The following is a problem about finding the shape of a domain such that the fundamental Helmholtz eigenmode has certain properties at the boundary of the domain. I search for an analytical or numerical method to solve the problem, e.g. a reference to a paper. I have been suggested to try conformal mapping in which I am not an expert (yet).

Problem statement
The function $\phi(x,y)$ in the domain $\Omega$ is the fundamental eigenfunction to the Helmholtz equation with real, positive wavenumber $k$, and boundary conditions at the three boundaries $\partial\Omega_1$,$\partial\Omega_2$, and $\color{red} {\partial \Omega_3}$, see the attached figure,

\begin{align} \nabla^2 \phi + k^2 \phi &=0, &&(x,y) \in \Omega,\\ \boldsymbol{n}\cdot\boldsymbol{\nabla} \phi &=0, \quad &&(x,y) \in \partial\Omega_1 \cup \color{red} {\partial \Omega_3},\\ \phi&=0, &&(x,y) = \partial\Omega_2. \end{align}

Here $\boldsymbol{n}$ is the outward-pointing normal vector. The boundaries $\partial\Omega_1$ and $\partial\Omega_2$ are fixed at $y=0$ and $x=0$, respectively, see the attached figure. The boundary $\color{red} {\partial \Omega_3}$ follows the curve $\color{green}{\boldsymbol{s}(t)}$, which go from the point $(0,y0)$ to $(x0,0)$ as the parameter $t$ goes from 0 to 1, that is, $$ \color{green}{\boldsymbol{s}(0)}=(0,y0), \qquad \color{green}{\boldsymbol{s}(1)}=(x0,0). $$

Find the curve $\color{green}{\boldsymbol{s}(t)}$ which form the boundary $\color{red} {\partial \Omega_3}$ such that an additional boundary condition on $\phi$ is met, namely

$$ \phi(\color{green}{\boldsymbol{s}(t)}) = g(t), \qquad t\in [0,1]. $$

Here $g(t)$ is a smooth real function. Any restriction can be put on $g(t)$. In particular, I am interested in functions $g(t)$ which are close to linear $g(t)\approx K t$.

The domain <span class=$\Omega$ and the three boundaries $\partial\Omega_1$,$\partial\Omega_2$, and $\color{red} {\partial \Omega_3}$. The green curve $\color{green}{\boldsymbol{s}(t)}$, $t\in [0,1]$ is unknown and form the boundary $\color{red} {\partial \Omega_3}$.">


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