Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 107 / 1 / a / ii

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- Solution ii

Solution

Choose

0 < r < dist (x, \partial Ω)

. Differentiating the ball mean-value formula with respect to its centre and then using the divergence theorem gives

D_{i} u (x) = \frac{1}{∣ B _{r} ∣} \int_{B_{r} (x)} D_{i} u (y) d y = \frac{1}{∣ B _{r} ∣} \int_{\partial B_{r} (x)} u (y) ν_{i} (y) d S_{y} .

Consequently the interior derivative estimate for a harmonic function yields

∣\nabla u (x) ∣ \leq \frac{C _{n}}{r} sup_{B_{r} (x)} ∣ u ∣ \leq \frac{C _{n}}{r} sup_{Ω} ∣ u ∣.

The constant may depend on

x

and its distance from the boundary, but it is independent of

u

.

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