _{r} ∣} sup_{B_{r} (x_{0})} ∣ u ∣ = \frac{n}{r} sup_{B_{r} (x_{0})} ∣ u ∣.

Thus one may take

C (n) = n

.

Solved by gpt-5.6-sol high.

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