# Berenstein rigidity — H5

There exist a bounded simply connected domain \(\Omega\subset\mathbb R^2\) with \(C^\infty\) boundary, a real number \(k>0\), and a nonzero real-valued function \(u\in C^2(\overline\Omega)\), such that \(\Omega\) is not a disk and
\[
\Delta u+k^2u=0\quad\text{in }\Omega,
\qquad
u=0\quad\text{on }\partial\Omega,
\qquad
\partial_\nu u=1\quad\text{on }\partial\Omega,
\]
where \(\nu\) is the outward unit normal.