Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 107 / 1 / v

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- Solution v

Solution

Fix

x \in Ω^{'}

and

0 < ρ < R (Ω^{'})

. Differentiate the ball mean-value formula and apply the divergence theorem:

\partial_{x_{j}} u (x) = \frac{1}{ω _{d} ρ ^{d}} \int_{B (x, ρ)} \partial_{j} u = \frac{1}{ω _{d} ρ ^{d}} \int_{\partial B (x, ρ)} u ν_{j} d S .

Since

∣ ν_{j} ∣ \leq 1

and

∣ \partial B (x, ρ) ∣ = d ω_{d} ρ^{d - 1}

,

∣ \partial_{x_{j}} u (x) ∣ \leq \frac{d}{ρ} max_{Ω} ∣ u ∣.

Letting

ρ ↑ R (Ω^{'})

and taking both maxima proves

1 \leq j \leq d max Ω^{'} max ∣ \partial_{x_{j}} u ∣ \leq \frac{d}{R ( Ω ^{'} )} Ω max ∣ u ∣.

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