Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 107 / 1 / b

Table of contents
- Solution b

Solution

For

x, z \in B_{ρ} (y)

, the open balls satisfy

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

. Nonnegativity and the mean value property for harmonic functions give

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

proves the Harnack inequality for harmonic functions:

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.

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