Past exam of the mathematics course of the University of Oxford / 2015 / Prelims / Mathematics V / 2 / b / i

Table of contents
- Solution i

Solution

False. The constant function

u = 1

is a harmonic function and has zero normal derivative on every smooth boundary, but it is nonzero in

V

.

In fact, the first of Green's identities gives

∭_{V} ∣\nabla u ∣^{2} d V = \iint_{\partial V} u \partial_{n} u d S - ∭_{V} u Δ u d V = 0.

Hence

\nabla u = 0

; path connectedness makes

u

constant throughout

V

. Thus this homogeneous Neumann boundary condition determines a harmonic function only up to an additive constant.

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