Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics V resit / Question 6 / c

A_{0} = 1

T (r, θ) = 1 .

ii

The boundary data consist of the third angular Fourier mode. Its regular radial factor in the harmonic expansion on a disk is

r^{3}

, so

T (r, θ) = r^{3} sin (3 θ) .

For a direct check of Laplace's equation,

T_{rr} + \frac{1}{r} T_{r} + \frac{1}{r ^{2}} T_{θθ} = 6 r sin (3 θ) + 3 r sin (3 θ) - 9 r sin (3 θ) = 0.

Also

T (1, θ) = sin (3 θ)

. At the origin the apparent coordinate singularity is harmless: this is the harmonic polynomial

3 x^{2} y - y^{3}

. The allowed uniqueness for the Dirichlet problem completes the identification.

sin^{3} θ = \frac{3}{4} sin θ - \frac{1}{4} sin (3 θ)

shows that only angular Fourier modes

1

and

3

are required. Differentiating the harmonic expansion on a disk at

r = 1

multiplies the coefficient of each nonconstant Fourier mode by

n

. Thus

B_{1} = 3/4

and

3 B_{3} = - 1/4

T (r, θ) = C + \frac{3}{4} r sin θ - \frac{1}{12} r^{3} sin (3 θ), C \in R .

Each term is a regular harmonic function, and the radial derivative is

T_{r} (1, θ) = \frac{3}{4} sin θ - \frac{1}{4} sin (3 θ) = sin^{3} θ .

\int_{0}^{2 π} sin^{3} θ d θ = 0,

The additive constant is undetermined. This Neumann boundary condition is unchanged by

T \mapsto T + C

, so the paper's blanket uniqueness assertion requires a normalization in this case. The uniqueness of the Neumann problem gives uniqueness up to a constant; prescribing

T (0) = 0

, or zero mean temperature, selects the displayed answer with

C = 0

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