Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2025 / Paper 2 / 2

For an axisymmetric harmonic function, put $\Psi =R(r)\Theta (\theta )$ in the Laplacian in spherical coordinates and multiply by $r^2/(R\Theta )$. Separation of variables givesWith $u=\cos \theta$, the angular equation becomes the Legendre differential equation. Regularity at both poles selects the Legendre polynomials $P_{\ell }(u)$ with integer $\ell \ge 0$; the singular second solutions are excluded. The radial Euler-Cauchy equation has solutions $r^{\ell }$ and $r^{-\ell -1}$. Superposition therefore giveswith coefficients chosen to ensure convergence and the required boundary regularity. The axisymmetric harmonic extension from a sphere gives these coefficients directly from boundary data.

Set $u=\cos \theta$. The boundary data simplify to $u+2u^2=\frac{2}{3}{P}_0(u)+P_1(u)+\frac{4}{3}{P}_2(u)$. The axisymmetric harmonic extension from a sphere keeps only decaying radial powers outside and regular powers inside. HenceandEach Legendre polynomial coefficient agrees at $r=a$. The exterior solution tends to zero, and the interior solution is finite at $r=0$. The maximum principle for harmonic functions proves uniqueness in these respective classes.

The plane separates the exterior into two half-domains. Use an odd Legendre extension for an equatorial Dirichlet plane in each. On the upper hemisphere, extend $u+2u^2$ to the odd function $u+2u|u|$ on the whole sphere. On the lower hemisphere, the appropriate odd extension is $u-2u|u|$. Define, for odd $\ell$,The orthogonality and parity of the Legendre polynomials give the solution in both half-domains asSet its value on $u=0$ to zero. The first coefficients are $b_1=3/2$ and $b_3=7/12$. All odd Legendre polynomials vanish at zero, so the plane condition holds. On the sphere the sum is $2u|u|$, and multiplication by $\mathrm{sgn}(u)$ recovers $2u^2$ on either side. Thus the original sphere data and decay at infinity both hold. A jump of the normal derivative at the sheet is permitted; the sheet can support induced surface charge density.