= Mixed Dirichlet-Neumann problem in an elliptic annulus
In <elliptic coordinates> $x=\cosh\mu\cos\nu$, $y=\sinh\mu\sin\nu$, a <harmonic function> in $a<\mu<b$, with $0<a<b$ and <period> $2\pi$ in $\nu$, satisfies $u_{\mu\mu}+u_{\nu\nu}=0$. Prescribe $u(b,\nu)=f(\nu)$ and $u_\mu(a,\nu)=g(\nu)$. Expand both boundary functions in <Fourier series>, writing their constant terms as $f_0,g_0$. For either the <sine> or <cosine> coefficient of order $n\geq1$, <separation of variables> gives
$$
u_n(\mu)=f_n\frac{\cosh[n(\mu-a)]}{\cosh[n(b-a)]}-g_n\frac{\sinh[n(b-\mu)]}{n\cosh[n(b-a)]}.
$$
The constant angular mode is $u_0(\mu)=f_0+g_0(\mu-b)$. These expressions follow by solving $u_n''-n^2u_n=0$ and checking $u_n(b)=f_n$, $u_n'(a)=g_n$. Their <Fourier series> gives the solution for sufficiently regular boundary data.
The phrase \i[normal derivative] needs care: the <scale factor> is $h=\sqrt{\sinh^2\mu+\sin^2\nu}$, so the outward <unit normal> derivative on the inner ellipse is $\partial_nu=-h^{-1}u_\mu$. Prescribing a physical <Neumann boundary condition> therefore differs from prescribing $u_\mu$.
For <uniqueness>, subtract two solutions to obtain $v$. Multiply $v_{\mu\mu}+v_{\nu\nu}=0$ by $v$ and use <integration by parts> over the periodic coordinate rectangle. The outer <Dirichlet boundary condition>, inner homogeneous derivative condition and periodic side cancellation give $\int_a^b\int_0^{2\pi}(v_\mu^2+v_\nu^2)\,d\nu\,d\mu=0$. Thus $v$ is constant, and the outer condition makes it zero.
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