The data are rotationally symmetric, so seek a radial solution. The radial Laplacian in two dimensions gives

\frac{1}{r} (r u^{'})^{'} = 16 r^{2} .

Twice integrating,

u (r) = r^{4} + C lo g r + D .

Regularity at the center forces

C = 0

, and

u (1) = 1

gives

D = 0

. Thus

u (r) = r^{4} .

ii

Solution

On the annulus the logarithmic term is allowed, so

u (r) = r^{4} + C lo g r + D .

The condition

u (1) = 1

gives

D = 0

, while

u (2) = 1

gives

16 + C lo g 2 = 1.

Therefore

u (r) = r^{4} - \frac{15 lo g r}{lo g 2} .

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