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 $n\ge 1$, separation of variables givesThe constant angular mode is $u_0(\mu )=f_0+g_0(\mu -b)$. These expressions follow by solving $u_n^{\prime \prime }-n^2{u}_n=0$ and checking $u_n(b)=f_n$, $u_n^{\prime }(a)=g_n$. Their Fourier series gives the solution for sufficiently regular boundary data.

The phrase 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 ${\partial }_{n}u=-h^{-1}{u}_{\mu }$. 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.