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

Fix

x \in Ω

and

R < dist (x, \partial Ω)

u (x) = \frac{1}{∣ \partial B _{r} ∣} \int_{\partial B_{r} (x)} u d S = \frac{1}{∣ B _{r} ∣} \int_{B_{r} (x)} u d y (0 < r \leq R) .

To prove the spherical identity, translate

x

to the origin and put

F (r) = ∣ S^{n - 1} ∣^{- 1} \int_{S^{n - 1}} u (r θ) d S_{θ}

F^{'} (r) = \frac{1}{∣ S ^{n - 1} ∣ r ^{n - 1}} \int_{B_{r}} Δ u d y = 0.

Hence

F (r) = lim_{s ↓ 0} F (s) = u (0)

. Integrating the spherical identity in polar coordinates gives the ball identity.

 Ancestors (11)

There are no discussions about this article yet.

There are currently no matching articles.