# Finding the domain shape, where the fundamental Helmholtz Eigenmode has certain properties at a boundary

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 $$g(t)\approx K t$$. $\Omega$ 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 $$\color{red} {\partial \Omega_3}$$.">